researcharticle gram-charlier processes and applications

Research ArticleGram-Charlier Processes and Applications to Option Pricing

Jean-Pierre Chateau1 and Daniel Dufresne2

1Faculty of Business Administration University of Macau Macau2Montreal QC Canada

Correspondence should be addressed to Daniel Dufresne dufresneunimelbeduau

Received 17 August 2016 Accepted 1 November 2016 Published 8 February 2017

Academic Editor Aera Thavaneswaran

Copyright copy 2017 Jean-Pierre Chateau and Daniel DufresneThis is an open access article distributed under theCreativeCommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

A Gram-Charlier distribution has a density that is a polynomial times a normal density For option pricing this retains thetractability of the normal distribution while allowing nonzero skewness and excess kurtosis Properties of the Gram-Charlierdistributions are derived leading to the definition of a process with independent Gram-Charlier increments as well as formulasfor option prices and their sensitivities A procedure for simulating Gram-Charlier distributions and processes is given Numericalillustrations show the effect of skewness and kurtosis on option prices

1 Introduction

Gram-Charlier series are expansions of the form

119891 (119909) = 120601 (119886 119887 119909)sdot [1 + 11988811198671198901 (119909 minus 119886119887 ) + 11988821198671198902 (119909 minus 119886119887 ) + sdot sdot sdot] (1)

where

120601 (119886 119887 119909) = 1119887radic2120587119890minus(121198872)(119909minus119886)2 119909 isin R (2)

is the usual normal density and 119867119890119896 is the Hermite polyno-mial of order 119896The expressionwithin square brackets in (1) isan orthogonal polynomial expansion for the ratio 119891(119909)120601(119909)given an arbitrary function the expansion may or may notconverge to the true value of 119891(119909) In this paper we focus onthe properties of the Gram-Charlier distributions obtainedby truncating the series after a finite number of termsWhat is obtained is a family of distributions parametrized by119886 119887 1198880 119888119873 as is explained in detail below

This paper has three main goals (1) define and study theproperties of the family of Gram-Charlier distributions (2)define a Gram-Charlier process and derive its basic proper-ties (3) apply those to European options The formulas we

give for European option prices and Greeks apply to Gram-Charlier distributions of any order and we use four- and six-parameter Gram-Charlier distributions in our examples Twonumerical illustrations show how option prices are affectedby the skewness and kurtosis of returns This paper cana reference for those using Gram-Charlier distributions inoption pricing but also in statistics

Most previous applications to option pricing haveassumed that 1198881 = 1198882 = 0 We believe this restriction is notnecessary in a general theory of Gram-Charlier distributionsas there may well be situations where the extra degrees offreedom given by 1198881 and 1198882 will be useful We discuss it indetail at the end of Section 2

It has been observed that option prices have nonconstantimplied volatilities meaning that log returns do not have anormal distribution under the risk-neutral measure Thereis a wide literature on modelling log returns to fit observedoption prices the main alternatives to Brownian motionbeing stochastic volatility models (where the parameter 120590 inBlack-Scholes is replaced with a continuous-time stochasticprocess) GARCH time series and Levy processes Gram-Charlier distributions are mathematically simpler than themodels just mentioned while allowing a better fit to datathan the normal distribution Several authors have usedGram-Charlier distributions in option pricing as a model for

HindawiJournal of Probability and StatisticsVolume 2017 Article ID 8690491 19 pageshttpsdoiorg10115520178690491

2 Journal of Probability and Statistics

log returns among others [1ndash10] The majority of previousauthors assumed a density of the form

120601 (119909) (1 + 11990461198671198903 (119909) + 119896241198671198904 (119909)) 120601 (119909) = 120601 (0 1 119909) = 1radic2120587119890minus119909

22(3)

for the normalized log return (In our notation this isa GC(0 1 0 0 1199046 11989624) distribution see Section 2) Thenotation emphasizes that in this case the coefficient of1198671198903(119909)turns out to be Pearsonrsquos skewness coefficient 119904 divided by 6and the coefficient of 1198671198904(119909) the excess kurtosis coefficient119896 divided by 24 The distribution of log returns is thena four-parameter Gram-Charlier distribution (since thereare two other parameters the mean and variance besides119904 and 119896) This distribution allows nonzero skewness andexcess kurtosis unlike the normal distribution found inBlack-Scholes In (3) the parameters (119904 119896) are restricted toa specific region (see Figure 3) because outside that regionthe function in (3) becomes negative for some values of 119909(see Section 21)

Jurczenko et al [8] specify the martingale restriction

1198780 = 119890minus119903119879EQ (119878119879) (4)

that the four-parameter Gram-Charlier density must sat-isfy in pricing options (previous authors had not taken itinto account) The martingale condition for general Gram-Charlier distributions is described in Section 5 Our Gram-Charlier distribution with parameters 119886 119887 1198881 119888119873 denotedby GC(119886 119887 1198881 119888119873) has density

120601 (119886 119887 119909)sdot [1 + 11988811198671198901 (119909 minus 119886119887 ) + sdot sdot sdot + 119888119873119867119890119873 (119909 minus 119886119887 )] (5)

As stated above we do not set 1198881 = 1198882 = 0 and 119873 canbe any even positive integer The question whether (5) isnonnegative for all 119909 is an important one Several authorshave disregarded this issue and in fact some have come upwith parameters that do not yield a true probability densityfunction One might argue that this is the price to pay fortruncating an infinite Gram-Charlier series However if thesame log return distribution is used to price many optionsthen a true probability density function is the only safe choicebecause otherwise there might be inconsistencies amongoption prices For instance if the function used as densityis negative over the interval (120572 120573) then a digital option thatpays off only when the log return is in that interval will havea negative price In this paper we consider Gram-Charlierdistributions not expansions and insist that the densitiesintegrate to one and be nonnegative Our goal is to define afamily of proper probability distributions nevertheless ourformulas do apply to truncated Gram-Charlier expansions aswell

The paper by Leon et al [11] presents an alternative tothe general Gram-Charlier distributions we study in thispaper Those authors consider the subclass of Gram-Charlierdistributions consisting of densities

119891119883 (119909) = 120601 (119886 119887 119909) 119901 (119909) (6)

where the polynomial 119901(119909) is the square of another polyno-mial 119901(119909) = 119902(119909)2 This has the obvious advantage that thenonnegativity restriction on 119891119883(sdot) is automatically satisfiedWe discuss that subclass of ldquosquaredrdquo Gram-Charlier distri-butions in the Conclusion

Almost all previous authors have used Gram-Charlierdistributed log returns over a single time period This hasan obvious downside in that it becomes tricky if not im-possible to preserve consistency between the prices ofoptions with different maturities Section 3 shows that aLevy process with Gram-Charlier increments does not existhowever it also shows that the sum of independent Gram-Charlier distributed variables also has a Gram-Charlier dis-tributionThis opens the way for multiperiod Gram-Charlieroption pricing using a discrete-time random walk model forwhich the log return over any period has a Gram-Charlierdistribution The Gram-Charlier distribution of the multi-period return has a larger number of parameters though themodel is still simpler than almost any (if not all) alternativestochastic volatility models There is no problem computa-tionally since we give explicit formulas for options underGram-Charlier distributions with an arbitrary number ofparameters

The layout of the paper is as follows In Section 2 weextend the study of Gram-Charlier distributions to all possi-ble polynomials 119901(sdot) and derive their properties (momentscumulants moment determinacy properties of the set ofvalid parameters tail and so on) Some of the formulasand properties in Theorem 2 appear to be new In Section 3we show that there is no Levy process with Gram-Charlierdistributed increments apart from Brownian motion andwe define a discrete-time process with independent Gram-Charlier increments that is suitable for option pricing InSection 4 we show that the log Gram-Charlier distribution isnot determined by itsmoments just like the lognormal NextSection 5 gives formulas for European call and put priceswhen the log-price returns of the underlying have a generalGram-Charlier distribution the previous literature mostlyconsidered the GC(119886 119887 0 0 1198883 1198884) family and the squaredGram-Charlier distributions (an exception is [10] where itis assumed that 1198881 = 1198882 = 0) In particular we derive achange ofmeasure formula that extends theCameron-Martinformula for the normal distribution the latter is used inpricing European options in the Black-Scholes model Wealso derive formulas for the sensitivities (Greeks) of thoseoption prices with respect to all parameters A technique forsimulating Gram-Charlier distributions is described Parts(c) (d) (i) (j) (k) and (l) of Theorem 2 and part (b) ofTheorem 3 and Theorems 6 7 9 and 11 appear to be newTheorem 3(a) is for the first time formulated for generalGram-Charlier distributions Theorems 3(a) 4 and 8 aregiven more or less explicitly for the subclass of squared

Journal of Probability and Statistics 3

Gram-Charlier distributions in [11] and some of the Greeksin Theorem 9 had been calculated for the GC(119886 119887 0 0 1198883 1198884)distribution by previous authors

In Section 6 we give two applications that show howoption prices depend on skewness and kurtosis of the logreturns (this of course cannot be done in the classical Black-Scholes setting while it appears quite complicated to doso in stochastic volatility or Levy driven models) The firstexample is equity indexed annuities (EIAs in the sequel)premium options The pricing of EIAs has been studied bymany authors including Hardy [12] Gaillardetz and Lin [13]and Boyle and Tian [14] The second example is lookbackoptions which illustrates the use of the simulation methodin Section 25 Lookback options have been studied bynumerous authors but there is no closed form formula forthe price of discretely monitored lookback options withinthe Black-Scholes model We refer the reader to Kou [15]In those examples all parameters are estimated by maximumlikelihood with the range of the parameters restricted towhere they correspond to a true probability distributionMoment estimation is of course possiblewhen 1198881 = 1198882 = 0 butit is not trustworthy because of the range restriction on theparameters 1198883 1198884 for instance the empirical skewness andexcess kurtosis have a positive probability of falling outsidethe feasible region (see Section 6) Theorem 2(f) showsthat the first 119896 moments of the Gram-Charlier distributiondepend on more than 119896 parameters which probably rulesout moment estimation unless one a priori fixes two of theparameters (which again is not trustworthy) We apply max-imum likelihood to the six-parameterGC(119886 119887 1198881 1198882 1198883 1198884) inSection 6 but we do realize that maximum likelihood esti-mation for higher order Gram-Charlier distributions posescomputational problems which are an interesting avenue forfurther research (A referee pointed out that fixing 119886 and 119887would make estimation easier our guess is that if one wishesto simplify maximum likelihood estimation then fixing 119886 and119887might be a better idea than fixing 1198881 and 1198882 though we havenot looked at this in any depth A two-step process mightbe imagined whereby the data first give information about119886 and 119887 and then maximum likelihood is applied using thatpreliminary information This would help by constrainingthe optimization to a smaller region while agreeing with theintuitive idea that 119886 and 119887 are location and scale parametersresp)

Notation 1 The normal density function is denoted as

120601 (119886 119887 119909) = 1119887radic2120587119890minus(121198872)(119909minus119886)2

120601 (0 1 119909) = 120601 (119909) = 119890minus11990922radic2120587 (7)

and its distribution function is

Φ (119909) = int119909minusinfin

120601 (119910) 119889119910 (8)

Two equivalent versions of the Hermite polynomials may befound in the literature for 119899 = 0 1 2

119867119899 (119909) = (minus1)119899 1198901199092 119889119899119889119909119899 119890minus1199092119867119890119899 (119909) = (minus1)119899 11989011990922 119889119899119889119909119899 119890minus11990922

(9)

The first one is common in mathematics and physics butin probability and statistics there is an obvious advantage inusing the second one (The conversion formula is 119867119890119899(119909) =2minus1198992119867119899(119909radic2)) The first few Hermite polynomials are

1198671198900 (119909) = 11198671198901 (119909) = 1199091198671198902 (119909) = 1199092 minus 11198671198903 (119909) = 1199093 minus 31199091198671198904 (119909) = 1199094 minus 61199092 + 31198671198905 (119909) = 1199095 minus 101199093 + 151199091198671198906 (119909) = 1199096 minus 151199094 + 451199092 minus 15

(10)

2 Gram-Charlier Distributions

For a fixed 119873 consider the class of distributions that have apdf of the form

119891 (119909) = 120601 (119909) 119873sum119896=0

119888119896119867119890119896 (119909) 119909 isin R (11)

with 119888119873 = 0 Noting that the leading term of 119867119890119896(119909) is 119909119896we conclude that 119873 must necessarily be even because if 119873were odd then the polynomial thatmultiplies120601(119909)would takenegative values for some 119909 For the same reason 119888119873 cannot benegative

Definition 1 Let 119886 isin R 119887 gt 0 119888119896 isin R 1198880 = 1 and 119873 isin 0 24 We write 119884 sim GC(119886 119887 1198881 119888119873) (or 119884 sim GC(119886 119887 ))if the variable (119884 minus 119886)119887 has probability density function

120601 (119909) 119873sum119896=0

119888119896119867119890119896 (119909) (12)

This will be called a Gram-Charlier distribution with param-eters 119886 119887 with = (1198881 119888119873) The largest 119873 such that119888119873 gt 0 is called the order of the Gram-Charlier distributionThe normal distribution with mean 119886 and variance 1198872 is aGC(119886 119887 (0 0)) (or GC(119886 119887 minus)) with order 0

The class of Gram-Charlier distributions just definedincludes all distributions with density

1119887120601 (119910 minus 119886119887 )119901 (119910) (13)


where 119901(119910) is a polynomial of degree 119873 since 119901(119910) can berewritten as a combination of

119867119890119896 (119910 minus 119886119887 ) 119896 = 0 1 119873 (14)

The condition 1198880 = 1 ensures that function (12) integrates toone since

intinfinminusinfin

120601 (119909)119867119890119896 (119909) 119889119909 = (minus1)119896 intinfinminusinfin

119889119896119889119909119896 120601 (119909) 119889119909 = 0119896 = 1 2

(15)

There are no simple conditions that ensure that a polynomialremains nonnegative everywhere though in some casesprecise conditions on are known see below If a vector leads to a true Gram-Charlier pdf then we will say that isvalid

Generating functions are convenient when dealing withorthogonal polynomials One is

1199081 (119905 119909) = infinsum119899=0

119905119899119899119867119890119899 (119909) = 119890119905119909minus(11990522) (16)

Another one is

1199082 (119905 119906 119909 119910)= infinsum

119896=0

infinsum119899=0

119905119896119896 119906119899

119899 intinfin

minusinfin119890minus11990922119867119890119896 (119909)119867119890119899 (119909 + 119910) 119889119909

= radic2120587119890119906(119905+119910)(17)

Letting 119910 = 0 leads to1radic2120587 int

infin

minusinfin119890minus11990922119867119890119896 (119909)119867119890119899 (119909) 119889119909 = 120597119896119905119896 120597

119899

119906119899 119890119905119906100381610038161003816100381610038161003816100381610038161003816119905=119906=0

= 0 if 119896 = 119899119899 if 119896 = 119899

(18)

This proves the orthogonality of theHermite polynomials andgives us the value of

1radic2120587 intinfin

minusinfin119890minus119909221198671198902119899 (119909) 119889119909 (19)

which is essential in deriving Gram-Charlier series Since1198671198900 = 1 this also implies

intinfinminusinfin

120601 (119909)119867119890119896 (119909) 119889119909 = 0 119896 = 1 2 (20)

Another formula is the Laplace transform of 120601(119909)119867119890119899(119909)which may be found by integrating by parts 119899 times

intinfinminusinfin

119890119905119909120601 (119909)119867119890119896 (119909) 119889119909 = 11990511989611989011990522 119896 = 0 1 (21)

Let us calculate the moments of a distribution with density(11) First consider

intinfinminusinfin

119909119899120601 (119909)119867119890119896 (119909) 119889119909= (minus1)119896 intinfin

minusinfin119909119899 ( 119889119896119889119909119896120601 (119909)) 119889119909

(22)

Integrating by parts repeatedly yields

intinfinminusinfin

119909119899120601 (119909)119867119890119896 (119909) 119889119909=

1198992(119899minus119896)2 ((119899 minus 119896) 2) if 119899 minus 119896 is even and nonnegative

0 if 119899 minus 119896 is odd and nonnegative(23)

Hence the 119899th moment of the distribution in (11) is

119873and119899sum119896=0

119888119896 1198992(119899minus119896)2 ((119899 minus 119896) 2)1119899minus119896 even (24)

This says in particular that the parameter 119888119896 only affects themoments of order 119896 and higher of theGC(119886 119887 ) distribution(see part (i) of Theorem 2) This is confirmed by the momentgenerating function which may be found from (21)

intinfinminusinfin

119890119905119909119891 (119909) 119889119909 = 11989011990522 119873sum119896=0

119888119896119905119896 (25)

It can be checked that differentiating this expression 119899 timesand setting 119905 = 0 give the same expression found for the 119899thmoment of (11)

Theorem 2 Suppose 119884 sim GC(119886 119887 ) isin R119873 with 119887 gt 01198880 = 1 and 119888119873 gt 0The order119873 of the distribution is necessarilyeven

(a)

E (119884 minus 119886)119899 = 119887119899 119873and119899sum119896=0


(b)

E119890119905119884 = 119890119886119905+(119887211990522) 119873sum119896=0

119887119896119888119896119905119896 119905 isin R (27)

(c) The representation of the GC(119886 119887 ) distribution interms of the parameters 119886 119887 and is unique

(d) All Gram-Charlier distributions are determined by theirmoments

(e)The set of valid inR119873 includes the origin is not reducedto a single point and is convex


(f) The first six moments of theGC(119886 119887 ) distribution are1198981 = 119886 + 11988711988811198982 = 1198862 + 1198872 + 2 (1198861198871198881 + 11988721198882) 1198983 = 1198863 + 31198861198872 + 31198871198881 (1198862 + 1198872) + 6 (11988611988721198882 + 11988731198883) 1198984 = 1198864 + 411988631198871198881 + 12119886211988721198882 + 611988621198872 + 1211988611988731198881

+ 2411988611988731198883 + 1211988741198882 + 2411988741198884 + 311988741198985 = 1198865 + 511988641198871198881 + 20119886311988721198882 + 1011988631198872 + 30119886211988731198881

+ 60119886211988731198883 + 6011988611988741198882 + 12011988611988741198884 + 151198861198874+ 1511988751198881 + 6011988751198883 + 12011988751198885

1198986 = 1198866 + 611988651198871198881 + 30119886411988721198882 + 1511988641198872 + 60119886311988731198881+ 120119886311988731198883 + 180119886211988741198882 + 360119886211988741198884+ 4511988621198874 + 9011988611988751198881 + 36011988611988751198883 + 72011988611988751198885+ 9011988761198882 + 36011988761198884 + 72011988761198886 + 151198876

(28)

(g) The first six cumulants of the GC(119886 119887 ) distributionare

1205811 = 119886 + 11988711988811205812 = 1198872 (1 minus 11988821 + 21198882) 1205813 = 21198873 (11988831 minus 311988811198882 + 31198883) 1205814 = minus61198874 (11988841 minus 411988821 1198882 + 211988822 + 411988811198883 minus 41198884) 1205815 = 241198875 (11988851 minus 511988831 1198882 + 5119888111988822 + 511988821 1198883 minus 511988821198883 minus 511988811198884+ 51198885)

1205816 = minus1201198876 (11988861 minus 611988841 1198882 + 911988821 11988822 minus 211988832 + 611988831 1198883minus 12119888111988821198883 + 311988823 minus 611988821 1198884 + 611988821198884 + 611988811198885 minus 61198886)

(29)

(h) The following hold for any GC(119886 119887 ) distributionmean 119886 + 1198871198881

variance 1198872 (1 minus 11988821 + 21198882)skewness coefficient

2 (11988831 minus 311988811198882 + 31198883)(1 minus 11988821 + 21198882)32

excess kurtosis coefficient minus 6 (11988841 minus 411988821 1198882 + 211988822 + 411988811198883 minus 41198884)(1 minus 11988821 + 21198882)2

(30)

(i) Suppose 119883 sim GC(119886 119887 119883) 119884 sim GC(119886 119887 119884) Then thefirst 119870moments of119883 and 119884 are the same that is

E119883119895 = E119884119895 119895 = 1 119870 (31)

if and only if 119888119883119895 = 119888119884119895 119895 = 1 119870 This implies that if 1198881up to 1198884 are equal to 0 then the distribution has zero skewnessand excess kurtosis hence this shows how to construct aninfinite number of distributions that share this property withthe normal

(j) Suppose119883 sim GC(119886 119887 119883) Then

119886 = E119883 lArrrArr 1198881198831 = 01198872 = E (119883 minus 119886)2 lArrrArr 1198881198832 = 0

119886 = E119883 1198872 = Var119883 lArrrArr 1198881198831 = 1198881198832 = 0 (32)

When 1198881198831 = 1198881198832 = 0 the skewness and excess kurtosis coefficientsof119883 are 61198881198833 and 241198881198834 respectively for any119873 = 0 2 4

(k) If119883 sim GC(119886 119887 1198881 119888119873) and 119902 is a constant then 119884 =119902119883 sim GC(119886119902 119887|119902| 11988810158401 1198881015840119873) where 1198881015840119896 = (sign(119902))119896119888119896 119896 ge 1In particular minus119883 sim GC(minus119886 119887 minus1198881 (minus1)119873minus1119888119873minus1 119888119873)

(l) The law of the square of 119883 sim GC(0 1 1198881 119888119873) is acombination of chi-square distributions with 1 3 119873 + 1degrees of freedom that has density

119892 (119910) = 119890minus1199102radic21205871199101198992sum119895=0

120572119895119910119895

119908ℎ119890119903119890 1198992sum119895=0

120572119895119910119895 = 1198992sum119895=0

11988821198951198671198902119895 (radic119910) (33)

Proof Parts (a) and (b) were proved above and (c) followsdirectly from (b) To prove (d) it is sufficient to note theexistence of E119890119905119884 for 119905 in an open neighbourhood of 119904 = 0 For(e) if = 0 isin R119873 then the distribution is the standard normaland this is a Gram-Charlier distribution Since 119888119873 gt 0 and119873is even the polynomial

119873sum119896=1

119867119890119896 (119909) (34)


tends toinfin when |119909| tends toinfin Hence there is 1205980 gt 0 suchthat

1 + 120598 119873sum119896=0

119867119890119896 (119909) gt 0 119909 isin R 0 le 120598 le 1205980 (35)

(recall that 11988801198671198900(119909) = 1 for all Gram-Charlier distributions)The set of valid vectors thus includes (120598 120598) isin R119873 for each0 le 120598 le 1205980 If (119895) 119895 = 1 2 are valid then 119901(1) + (1 minus 119901)(2) isalso valid for any 119901 isin (0 1)

Part (f) is found by expanding the mgf in (b) as a series in119904 around the origin Part (g) is found by expanding

119905 997891997888rarr log(119890119886119905+(119887211990522) 119873sum119896=0

119887119896119888119896119905119896) (36)

The formulas in (h) follow (f) and (g)For part (i) it is sufficient to consider the case 119886 = 0 119887 = 1

only Write

120595 (119905) = 11989011990522119902119883 (119905) = 119873sum

119896=0

119888119883119896 119905119896119902119884 (119905) = 119873sum

119896=0

119888119884119896 119905119896(37)

and calculate the moments of 119883 and 119884 by successivelydifferentiating the mgf rsquos of 119883 and 119884 (note that 1198881198830 = 119902119883(0) =1198881198840 = 119902119884(0) = 1) The first 119870 moments of 119883 and 119884 are thesame if and only if

1205951015840 (0) 119902119883 (0) + 120595 (0) 1199021015840119883 (0) = 1205951015840 (0) 119902119884 (0) + 120595 (0) 1199021015840119884 (0) 119870sum119895=0

(119870119895)120595(119895) (0) 119902(119870minus119895)119883 (0) = 119870sum119895=0

(119870119895)120595(119895) (0) 119902(119870minus119895)119884 (0) (38)

Suppose that 119888119883119895 = 119888119884119895 119895 = 1 119870 Then 119902(119895)119883 (0) = 119902(119895)119884 (0)119895 = 1 119870 and thus the first119870moments of119883 and119884 are thesame Conversely suppose E119883119895 = E119884119895 119895 = 1 119870 Thenthe first identity above implies that 1198881198831 = 1199021015840119883(0) = 1199021015840119884(0) = 1198881198841 since 120595(0) is not zero The second identity implies that 1198881198832 =1198881198842 and so on up to 119888119883119870 = 119888119884119870

Turning to (j) the first equivalence

119886 = E119883 lArrrArr 1198881198831 = 0 (39)

is an immediate consequence of property (i) with 119870 = 1 Toprove the second equivalence suppose 119883 sim GC(0 1 ) andlet 119884 sim GC(0 1 minus) = N(0 1) Then

E1198832 = 12059510158401015840 (0) 119902119883 (0) + 21205951015840 (0) 1199021015840119883 (0) + 120595 (0) 11990210158401015840119883 (0) E1198842 = 12059510158401015840 (0) 119902119884 (0) + 21205951015840 (0) 1199021015840119884 (0) + 120595 (0) 11990210158401015840119884 (0) (40)

Here 119902119883(0) = 119902119884(0) = 1 and 1205951015840(0) = 0 hence E1198832 = E1198842if and only if 11990210158401015840119883(0) = 11990210158401015840119884(0) The last equality is 1198881198832 = 1198881198842 For119883 sim GC(119886 119887 119883) with 119886 and 119887 being arbitrary this meansthat

E(119909 minus 119886119887 )2 = 1 lArrrArr 1198881198832 = 0 (41)

For (k) observe that if 119902 = 0 then the mgf of 119884 is

119890119886119902119904+(12)119887211990221199042 119873sum119896=0

119888119896119887119896119902119896119904119896

= 119890119886119902119904+(12)119887211990221199042 119873sum119896=0

119888119896 ( 11990210038161003816100381610038161199021003816100381610038161003816)119896 (119887 10038161003816100381610038161199021003816100381610038161003816)119896 119904119896

(42)

Finally turn to (l) Routine calculations show that the densityof the square is

12radic119910 (119891119883 (radic119910) + 119891119883 (minusradic119910))= 120601 (radic119910)2radic119910

119873sum119896=0

119888119896 [119867119890119896 (radic119910) + 119867119890 (minusradic119910)] (43)

The Hermite polynomials of odd order are odd functionsand so disappear from that expression while the even orderHermite polynomials are even functions

When 119873 = 0 the GC(119886 119887 ) distribution is the normaldistribution with mean 119886 and variance 1198872 However part (b)of the theorem says that when119873 gt 0 the parameters 119886 and 1198872are not necessarily the mean and variance of the distributionSimple calculations lead to the following result

Theorem 3 (a) If 119883 sim GC(0 1 1198881 119888119873) thenP (119883 le 119909) = Φ (119909) minus 120601 (119909) 119873sum

119896=1

119888119896119867119890119896minus1 (119909) P (119883 gt 119909) = Φ (minus119909) + 120601 (119909) 119873sum

119896=1

119888119896119867119890119896minus1 (119909) (44)

(b) The tails of the GC(119886 119887 1198881 119888119873) distribution are

P (119883 gt 119909) sim 119888119873 (119909 minus 119886119887 )119873minus1 120601(119909 minus 119886119887 )as 119909 997888rarr infin

P (119883 lt 119909) sim 119888119873 (|119909 minus 119886|119887 )119873minus1 120601(119909 minus 119886119887 )as 119909 997888rarr minusinfin

(45)

The tails of the Gram-Charlier distributions are thickerthan those of the normal distribution but are still ldquothinrdquobecause they are in the limit smaller than any exponentialfunction exp(minus120572119909)


21 The GC(119886 119887 0 0 1198883 1198884) Family Here the exact region forthe (1198883 1198884) that lead to a true probability distribution has beenfound This goes back to Barton and Dennis [16] but a moredetailed explanation is given in Jondeau and Rockinger [6]The region is shown in Figure 3 (use the correspondence119904 = 61198883 119896 = 241198884 from Theorem 2(h)) An important factabout this region is that it is not rectangular the possibleexcess kurtosis values depend on skewness and conversely

22 GC(119886 119887 ) Distributions of Order 4 and Higher TheGC(119886 119887 1198881 1198882 1198883 1198884) family has six parameters rather thanfour and thus has more degrees of freedom in fitting datato the authorsrsquo knowledge the general six-parameter Gram-Charlier distribution has been used in financial applica-tions by Leon et al [11] only (those authors use the sub-family consisting of polynomials 119901(sdot) that are squares ofsome second-degree polynomial) Schlogl [10] fits the six-and eight-parameter families GC(119886 119887 0 0 1198883 1198884 1198885 1198886) andGC(119886 119887 0 0 1198883 1198884 1198885 1198886 1198887 1198888) to data

The set of (1198881 1198882 1198883 1198884) that yield true probability distri-butions has not been identified but it is possible to fit the sixparameters 119886 119887 1198881 1198882 1198883 1198884 by maximum likelihood

23 Why We Do Not Assume That 1198881 = 1198882 = 0 Almost allprevious authors have assumed that 1198881 = 1198882 = 0 because theyused normalized data

119884 = 119883 minus E119883radicVar119883 (46)

(see part (j) ofTheorem 2) as explained below In this sectionwe explain why it is important not to restrict Gram-Charlierdistributions or series in that way The first reason for notdoing so is that enlarging the parameter space can onlybe a good thing The second one is that in fitting thosedistributions to data there may be very real advantages inletting 1198881 and 1198882 be different from zero The only downsideof letting 1198881 and 1198882 take nonzero values and it is of no realimportance is that 1198883 and 1198884 lose their simple relationshipwith skewness and excess kurtosis (see part (h) ofTheorem2)A third reason is that after an exponential change of measurea Gram-Charlier distribution will rarely have 1198881 = 1198882 = 0 seeSection 24

We now show using an example that can be worked outexplicitly that it is not always best to use normalized datawhen fitting Gram-Charlier distributions because choosinganother affine transformation of the data may well yield amuch better fit

The ldquostandardrdquo Gram-Charlier expansion for a function119891(sdot) is120601 (119909) infinsum

119896=0

119888119896119867119890119896 (47)

where

119888119896 = 1119896 intinfin

minusinfin119867119890119896 (119909) 119891 (119909) 119889119909 = 1119896E119867119890119896 (119883) (48)

A classical result about Hermite series proved by Cramer[17] is that sufficient conditions for the Gram-Charlierexpansion (47) to converge to

12 (119891 (119909minus) + 119891 (119909+)) (49)

for all 119909 isin R are that (i) 119891(sdot) has finite variation in everybounded interval and (ii) satisfies

intinfinminusinfin

11989011990924119891 (119909) 119889119909 lt infin (50)

If 119883 has density 119891(sdot) then the last condition is E11989011988324 lt infinThis condition cannot be improved upon in the sense thatthere are cases where the Gram-Charlier series defined abovediverges althoughE1198901205881198832 lt infin for all 120588 lt 14 (it will be shownbelow that one such case is the normal distributionwithmean0 and variance 2)

Let us first calculate

ℎ119902119896 = E119867119890119896 (119883) 119883 sim N (0 1199022) (51)

Using the generating function (16) we find

ℎ119902119896 = 0 if 119896 is odd

11989621198962 (1198962) (1199022 minus 1)1198962 if 119896 is even (52)

The Gram-Charlier expansion corresponding to the N(0 1199022)density is thus

120601 (119909) infinsum119896=0

ℎ119902119896119896 119867119890119896 (119909)= 120601 (119909) infinsum

119899=0

12119899119899 (1199022 minus 1)1198991198671198902119899 (119909) (53)

It is possible to determinewhether this converges or notwhen119909 = 0 the Gram-Charlier series for 119891(0) is1radic2120587

infinsum119899=0

(minus1)119899 (2119899)22119899 (119899)2 (1199022 minus 1)119899 (54)

From Stirlingrsquos formula

(2119899)22119899 (119899)2 sim 1radic120587119899 (55)

as 119899 tends to infinity which implies that there are threepossibilities expansion (54) (i) converges absolutely if 0 lt1199022 lt 2 (ii) converges simply if 1199022 = 2 and diverges if 1199022 gt 2(The preceding calculations are from Cramer [17])

Let us now consider a random variable 119883119901 0 le 119901 le 1with density

119892119901 (119909) = 119901120601 (119909) + (1 minus 119901) 1119901120601(119909119901) (56)


In words 119883119901 has a N(0 1) distribution with probability 119901and aN(0 1199012) distribution with probability 1 minus 119901 Define thenormalized variable

119884119901 = 119883119901

radicVar119883119901

= 119883119901120590119901 1205902119901 = Var119883119901 = 119901 + 1199012 minus 1199013

(57)

The coefficients 119888119901119896 of the Gram-Charlier expansion for 119884119901are

119888119901119896 = 1119896E119867119890119896 (119884119901)= 1119896 (119901ℎ1120590119901 119896 + (1 minus 119901) ℎ119901120590119901 119896)

(58)

We have shown that (53) diverges if 1199022 gt 2 Here we see that119888119901119896 involves ℎ119902119896 with 119902 = 1120590119901 and 119902 = 119901120590119901 in the secondcase there is no problem as (119901120590119901)2 lt 2 for all 119901 isin (0 1] Inthe first case

11205902119901 gt 2 lArrrArr 119901 + 1199012 minus 1199013 lt 12 (59)

This condition is satisfied for 119901 smaller than 119901lowast asymp 0403and so the Gram-Charlier expansion for 119884119901 diverges for all119901 lt 119901lowast This is a sad state of affair a series designed to workfor distributions that are ldquoclose to the normalrdquo that fails forcombinations of two normal densities

There is an easy solution use a different scaling for 119883119901Rather than multiplying 119883119901 by 1120590119901 use a factor 119903 such thatthe series converges In this example we know that the seriesconverges if and only if 119903 lt radic2 Say we choose 119901 = 3 thisimplies

1Var119883119901

= 103 + 032 minus 033 = 275482 gt 2 (60)

Figure 1 shows how the true density for11988303 is approximatedby order 10 and 20 Gram-Charlier truncated series if 119903 =112059003 ≐ 166 is chosen the graph is not surprising theinfinite Gram-Charlier expansion diverges Figure 2 showsthe same except that 119903 is set to 14 The latter is smaller thanradic2 so the infinite expansion converges In this example thedensity to be approximated is symmetrical about 0 so 1198881 isalways 0 while

119902 = 112059003 997904rArr 1198882 = 0119902 = 14 997904rArr 1198882 = minus014426

(61)

minus2 minus1 1 2

minus4

minus2

2

4

6

Figure 1 Gram-Charlier approximations to the density 11989203(119909)(solid line) when 119902 = 112059003 ≐ 166 resulting in 1198881 = 1198882 = 0 Thecurve with long dashes is the order 10 approximation and the onewith short dashes is the order 20 approximation

minus2 minus1 1 2

02

04

06

08

10

Figure 2 Gram-Charlier approximations to the density 11989203(119909)(solid line) when 119902 = 14 The curve with long dashes is the order10 approximation and the one with short dashes is the order 20approximation

24 Exponential Change of Measure The one-dimensionalCameron-Martin formula may be stated as follows if 119883 simN(120583 1205902) then for 119902 isin R and 119891 ge 0

E119890119902119883119891 (119883) = 119890120583119902+(12)12059021199022E119891 (119883 + 1205902119902) (62)

The same property may be expressed in terms of a change ofmeasure If 119883 Psim N(120583 1205902) and a change of measure is definedby

1198751015840 = 119890119902119883EP119890119902119883 sdot 119875 (63)

then 119883 P1015840sim N(120583 + 1205902119902 1205902) The next result is an extension ofthe Cameron-Martin formula for the normal distribution tothe Gram-Charlier distributions There are different thoughequivalent formulas in Schlogl [10] (for the special case 1198881 =1198882 = 0)Theorem 4 Suppose that 119883 Psim GC(119886 119887 1198881 119888119873) and thatP1015840 is defined by (63) for 119902 isin R Then 119883 P1015840sim GC(119886 + 1198872119902 11988711988810158401 1198881015840119873) where 11988810158401 1198881015840119873 are found from

119873sum119896=0

1198871198961198881015840119896119904119896 = sum119873119896=0 119887119896119888119896 (119902 + 119904)119896sum119873

119895=0 119887119895119888119895119902119895 119904 isin R (64)


or more precisely

1198881015840119896 = 1sum119873119895=0 119887119895119888119895119902119895

119873sumℓ=119896

(ℓ119896) 119887ℓminus119896119888ℓ119902ℓminus119896 (65)

Proof It is sufficient to calculate the mgf of119883 under P1015840

E1198751015840119890119904119883 = 1

EP119890119902119883EP119890(119902+119904)119883 = (sdot sdot sdot)= 119890(119886+1198872119902)119904+(12)11988721199042 119873sum

119896=0

119887119896119904119896 1sum119873119895=0 119887119895119888119895119902119895

sdot 119873sumℓ=119896

(ℓ119896) 119887ℓminus119896119888ℓ119902ℓminus119896(66)

Example 5 If 119883 Psim GC(119886 119887 0 0 1198883 1198884) and 119902 = 1 the changeof measure (63) leads to

11988810158401 = 311988721198883 + 4119887311988841 + 11988731198883 + 11988741198884 11988810158402 = 31198871198883 + 6119887211988841 + 11988731198883 + 11988741198884 11988810158403 = 1198883 + 411988711988841 + 11988731198883 + 11988741198884 11988810158404 = 11988841 + 11988731198883 + 11988741198884

(67)

Hence assuming 1198881 = 1198882 = 0 does not imply that 11988810158401 and 11988810158402are also zero In other words the familyGC(119886 119887 0 0 1198883 1198884) isnot closed under a change of measure that occurs naturally inoption pricing (see Section 5)

25 Simulating Gram-Charlier Distributions Simulation isrequired for many kinds of options and it turns out that theGram-Charlier distributions are very easy to generate as wenow show there is no need to invert their distribution func-tions When estimating some quantity 119898 = E119892(1198831 119883119904)by simulation one generates 119899 independent vectors

(119883(119895)1 119883(119895)

119904 ) 119895 = 1 119899 (68)

with the same distribution Suppose all the 119883rsquos are inde-pendent and have the same Gram-Charlier distribution withdensity

119891119883 (119909) = 120601 (119886 119887 119909) 119901 (119909) (69)

where the polynomial 119901(119909) is given in (5) Then

E119892 (1198831 119883119904) = intR119904119892 (1199091 119909119904)

sdot 119904prod119896=1

[120601 (119886 119887 119909119896) 119901 (119909119896)] 1198891199091 sdot sdot sdot 119889119909119904= EQ [119892 (1198831 119883119904) 119904prod

119896=1

119901 (119883119896)] (70)

where under the measure Q the 119883rsquos have a normal distri-bution N(119886 1198872) Hence estimating E119892(1198831 119883119904) by sim-ulation can be performed by generating ordinary normalrandom variables (119883(119895)

1 119883(119895)119904 ) 119895 = 1 119899 and then

using the estimator

119899 = 1119899119899sum119895=1

[119892 (119883(119895)1 119883(119895)

119904 ) 119904prod119896=1

119901 (119883(119895)

119896)] (71)

This is an application of the likelihood ratio method

3 Convolution of Gram-CharlierDistributions Gram-Charlier Processes

The simplest way to find the distribution of the sum oftwo independent Gram-Charlier distributed variables is tomultiply their moment generating functions Suppose 119883119895 simGC(119886119895 119887119895 119888(119895)1 119888(119895)119873 ) 119895 = 1 2 are independent Then

E119890119911(119883+119884)= 119890(1198861+1198862)119905+(11988721+11988722 )(11990522)(119873(1)sum

119896=0

119888(1)119896 119905119896)(119873(2)sum119896=0

119888(2)119896 119905119896) (72)

Expanding the product this says that

119883 + 119884 sim GC(1198861 + 1198862 radic11988721 + 11988722 1198881 119888119873(1)+119873(2)) 119888119896 = 119896sum

119895=0

119888(1)119895 119888(2)119896minus119895(73)

It is then possible to know the explicit distribution of

119885119899 = 1198831 + sdot sdot sdot + 119883119899 (74)

where the 119883rsquos are independent and have a Gram-Charlierdistribution constituting a discrete-timeGram-Charlier pro-cess If the119883rsquos have the sameGC(119886 119887 1198881 119888119873) distributionthen 119885119899 119899 ge 0 is a random walk The derivation of thedistribution of 119885119899 can be done recursively using (73) or itcan be done by finding the Taylor expansion of

( 119873sum119896=0

119888119896119905119896)119899 = 119899119873sum

119896=0

119888(119899)119896 119905119896= 1 + 1198881119899119905 + (1198882119899 + 1211988821119899 (119899 minus 1)) 1199052 + sdot sdot sdot+ 119888119899119873119905119899119873

(75)


and thus

119885119899 sim GC(119886119899 119887radic119899 1198881119899 1198882119899 + 1211988821119899 (119899 minus 1) 119888119899119873) (76)

These computations are simple using symbolic mathematicssoftware An example is given at the end of Section 61

The above raises the question of whether there is acontinuous-time process that has Gram-Charlier distributedincrements There is such a process with normal increments(Brownian motion) and it is moreover a Levy process

Theorem 6 The only Levy process with Gram-Charlier dis-tributed increments is Brownian motion

Proof It is sufficient to show that besides the normal distri-bution any Gram-Charlier distribution cannot be infinitelydivisible If 119883 has a Gram-Charlier distribution then itsmoment generating function is of the form

119872(119911) = E119890119911119883 = 119890119886119911+1198872(11991122)120587 (119911) (77)

where 120587(sdot) is a polynomial Suppose 1198831198991 119883119899119899 are inde-pendent have the same distribution and add up to119883 (in law)Fubinirsquos theorem implies that E1198901199051198831198991 is finite for all real 119905 andthus

119872119899 (119911) = E1198901199111198831198991 (78)

is an analytic function of 119911 in the whole complex plane Wethen have the identity

119872119899 (119911)119899 = 119872(119911) 119911 isin C (79)

for every 119899 = 1 2 3 This means that the function 120587(119911)1119899which is well defined for all 119911 that is not a zero of 120587 has ananalytic continuation over the whole of C for each 119899 ge 1which is impossible for instance take 119899 larger than the degreeof 120587

More precisely this says that if we exclude Brownianmotion no increment of any Levy process can have a Gram-Charlier distribution Any Gram-Charlier process with inde-pendent increments must be discrete-time

4 The Log Gram-Charlier Distribution

The distribution of the exponential of a Gram-Charlier dis-tributed variable will naturally be called ldquologGram-Charlierrdquoas we do for the lognormal if 119884 sim GC(119886 119887 ) then 119871 =exp(119884) sim LogGC(119886 119887 ) The density of 119871 is

119891119871 (119911) = 1119911119891119884 (log 119911) 119911 gt 0 (80)

This distribution has all moments finite and they are givenby Theorem 2(b) The log Gram-Charlier distribution sharesone property with the lognormal it is ldquomoment indetermi-naterdquo

Theorem 7 The log Gram-Charlier distribution is not deter-mined by its moments More precisely there is a noncountablenumber of other distributions that have the same moments asany particular log Gram-Charlier distribution

Proof There is a well-known way to construct a family ofdistributions that have the same moments as the lognormal(Feller [18] p 227) the trick works for arbitrary parametersbut for simplicity let 119871 sim LogN(0 1) that is

119891119871 (119911) = 1119911radic2120587119890minus(12)log2119911 119911 gt 0 (81)

All the functions

119891120572 (119911) = 119891119871 (119911) (1 + 120572 sin (2120587 log 119911)) minus 1 le 120572 le 1 (82)

are nonnegative integrate to 1 and have the same momentsas 119871 (as a result of the symmetry of the standard normaldistribution after an obvious change of variable) Nowsuppose 119871 sim LogGC(0 1 ) (once again the same argumentswork for other values of 119886 and 119887) and consider

119892120572 (119911) = 119891119871 (119911) + 120572120601 (119911) sin (2120587 log 119911) (83)

By the same change of variables used for the lognormalone immediately finds that 119892120572 integrates to one and has thesame moments as 119871 The only difference with the case ofthe lognormal is that 119892119886 is not necessarily nonnegative forminus1 le 120572 le 1 Two cases may arise The first one is that thepolynomial

119901 (119910) = 119873sum119896=0

119888119896119867119890119896 (119910) (84)

has no real zero In that case its infimum is strictly greaterthan zero and one can find a nonempty interval 119868 = (minus120598 120598)such that 119892120572 is nonnegative for all 120572 isin 119868 In the second case119901(sdot) has one or more zeros and the previous argument breaksdown but can be modified to yield the same conclusion if thefunction sin(2120587119910) is replaced with

119904 (119910) = sin (2120587119910) 1119910notin119869 (85)

where 119869 is a collection of intervals that include the zeros of119901(sdot) so defined that 119904(sdot) is not identically zero and satisfies

119904 (119910 + 2119896120587) = 119904 (119910) 119904 (minus119910) = minus119904 (119910) (86)

(These two conditions are sufficient for

intinfinminusinfin

119890119896119910120601 (119910) 119904 (119910) 119889119910 = 0 (87)


to hold for 119896 = 1 2 )The details are omitted Another wayto prove that the log Gram-Charlier distribution is momentindeterminate when 119901(119910) has no zero (and thus the density119891119871(119911) does not take the value zero for any 119911 gt 0) is to usea Krein condition (Stoyanov [19] p 941) which says that acontinuous distribution on R+ with positive density 119891(sdot) isnot determined by its moments if

minusintinfin0

log119891 (1199112)1 + 1199112 119889119911 lt infin (88)

5 Option Pricing Formulas

The formulas below hold for any vector (1198881 119888119873) and thusextend those that have been derived by previous authorsfor the case where the log return has a GC(119886 119887 0 0 1198883 1198884)distribution or a ldquosquaredrdquoGram-Charlier distribution (Leonet al [11]) Schlogl [10] derives a formula equivalent to (90)but for densities expressed as an infinite Gram-Charlier serieswith 1198881 = 1198882 = 0

As previous authors have done we consider amarket witha risky security 119878 and a risk-free security with annual return 119903and suppose that the log return for period [0 119879] (denoted as119883119879) has a Gram-Charlier distribution under the risk-neutral(or ldquopricingrdquo) measure which we denote as Q The physicalmeasure (usually denoted as P) is not specified (nothing saysthat the log return has or does not have a Gram-Charlierdistribution under the physical measure)

The market model may have one or more periods butsince we consider the pricing of ordinary European putsand calls only the distribution of the log return for thewhole period [0 119879] is required For other types of optionsin particular the applications presented in the next section itmay be necessary to use the one-period returns separately asis done inTheorem 11

The risky security has price 1198780 at time 0 and 119878119879 = 1198780119890119883119879 Theorem 8 Suppose that a risky security pays dividends ata constant rate 120575 over [0 119879] and that the risk-free rate ofinterest is 119903 Suppose also that under the risk-neutral measureQ the log return of the risky security over [0 119879] is 119883119879

QsimGC(119886 119887 1198881 119888119873) which satisfies the martingale condition

119890119886+(11988722) 119873sum119896=0

119887119896119888119896 = 119890(119903minus120575)119879 (89)

Then the time-0 price of a European call option with maturity119879 and strike price 119870 is

1198620 = 1198780119890minus120575119879Φ(1198891) minus 119870119890minus119903119879Φ(1198892) + 119870119890minus119903119879120601 (1198892)sdot 119873sum119896=1

[119888lowast119896119867119890119896minus1 (minus1198891) minus 119888119896119867119890119896minus1 (minus1198892)] (90)

where 1198892 1198891 and 119888lowast119896 are given by (97) (101) and (105) Theprice of the corresponding European put is

1198750 = 119870119890minus119903119879Φ(minus1198892) minus 1198780119890minus120575119879Φ(minus1198891) minus 119870119890minus119903119879120601 (1198892)sdot 119873sum119896=1


If119873 = 4 then the above simplify to

1198620 = 1198780119890minus120575119879Φ(1198891) minus 119870119890minus119903119879Φ(1198892) + 119887119870119890minus119903119879120601 (1198892)sdot [1198882 + (119887 minus 1198892) 1198883 + (1198872 minus 1198871198892 + 11988922 minus 1) 1198884]

1198750 = 119870119890minus119903119879Φ(minus1198892) minus 1198780119890minus120575119879Φ(minus1198891) minus 119887119870119890minus119903119879120601 (1198892)sdot [1198882 + (119887 minus 1198892) 1198883 + (1198872 minus 1198871198892 + 11988922 minus 1) 1198884]

(92)

Proof Absence of arbitrage implies that EQ119878119879 = 119890(119903minus120575)1198791198780 or1198780119890119886+(11988722) 119873sum

119896=0

119887119896119888119896 = 119890(119903minus120575)1198791198780 (93)

This justifies (89) The price of the call is

1198620 = 119890minus119903119879EQ (119878119879 minus 119870)+ = 119862+0 minus 119862minus0119862+0 = 119890minus119903119879EQ119878119879 (1119878119879gt119870) 119862minus0 = 119870119890minus119903119879EQ (1119878119879gt119870)

(94)

The second part is easier to deal with

119862minus0 = 119870119890minus119903119879Q(119883119879 gt log(1198701198780)) (95)

Theprobability of the event119883119879 gt log(1198701198780) can be calculatedexplicitly by recalling that (119883119879 minus 119886)119887 Qsim GC(0 1 1198881 119888119873)and usingTheorem 3

Q(119883119879 gt log(1198701198780))= Q(119883119879 minus 119886119887 gt 1119887 (log(1198701198780) minus 119886))= Φ (1198892) + 120601 (1198892) 119873sum

119896=1

119888119896119867119890119896minus1 (minus1198892) (96)

where

1198892 = 1119887 (log(1198780119870) + 119886) (97)

To calculate 119862+0 use the exponential change of measureformula defining

Q1015840 = 119878119879EQ119878119879 sdotQ = 119890119883119879

EQ119890119883119879 sdotQ (98)


Then

EQ (1198781198791119878119879gt119870) = (EQ119878119879)EQ1015840 (1119878119879gt119870) (99)

Since119883119879

Q1015840sim GC(119886 + 1198872 119887 11988810158401 1198881015840119873)(EQ119878119879)EQ1015840 (1119878119879gt119870) = 1198780119890(119903minus120575)119879Q1015840 (119883119879 minus 119886 minus 1198872119887gt 1119887 (log(1198701198780) minus 119886 minus 1198872)) = 1198780119890(119903minus120575)119879Φ(1198891)+ 1198780119890(119903minus120575)119879120601 (1198891) 119873sum

119896=1

1198881015840119896119867119890119896minus1 (minus1198891) (100)

where

1198881015840119896 = 1sum119873119895=0 119887119895119888119895

119873sumℓ=119896

(ℓ119896) 119887ℓminus119896119888ℓ 119896 = 1 1198731198891 = 1119887 [log(1198780119870) + 119886 + 1198872] = 1198892 + 119887

(101)

Hence

1198620 = 1198780119890minus120575119879Φ(1198891) minus 119870119890minus119903119879Φ(1198892)+ 1198780119890minus120575119879120601 (1198891) 119873sum

119896=1

1198881015840119896119867119890119896minus1 (minus1198891)minus 119870119890minus119903119879120601 (1198892) 119873sum

119896=1

119888119896119867119890119896minus1 (minus1198892) (102)

Using

1198780119890minus120575119879120601 (1198891)sum119873119896=0 119887119896119888119896 = 1198780119890119886+(11988722)minus119903119879120601 (1198891) = 119870119890minus119903119879120601 (1198892) (103)

(a consequence of (89) and (101)) it is possible to write

1198780119890minus120575119879120601 (1198891) 1198881015840119896 = 119870119890minus119903119879120601 (1198892) 119888lowast119896 (104)

where

119888lowast119896 = 1198881015840119896 119873sum119896=0

119887119896119888119896 = 119873sumℓ=119896

(ℓ119896) 119887ℓminus119896119888ℓ 119896 = 1 119873 (105)

This proves (90)The price of a European put can be found from the put-

call parity identity

1198750 = 1198620 minus 1198780 + 119870119890minus119903119879 (106)

Finally if119873 = 4 then the summation in (90) becomes

4sum119896=1

119888lowast119896119867119890119896minus1 (minus1198891) minus 4sum119896=1

119888119896119867119890119896minus1 (minus1198892)= 1198871198882 + (1198872 minus 1198871198892) 1198883 + (1198873 minus 11988721198892 + 11988711988922 minus 119887) 1198884

(107)

This ends the proof

The option price formulas are of the form ldquoBlack-Scholesplus correction termrdquo Observe however that the values of1198891 and 1198892 are different from what they would be in theBlack-Sholes formula More precisely in the Black-Scholesmodel (where 119888119896 = 0 for all 119896 ge 1) we have 119886 =EQ119883119879 and 1198872 = VarQ119883119879 but this does not happen withGram-Charlier distributed log returns first because of themartingale condition (89) and second because of the resultin part (j) of Theorem 2

51 Sensitivities (ldquoGreeksrdquo) The previous literature includesformulas for sensitivities of Gram-Charlier option prices butonly in the case of the four-parameter GC(119886 119887 0 0 1198883 1198884)Gram-Charlier distributions see Jurczenko et al [7] Rouahand Vainberg [20] and Chateau [3] Below we give sensi-tivities of the option prices calculated above with respect toall the parameters for a general Gram-Charlier distributiontaking the martingale condition (89) into account Thismeans that 119886 is a function of 120575 119887 119888119896 119903 and 119879 (we might havewritten 119886 = 119886(120575 119887 119888119896 119903 119879)) However 119886 is not a functionof 1198780Theorem 9 Let 1198620 be the price of the European call optiondescribed in Theorem 8 Then

Δ = 12059711986201205971198780= 119890minus120575119879(Φ(1198891) + 120601 (1198891) 119873sum

119896=1

1198881015840119896119867119890119896minus1 (minus1198891))

120574 = 1205972119862012059711987820 = 119890119886+(11988722)minus1199031198791198871198780 120601 (1198891) 119873sum119896=0

119888119896119867119890119896 (minus1198892) 120588 = 1205971198620120597119903= 119870119879119890minus119903119879(Φ(1198892) + 120601 (1198892) 119873sum

119896=1

119888119896119867119890119896minus1 (minus1198892)) 120581 = 1205971198620120597119887 = 1198780119890minus120575119879120601 (1198891)sdot 119873sum119896=0

[1198881015840119896 minus 119888101584011198881015840119896+1 + (119896 + 2) 1198881015840119896+2]119867119890119896 (minus1198891) 1205971198620120597119888119895 = 119870119890minus119903119879120601 (1198892)

sdot (119895minus1sum119896=1

119887119896119867119890119895minus1minus119896 (minus1198892) minus 119887119895 119873sum119896=1

1198881015840119896119867119890119896minus1 (minus1198891)) 119895 ge 1

(108)

In the formula for 120581 the symbols 1198881015840119873+1 and 1198881015840119873+2 are equal to zero


Proof The following lemma is obtained by elementary calcu-lations

Lemma 10 For any integrable random variable 119880 and anyconstant 119870

120597120597119904E (119904119880 minus 119870)+ = E (1198801119904119880gt119870) (109)

If 119880 has a continuous density 119891119880 then12059721205971199042E (119904119880 minus 119870)+ = 119870

2

1199043 119891119880 (119870119904 ) (110)

From 119878119879 = 1198780119890119883119879 and (100)

Δ = 1205971205971198780 [119890minus119903119879EQ (1198780119890119883119879 minus 119870)+] = 11198780119862+0= 119890minus120575119879(Φ(1198891) + 120601 (1198891) 119873sum

119896=1

1198881015840119896119867119890119896minus1 (minus1198891)) (111)

From the lemma

120574 = 1205972120597211987801198620 = 119890minus119903119879120597212059721198780EQ (1198780119880 minus 119870)+

= 1198702119890minus11990311987911987830 119891119880 (1198701198780) 119880 = 119890119883(112)

where 119883119879

Qsim GC(119886 119887 1198881 119888119873) Now the density of 119880 maybe expressed in terms of the density of 119866 = (119883119879 minus 119886)119887 QsimGC(0 1 1198881 119888119873)

119891119880 (119906) = 1119887119906119891119866 (1119887 (log 119906 minus 119886))= 1119887119906120601 (119910)

119873sum119896=0

119888119896119867119890119896 (119910)1003816100381610038161003816100381610038161003816100381610038161003816119910=(1119887)(log 119906minus119886)

(113)

Setting 119906 = 1198701198780 and then using (103) we obtain

120574 = 119870119890minus11990311987911988711987820 120601 (1198892) 119873sum119896=0

119888119896119867119890119896 (minus1198892)= 119890119886+(11988722)minus1199031198791198871198780 120601 (1198891) 119873sum

119896=0

119888119896119867119890119896 (minus1198892) (114)

Next

120588 = 120597120597119903EQ (1198780119890119886+119887119866minus119903119879 minus 119870119890minus119903119879)+ = 120597120597119903EQℎ (119866 119903) (115)

where 119866 Qsim GC(0 1 ) For fixed 119866 the function 119892(119903) =ℎ(119866 119903) is absolutely continuous and thus

120588 = EQ [ 120597120597119903 (1198780119890119886+119887119866minus119903119879 minus 119870119890minus119903119879)] 11198780119890119886+119887119866minus119903119879gt119870119890minus119903119879 (116)

Now from (89)

119890119886+119887119866minus119903119879 = 119890119887119866minus(11988722)minus120575119879sum119873119896=0 119887119896119888119896 (117)

does not depend on 119903 so120588 = EQ (119870119879119890minus11990311987911198780119890119886+119887119866minus119903119879gt119870119890minus119903119879)= 119870119879119890minus119903119879 [Φ (1198892) + 120601 (1198892) 119873sum

119896=1

119888119896119867119890119896minus1 (minus1198892)] (118)

The same line of reasoning shows that the sensitivity to 119887 is120581 = 1198780119890minus119903119879EQ [( 120597120597119887119890119886+119887119866) 1S0119890119886+119887119866gt119870] (119)

Define

119898(119887) = E119890119887119866minus(11988722) = sum119896=0

1198871198961198881198961198981015840 (119887) = sum

119896=1

119896119887119896minus1119888119896 = 119888lowast1 (120)

In order to shorten the formulas we will write (from (89))

119890119886 = 119890(119903minus120575)119879minus(11988722)119898(119887) (121)

and thus find an expression for

1198780119890minus120575119879EQ [( 120597120597119887 119890119887119866minus(11988722)

119898(119887) ) 1119866gtminus1198892]

= 1198780119890minus120575119879minus(11988722)119898(119887)2sdot EQ [(119866119890119887119866119898(119887) minus 119890119887119866 (1198981015840 (119887) + 119887119898 (119887)))sdot 1119866gtminus1198892]

(122)

For arbitrary 119910 isin R

EQ (1198901198871198661119866gtminus119910) = 11989011988722 [119898 (119887)Φ (119910 + 119887)

+ 120601 (119910 + 119887) 119873sum119896=1

119888lowast119896119867119890kminus1 (minus119910 minus 119887)] (123)

To calculate

EQ (1198661198901198871198661119866gtminus119910) = 120597120597119887EQ (1198901198871198661119866gtminus119910) (124)

we need (see (105))

120597119888lowast119896120597119887 = 120597120597119887119873sumℓ=119896

(ℓ119896) 119887ℓminus119896119888ℓ = (119896 + 1) 119888lowast119896+1119896 = 1 119873 minus 1

120597119888lowast119873120597119887 = 0(125)


Then noting that 1198981015840(119887) = 119888lowast1 = 11988810158401119898(119887) and (120601119867119890119896minus1)1015840 =minus120601119867119890119896EQ (1198661198901198871198661119866gtminus119910) = 119887EQ (1198901198871198661119866gtminus119910) + 11989011988722119898(119887)sdot [11988810158401Φ(119910 + 119887) + 120601 (119910 + 119887)sdot 119873minus2sum119896=0

(119896 + 2) 1198881015840119896+2119867119890119896 (minus119910 minus 119887) + 120601 (119910 + 119887)sdot 119873sum119896=0

1198881015840119896119867119890119896 (minus119910 minus 119887)]

(126)

Setting 119910 = 1198892 and putting the pieces together yield theresult When = 0 this reduces to 1198780119890minus120575119879120601(1198891) which is thesensitivity of the Black-Scholes call price to 119887 = 120590radic119879

Finally turn to the sensitivities with respect to 119888119895 119895 ge 1First let us derive 120597119886120597119888119895 from (89)

120597119886120597119888119895 119890119886+(11988722)

119873sum119896=0

119887119896119888119896 + 119887119895119890119886+(11988722) = 0or 120597119886120597119888119895 = minus

119887119895sum119873119896=0 119887119896119888119896 = minus119887

119895119890119886+(11988722)+(120575minus119903)119879(127)

Next

119890minus119903119879 120597120597119888119895EQ (1198780119890119886+119887119866 minus 119870)+= 119890minus119903119879 120597120597119888119895 int

infin

minus1198892

(1198780119890119886+119887119910 minus 119870)120601 (119910) 119873sum119896=0

119888119896119867119890119896 (119910) 119889119910= 119890minus119903119879intinfin

minus1198892

1198780 120597119886120597119888119895 119890119886+119887119910120601 (119910)119873sum119896=0

119888119896119867119890119896 (119910) 119889119910+ 119890minus119903119879intinfin

minus1198892

(1198780119890119886+119887119910 minus 119870)120601 (119910)119867119890119895 (119910) 119889119910

(128)

The first integral reduces to 1198780Δ120597119886120597119888119895 (see the derivation ofΔ above)The second integral is evaluated by repeated partialintegration if 119892(119909) is continuously differentiable 119896 times anddoes not grow too fast

intinfin119910119892 (119909) 120601 (119909)119867119890119895 (119909) 119889119909= 120601 (119910) 119895minus1sum

119896=0

119892(119896) (119910)119867119890119895minus1minus119896 (119910)+ intinfin

119910119892(119895) (119909) 120601 (119909) 119889119909

(129)

Hence the second integral in (128) becomes

1198780119890119886minus1198871198892minus119903119879120601 (1198892) 119895minus1sum119896=1

119887119896119867119890119895minus1minus119896 (minus1198892)+ 1198780119890119886minus119903119879intinfin

minus1198892

119887119895119890119887119910120601 (119910) 119889119910= 1198780119890119886minus1198871198892minus119903119879120601 (1198892) 119895minus1sum

119896=1

119887119896119867119890119895minus1minus119896 (minus1198892)+ 1198871198951198780119890119886minus119903119879+(11988722)Φ(1198891)

(130)

Putting all this together (using (103)) we get the formula for12059711986201205971198881198956 Two Applications

61 Equity Indexed Annuities Hardy [12] describes the maintypes of equity indexed annuities (EIAs) point-to-pointannual ratchet and high water mark in which an embed-ded European call option has to be priced We considercompound ratchet EIAs without life-of-contract guaranteeto explore the dependence of ratchet premium options onskewness and kurtosis of returns

A single premium 119875 is paid by the policy-holder and thebenefit under the ratchet premium contract is ([12] p 248)

119861119899 = 119875 119899prod119905=1

[1 +max(120572( 119878119905119878119905minus1 minus 1) 0)] (131)

Here 119899 is the term of the contract in years 120572 is theparticipation rate (a number between 0 and 1) and 119878119905 is thevalue of the equity index usually the SampP500 index

In [12] p 249 the formula

119875 [119890minus119903 + 120572 (119890minus120575Φ(1198891) minus 119890minus119903Φ(1198892))]119899 (132)

for 119861119899 is proved for the value of the premium optionunder a compound annual ratchet contract in the Black-Scholes model The same reasoning will now be appliedwhen index log returns have a Gram-Charlier distributionTo find a formula for the price of this option we assume thatunder the risk-neutral measure the one-year log returns areindependent Gram-Charlier distributed random variables

119877119905 Qsim GC (119886 119887 ) 119905 = 1 119899 (133)

Independence of the variables across time implies that thevalue of the ratchet premium option is (if the annual risk-freerate of interest is 119903)1198810 = EQ (119890minus119903119899119861119899)= 119875 119899prod

119905=1

119890minus119903EQ [1 +max(120572( 119878119905119878119905minus1 minus 1) 0)] (134)

As Hardy notes ([12] p 249) each factor in the product is theprice of a one-year European call option on 119878119905 if initial indexprice and strike are both equal to one


Theorem 11 The no-arbitrage price of the EIA ratchet pre-mium option described above is

1198810 = 119875 (119890minus119903 + 1205721198620)119899 (135)

where 1198620 is the price of a one-year call (see (90)) with 1198780 = 1and 119870 = 1 The first-order sensitivities are

1205971198810120597120585 = 120572119875 (119890minus119903 + 1205721198620)119899minus1 1205971198620120597120585 (136)

with 120585 replaced with one of 1198780 119903 119887 119888119895 and 1205971198620120597120585 given inTheorem 9 The second-order sensitivity with respect to 1198780 is

1205972119862012059711987820 = 1205722119875 (119890minus119903 + 1205721198620)119899minus2 (12059711986201205971198780 )2

+ 120572119875 (119890minus119903 + 1205721198620)119899minus1 1205972119862012059711987820 (137)

We now show the effect of skewness and kurtosis on theratchet premium option values this is done using the four-parameter distribution GC(119886 119887 0 0 1198883 1198884) for which this isespecially simple We also compute prices using the six-parameter GC(119886 119887 1198881 1198882 1198883 1198884) distribution

Our application involves an annual ratchet and so theoption 1198620 in Theorem 11 has a maturity of one year wethus need the distribution of one-year returns For pric-ing that is consistent with the (derivatives) market whatis needed is the one-year distribution of returns underthe risk-neutral measure which should be obtained fromobserved derivative prices see Leon et al [11] or Rompolisand Tzavalis [21] for more details For illustrative purposeswe estimated the four parameters of that distribution bymaximum likelihood ([6 22]) to obtain the parameters ofthe GC(119886 119887 0 0 1198883 1198884)

119886 = 00687119887 = 016851198883 = minus011501198884 = 003598

(138)

This says that the skewness and excess kurtosis of the fitteddistribution are

119904 = 61198883 = minus06898119896 = 241198884 = 08634 (139)

This point is on the boundary of the feasible (119904 119896) region seeFigure 3

We use the parameters 119886 and 119887 while varying the valuesof 1198883 1198884 to show how EIA premiums vary with skewness andexcess kurtosis Figure 4 shows the ratchet premium optionvalues (Theorem 11) of an EIA with participation rate 120572 = 06and single premium 119875 = 100 as a function of skewness 119904 and

0

1

2

3

4

k

minus10 minus05 00 05 10

s

Figure 3 Feasible region for the skewness and excess kurtosis of theGC(119886 119887 0 0 1198883 1198884) distributions and maximum likelihood estimate

minus1

0

1

0

2

4 104

106

108

110

Option

k s

Figure 4 Ratchet premium option prices as a function of skewnessand excess kurtosis withGC(119886 119887 0 0 1198883 1198884) distributions

excess kurtosis 119896 The risk-free rate is 3 the dividend rateis 2 and the term is 7 years The martingale condition (89)must hold and so 119886 is replaced with 1198861015840 so that (89) is satisfied(NBThe Black-Scholes model corresponds to 1198883 = 1198884 = 0 inwhich case 1198861015840 = (119903 minus 120575)119879 minus 11988722 this is minus0004193 with thevalues of 119903 120575 and 119887 we are using and 119879 = 1)

The effect of varying (1198883 1198884) is quite significant Thehighest and lowest ratchet premium options in the graphare 10930 to 10424 which is the range [95 100] as aproportion of the Black-Scholes premium option ($10926)which has skewness and excess kurtosis equal to zero (Themaximum value of the premiumoption is reached at (1198883 1198884) =(00230 000332567) and the minimum at (minus00836 0163)these points correspond to (119904 119896) equal to (138 0798) and


(0023 0003326) resp) The dependence on (119904 119896) is nearlylinear This is easily explained The price of a call is

1198620 = 119890minus119903119879EQ (119878119879 minus 119870)= 119890minus119903119879intinfin

(1119887)(log(1198701198780)minus1198861015840)(11987801198901198861015840+119887119909 minus 119870)120601 (119909)

sdot ( 119873sum119896=0

119888119896119867119890119896 (119909))119889119909(140)

The sensitivity of 1198620 to 119888119895 (119895 = 3 4) is thus119890minus119903119879 1205971198861015840120597119888119895 EQ (1198781198791119878119879gt119870)+ 119890minus119903119879intinfin

(1119887)(log(1198701198780)minus1198861015840)(11987801198901198861015840+119887119909 minus 119870)120601 (119909)

sdot 119867119890119895 (119909) 119889119909(141)

The first term above is relatively small and the second one isalmost constant since 1198861015840 does not change much with 119888119895 Thiscarries over to the sensitivity of the ratchet premium option1198810 to changes in 119888119895 (see the first formula inTheorem 11)

The parameters fitted by maximum likelihood for theGC(119886 119887 1198881 1198882 1198883 1198884) distribution are

119886 = 01174119887 = 015951198881 = minus030536756952010661198882 = 0095420793734891531198883 = minus0123839711263352431198884 = 006120331530131559

(142)

The martingale condition implies 1198861015840 = 00451 and theEIA premium option has value 10790 The skewness andexcess kurtosis of the fitted distribution are 119904 = minus05437and 119896 = 05092 (NB The feasible region for (119904 119896) inthe GC(119886 119887 1198881 1198882 1198883 1198884) case is naturally larger than theone for the GC(119886 119887 0 0 1198883 1198884) in this case (119904 119896) are justoutside the region in Figure 3) It was checked numeri-cally that the parameters 1198881 to 1198884 lead to a nonnegativefunction

Table 1 shows ratchet premium option values based onGram-Charlier with four parameters that are lower than theBlack-Scholes based ones Rows A and F are Black-Scholescases row B is our estimation of the four-parameter Gram-Charlier (as explained above) row C is the four-parameterGram-Charlier with the largest negative skewness possible(see Figure 1) row D is similar but with the largest positiveskewness and row E hasmaximum kurtosis but no skewnessRows A to E have the same parameter 119887 the one we estimatedfrom 119878amp119875500 for a four-parameter Gram-Charlier while rowF has the same 119887 as the six-parameter Gram-Charlier weestimated Skewness and kurtosis are shown followed by the

break-even (or ldquofairrdquo) 120572 (ldquoB-E 120572rdquo in the table) and then theratchet premium option values for two specific choices for120572 Percentage differences with the values in row A (resp F)are also displayed (ldquoΔ119860rdquo in the table) It is notable thatin all cases the effect of moving away from Black-Scholesis more pronounced for the break-even participation ratesthan for the option values themselves The scenario withthe greatest impact (compared to Black-Scholes) is the onezero skewness and maximal excess kurtosis (row E) Insurersare unlikely to accept a cost of contract (1198810) greater thanthe received initial premium ($100) they may thus wish toadjust the participation rate 120572 down to the break-even rate(to wit see Lin and Tan [23]) The Black-Scholes break-evenparticipation rate in row A is 419 but it increases from443 up to 493 in rows B to E These are significantincreases ranging from55 to 175 Rows F andG are basedon two quite different risk-neutral probability distributionsbut the ratchet premium options are surprisingly similarTheparameters we estimated may not be those that a practitionerwould use but what this example says is that skewness andkurtosis have a very real importance in the pricing of equityindexed annuities and that the fair participation rate is atleast here more sensitive to nonzero skewness and excesskurtosis than are option values (See also [24] regarding theseparate effects of skewness and excess kurtosis on optionvalues)

Let us compare this with the approach in [11] Thatpaper is about a subclass of the Gram-Charlier distributionsdescribed in this paper the authors call their distributionsldquoseminonparametricrdquo and have density (6) where 119901(sdot) is thesquare of the polynomial 119902(sdot) If 119902(119909) is of order 119898 then it ispossible to express 119901(119909) = 119902(119909)2 as

119901 (119909) = 2119898sum119895=0

120575119895119867119890119895 (119909) (143)

Howevermost nonnegative polynomials cannot be expressedas squares of another polynomial so the order-119898 SNP familyis a strict subset of the order-2119898 Gram-Charlier familyThe order 2119898 SNP family has 2119898 + 2 free parameters (thecoefficients of an order-2119898 polynomial plus the location anddispersion parameters 119886 119887 subject to density integrating toone which in effect removes one parameter) by comparisontheGC(119886 119887 1198881 1198882119898) distributions have 2119898+2 parametersbut they are not ldquofreerdquo because the resulting density must benonnegative It is then not a priori clear which of SNP(2119898)or GC(119886 119887 1198881 1198882119898) would do best We looked at thisproblem with the SampP500 annual prices comparing SNP(2)and GC(119886 119887 0 0 1198883 1198884) which both have four parametersThe likelihood function is written the sameway in both casesthere are just different restrictions on the parameters Theresults forGC(119886 119887 0 0 1198883 1198884) are given in (138) and those forthe SNP(2) are

119886 = minus02445119887 = 019401198881 = 16140


Table 1 Comparison of ratchet option values computed with the Black-Scholes formula and Gram-Charlier distributions Rows A to F arebased on the four-parameter GC(119886 119887 0 0 1198883 1198884) while row G is based on a six-parameter GC(119886 119887 1198881 1198882 1198883 1198884)

1198861015840 119887 1198881 1198882 1198883 1198884 119904 119896 B-E 120572 Δ(119860) 1198810 (120572 = 06) Δ(119860) 1198810 (120572 = 0419) Δ(119860)A minus00042 01685 0 0 0 0 0 0 0419 0 10926 0 10000 0B minus00035 01685 0 0 minus01150 00360 minus06898 08634 0443 55 10760 minus15 9892 minus11C minus00034 01685 0 0 minus01749 01021 minus10493 24508 0478 139 10542 minus35 9750 minus25D minus00051 01685 0 0 01749 01021 10493 24508 0446 63 10739 minus17 9878 minus12E minus00043 01685 0 0 0 01667 0 40000 0493 175 10459 minus43 9696 minus301198861015840 119887 1198881 1198882 1198883 1198884 119904 119896 B-E 120572 Δ(119865) 1198810 (120572 = 06) Δ(119865) 1198810 (120572 = 0441) Δ(119865)F 00027 01595 0 0 0 0 0 0 0441 0 10769 0 10000 0G 00451 01595 minus3054 09542 minus012384 006120 minus05437 05091 0438 minus07 10790 02 10014 014

1198882 = 117331198883 = 042531198884 = 007320

(144)

This translates into the following moments for the SNP(2)fitted distribution

mean 00687standard deviation 01671

skewness minus 06289excess kurtosis 25502

(145)

The excess kurtosis is strikingly far from the datarsquos 08903The likelihood function evaluated at that estimate was lowerthan the likelihood evaluated at the parameters of theGC(119886 119887 0 0 1198883 1198884) distribution This is a puzzling result thatdoes not mean that GC(119886 119887 0 0 1198883 1198884) will always do betterthan SNP(2) it does however show that restricting the searchto squares of polynomials does have consequences Leon etal [11] derive formulas for European option prices takingthe martingale restriction into account based on expression(143) in this paper we do the same calculation for an arbitraryGram-Charlier distribution It is easy to see that the SNPclass is not closed under convolution that is the distributionof the sum of two independent SNP variables is in generalnot an SNP distribution the general formulas we deriveare essential in order to add independent Gram-Charlierdistributed variables (including those in the SNP subset) andtherefore the general formulas are needed to define Gram-Charlier processes Leon et al [11] are correct in pointingout that there is an advantage in using 119901(119909) = 119902(119909)2 as faras the nonnegativity constraint is concerned but we believethat it is no more difficult to estimate the parameters ofa general Gram-Charlier distribution including the non-negativity constraint in the maximization procedure (ratherthan limiting the parameter space) A full comparison ofthe numerical and statistical advantagesdisadvantages of ourapproach versus the one in [11] is an interesting avenue forfurther research

k

s

01234

105

110

Option

0500

minus05

Figure 5 Lookback option prices as a function of skewness andexcess kurtosis

62 Lookback Options Consider a European lookback calloption with fixed strike with payoff (119878 minus 119870)+ where 119878 isthe maximum of the stock price between time 0 and time119879 We let the time points used to determine the maximumbe equally spaced There is no closed form formula for thedistribution of the maximum price nor for the price ofsuch a discretely monitored lookback option so simulationis used We generated normal random variables and usedthe technique described in Section 25 The assumptions arematurity119879 = 1 monthlymonitoring annual volatility 01685(where this number comes from is explained in the nextsubsection) The interest rate is set to 0 initial stock price is$100 and strike price is also $100

Figure 5 shows the option prices for values of skew-ness and excess kurtosis that can be achieved with theGC(119886 119887 0 0 1198883 1198884) family for monthly rates of returnRoughly speaking prices decrease when excess kurtosisincreases and also as kurtosis decreases though the depen-dence on 1198883 and 1198884 becomes less linear as 119896 increases When1198883 = 1198884 = 0 (Gaussian case) the option price is 11277The price is a little higher ($11286) when 119904 = 075 and119896 = 10 The lowest price ($10163) occurs when 119904 = 0and 119896 = 4 Skewness and kurtosis thus have a significanteffect on lookback option prices the effect will of course varydepending on the parameters (Observe that all those pricesare quite far from the Black-Scholes continuous-monitoringlookback price $1417)


7 Discussion and Conclusion

This paper tries to provide a framework for applying Gram-Charlier series to option pricing We have shown that optionprices may be significantly affected by varying skewnessor kurtosis away from their Gaussian values The practi-cal consequence is that using the Black-Scholes formula(ie zero skewness and excess kurtosis) may distort optionprices Gram-Charlier distributions capture skewness andkurtosis and retain a lot of the tractability of the normaldistribution Estimating parameters remains a challengefor Gram-Charlier distributions involving more than fourparameters

Convergence of Gram-Charlier and other expansionsin option pricing and other applied problems is an areafor further research The heuristic derivation of the Gram-CharlierEdgeworth series of a density 119892(sdot) in terms of thenormal density and its derivatives (see eg [25] Section 32)may give the impression that the inverted series should con-verge or at least have some asymptotic property this is not thecase in general Gram-Charlier (or Edgeworth) expansionsdo not always converge to the true probability distributionas Section 23 indicates see [26] Section 17 or [25] for moredetails In the context of the Central Limit Theorem theEdgeworth series for the density of the normalized sumor random variables have asymptotic properties when thenumber of summands 119899 tends to infinity Details are given in[25] Multivariate versions of the Gram-Charlier series existfor instance [10 27 28] apply such generalizations

The Gram-Charlier type B series are based on the Pois-son distribution rather than the normal There are severalother expansions that can be used For instance series ofLaguerre polynomials may be used for densities on thepositive half-line (a convergent Laguerre series has somenumerical success in pricing Asian options see [29]) TheJacobi polynomials [30] are orthogonal over the interval(minus1 1) but any other finite interval may be chosen bymakingthe obvious change of variable and this leads to seriesexpansions for densities with compact support based on thebeta distribution An application of log-transformed shiftedJacobi polynomials to arbitrary approximation of probabilitydistributions on the positive half-line may be found in [31]

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The authors thank the Committee for Knowledge Extensionand Research (CKER) of the Society of Actuaries for partiallyfunding this project

References

[1] P A Abken D B Madan and S Ramamurtie ldquoEstimation ofrisk-neutral and statistical densities by Hermite polynomialapproximationrdquo Working Paper 96-5 Federal Reserve Bank ofAtlanta 1996

[2] D Backus S Foresi K Li and L Wu ldquoAccounting for biases inBlack-Scholesrdquo Working Paper Stern School of Business NewYork University New York NY USA 1997

[3] J-P D Chateau ldquoMarking-to-model credit and operationalrisks of loan commitments a Basel-2 advanced internal ratings-based approachrdquo International Review of Financial Analysis vol18 no 5 pp 260ndash270 2009

[4] J D Chateau ldquoValuing european put options under skewnessand increasing (excess) kurtosisrdquo Journal of MathematicalFinance vol 4 no 3 pp 160ndash177 2014

[5] C J Corrado ldquoThe hidden martingale restriction in Gram-Charlier option pricesrdquo Journal of Futures Markets vol 27 no6 pp 517ndash534 2007

[6] E Jondeau and M Rockinger ldquoGram-Charlier densitiesrdquo Jour-nal of Economic Dynamics and Control vol 25 no 10 pp 1457ndash1483 2001

[7] E Jurczenko B Maillet and B Negrea ldquoSkewness and kurtosisimplied by option prices a second commentrdquo Discussion Paperof the LSE-FMG 419 2002

[8] E Jurczenko BMaillet and B Negrea ldquoA note on skewness andkurtosis adjusted option pricing models under the Martingalerestrictionrdquo Quantitative Finance vol 4 no 5 pp 479ndash4882004

[9] J L Knight and S E Satchell Return Distributions in FinanceButterworth Heineman 2001

[10] E Schlogl ldquoOption pricing where the underlying assets follow aGramCharlier density of arbitrary orderrdquo Journal of EconomicDynamics amp Control vol 37 no 3 pp 611ndash632 2013

[11] A Leon J Mencıa and E Sentana ldquoParametric properties ofsemi-nonparametric distributions with applications to optionvaluationrdquo Journal of Business amp Economic Statistics vol 27 no2 pp 176ndash192 2009

[12] M Hardy Investment Guarantees Modeling and Risk Manage-ment for Equity-Linked Life Insurance JohnWiley amp Sons NewYork NY USA 2003

[13] P Gaillardetz and X S Lin ldquoValuation of equity-linkedinsurance and annuity products with binomial modelsrdquo NorthAmerican Actuarial Journal vol 10 no 4 pp 117ndash144 2006

[14] P Boyle andW Tian ldquoThe design of equity-indexed annuitiesrdquoInsurance Mathematics amp Economics vol 43 no 3 pp 303ndash3152008

[15] S G Kou ldquoDiscrete barrier and lookback optionsrdquo in Hand-books in Operations Research and Management Science vol 15chapter 8 pp 343ndash373 2007

[16] D E Barton and K E Dennis ldquoThe conditions under whichGram-Charlier and Edgeworth curves are positive definite andunimodalrdquo Biometrika vol 39 no 3-4 pp 425ndash427 1952

[17] H Cramer On Some Classes of Series Used in MathematicalStatistics Skandinaviske Matematiker Congres CopenhagenDenmark 1925

[18] W Feller An Introduction to Probability Theory and Its Applica-tions vol 2 John Wiley amp Sons New York NY USA 1971

[19] J Stoyanov ldquoKrein condition in probabilistic moment prob-lemsrdquo Bernoulli vol 6 no 5 pp 939ndash949 2000

[20] F D Rouah and G Vainberg Option Pricing Models andVolatility Using Excel-VBA John Wiley amp Sons New York NYUSA 2007

[21] L S Rompolis and E Tzavalis ldquoRetrieving risk neutral densitiesbased on risk neutral moments through a Gram-Charlier seriesexpansionrdquoMathematical and Computer Modelling vol 46 no1-2 pp 225ndash234 2007


[22] E B Del Brio and J Perote ldquoGram-Charlier densities max-imum likelihood versus the method of momentsrdquo InsuranceMathematics and Economics vol 51 no 3 pp 531ndash537 2012

[23] X S Lin and K S Tan ldquoValuation of equity-indexed annuitiesunder stochastic interest ratesrdquo North American ActuarialJournal vol 7 no 4 pp 72ndash91 2003

[24] S R Das and R K Sundaram ldquoOf smiles and smirks a termstructure perspectiverdquo Journal of Financial and QuantitativeAnalysis vol 34 no 2 pp 211ndash239 1999

[25] J E Kolossa Series Approximation Methods in StatisticsSpringer New York NY USA 3rd edition 2006

[26] H Cramer Mathematical Methods of Statistics Princeton Uni-versity Princeton NJ USA 1946

[27] E B Del Brio T-M Nıguez and J Perote ldquoGram-Charlierdensities a multivariate approachrdquoQuantitative Finance vol 9no 7 pp 855ndash868 2009

[28] E B Del Brio T-M Nıguez and J Perote ldquoMultivariatesemi-nonparametric distributions with dynamic conditionalcorrelationsrdquo International Journal of Forecasting vol 27 no 2pp 347ndash364 2011

[29] D Dufresne ldquoLaguerre series for Asian and other optionsrdquoMathematical Finance vol 10 no 4 pp 407ndash428 2000

[30] N N Lebedev Special Functions andTheir Applications DoverNew York NY USA 1972

[31] D Dufresne ldquoFitting combinations of exponentials to proba-bility distributionsrdquo Applied Stochastic Models in Business andIndustry vol 23 no 1 pp 23ndash48 2007

Submit your manuscripts athttpswwwhindawicom

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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


log returns among others [1ndash10] The majority of previousauthors assumed a density of the form

120601 (119909) (1 + 11990461198671198903 (119909) + 119896241198671198904 (119909)) 120601 (119909) = 120601 (0 1 119909) = 1radic2120587119890minus119909

22(3)

for the normalized log return (In our notation this isa GC(0 1 0 0 1199046 11989624) distribution see Section 2) Thenotation emphasizes that in this case the coefficient of1198671198903(119909)turns out to be Pearsonrsquos skewness coefficient 119904 divided by 6and the coefficient of 1198671198904(119909) the excess kurtosis coefficient119896 divided by 24 The distribution of log returns is thena four-parameter Gram-Charlier distribution (since thereare two other parameters the mean and variance besides119904 and 119896) This distribution allows nonzero skewness andexcess kurtosis unlike the normal distribution found inBlack-Scholes In (3) the parameters (119904 119896) are restricted toa specific region (see Figure 3) because outside that regionthe function in (3) becomes negative for some values of 119909(see Section 21)

Jurczenko et al [8] specify the martingale restriction

1198780 = 119890minus119903119879EQ (119878119879) (4)

that the four-parameter Gram-Charlier density must sat-isfy in pricing options (previous authors had not taken itinto account) The martingale condition for general Gram-Charlier distributions is described in Section 5 Our Gram-Charlier distribution with parameters 119886 119887 1198881 119888119873 denotedby GC(119886 119887 1198881 119888119873) has density

120601 (119886 119887 119909)sdot [1 + 11988811198671198901 (119909 minus 119886119887 ) + sdot sdot sdot + 119888119873119867119890119873 (119909 minus 119886119887 )] (5)

As stated above we do not set 1198881 = 1198882 = 0 and 119873 canbe any even positive integer The question whether (5) isnonnegative for all 119909 is an important one Several authorshave disregarded this issue and in fact some have come upwith parameters that do not yield a true probability densityfunction One might argue that this is the price to pay fortruncating an infinite Gram-Charlier series However if thesame log return distribution is used to price many optionsthen a true probability density function is the only safe choicebecause otherwise there might be inconsistencies amongoption prices For instance if the function used as densityis negative over the interval (120572 120573) then a digital option thatpays off only when the log return is in that interval will havea negative price In this paper we consider Gram-Charlierdistributions not expansions and insist that the densitiesintegrate to one and be nonnegative Our goal is to define afamily of proper probability distributions nevertheless ourformulas do apply to truncated Gram-Charlier expansions aswell

The paper by Leon et al [11] presents an alternative tothe general Gram-Charlier distributions we study in thispaper Those authors consider the subclass of Gram-Charlierdistributions consisting of densities

119891119883 (119909) = 120601 (119886 119887 119909) 119901 (119909) (6)

where the polynomial 119901(119909) is the square of another polyno-mial 119901(119909) = 119902(119909)2 This has the obvious advantage that thenonnegativity restriction on 119891119883(sdot) is automatically satisfiedWe discuss that subclass of ldquosquaredrdquo Gram-Charlier distri-butions in the Conclusion

Almost all previous authors have used Gram-Charlierdistributed log returns over a single time period This hasan obvious downside in that it becomes tricky if not im-possible to preserve consistency between the prices ofoptions with different maturities Section 3 shows that aLevy process with Gram-Charlier increments does not existhowever it also shows that the sum of independent Gram-Charlier distributed variables also has a Gram-Charlier dis-tributionThis opens the way for multiperiod Gram-Charlieroption pricing using a discrete-time random walk model forwhich the log return over any period has a Gram-Charlierdistribution The Gram-Charlier distribution of the multi-period return has a larger number of parameters though themodel is still simpler than almost any (if not all) alternativestochastic volatility models There is no problem computa-tionally since we give explicit formulas for options underGram-Charlier distributions with an arbitrary number ofparameters

The layout of the paper is as follows In Section 2 weextend the study of Gram-Charlier distributions to all possi-ble polynomials 119901(sdot) and derive their properties (momentscumulants moment determinacy properties of the set ofvalid parameters tail and so on) Some of the formulasand properties in Theorem 2 appear to be new In Section 3we show that there is no Levy process with Gram-Charlierdistributed increments apart from Brownian motion andwe define a discrete-time process with independent Gram-Charlier increments that is suitable for option pricing InSection 4 we show that the log Gram-Charlier distribution isnot determined by itsmoments just like the lognormal NextSection 5 gives formulas for European call and put priceswhen the log-price returns of the underlying have a generalGram-Charlier distribution the previous literature mostlyconsidered the GC(119886 119887 0 0 1198883 1198884) family and the squaredGram-Charlier distributions (an exception is [10] where itis assumed that 1198881 = 1198882 = 0) In particular we derive achange ofmeasure formula that extends theCameron-Martinformula for the normal distribution the latter is used inpricing European options in the Black-Scholes model Wealso derive formulas for the sensitivities (Greeks) of thoseoption prices with respect to all parameters A technique forsimulating Gram-Charlier distributions is described Parts(c) (d) (i) (j) (k) and (l) of Theorem 2 and part (b) ofTheorem 3 and Theorems 6 7 9 and 11 appear to be newTheorem 3(a) is for the first time formulated for generalGram-Charlier distributions Theorems 3(a) 4 and 8 aregiven more or less explicitly for the subclass of squared






120601 (0 1 119909) = 120601 (119909) = 119890minus11990922radic2120587 (7)



120601 (119910) 119889119910 (8)



(9)



(10)



119891 (119909) = 120601 (119909) 119873sum119896=0

119888119896119867119890119896 (119909) 119909 isin R (11)



120601 (119909) 119873sum119896=0

119888119896119867119890119896 (119909) (12)



1119887120601 (119910 minus 119886119887 )119901 (119910) (13)



119867119890119896 (119910 minus 119886119887 ) 119896 = 0 1 119873 (14)


intinfinminusinfin


119889119896119889119909119896 120601 (119909) 119889119909 = 0119896 = 1 2

(15)



1199081 (119905 119909) = infinsum119899=0

119905119899119899119867119890119899 (119909) = 119890119905119909minus(11990522) (16)

Another one is

1199082 (119905 119906 119909 119910)= infinsum

119896=0

infinsum119899=0

119905119896119896 119906119899

119899 intinfin


= radic2120587119890119906(119905+119910)(17)


infin


119899

119906119899 119890119905119906100381610038161003816100381610038161003816100381610038161003816119905=119906=0

= 0 if 119896 = 119899119899 if 119896 = 119899

(18)





intinfinminusinfin

120601 (119909)119867119890119896 (119909) 119889119909 = 0 119896 = 1 2 (20)


intinfinminusinfin

119890119905119909120601 (119909)119867119890119896 (119909) 119889119909 = 11990511989611989011990522 119896 = 0 1 (21)


intinfinminusinfin

119909119899120601 (119909)119867119890119896 (119909) 119889119909= (minus1)119896 intinfin

minusinfin119909119899 ( 119889119896119889119909119896120601 (119909)) 119889119909

(22)


intinfinminusinfin

119909119899120601 (119909)119867119890119896 (119909) 119889119909=




119873and119899sum119896=0



intinfinminusinfin

119890119905119909119891 (119909) 119889119909 = 11989011990522 119873sum119896=0

119888119896119905119896 (25)



(a)

E (119884 minus 119886)119899 = 119887119899 119873and119899sum119896=0


(b)

E119890119905119884 = 119890119886119905+(119887211990522) 119873sum119896=0

119887119896119888119896119905119896 119905 isin R (27)






+ 2411988611988731198883 + 1211988741198882 + 2411988741198884 + 311988741198985 = 1198865 + 511988641198871198881 + 20119886311988721198882 + 1011988631198872 + 30119886211988731198881

+ 60119886211988731198883 + 6011988611988741198882 + 12011988611988741198884 + 151198861198874+ 1511988751198881 + 6011988751198883 + 12011988751198885

1198986 = 1198866 + 611988651198871198881 + 30119886411988721198882 + 1511988641198872 + 60119886311988731198881+ 120119886311988731198883 + 180119886211988741198882 + 360119886211988741198884+ 4511988621198874 + 9011988611988751198881 + 36011988611988751198883 + 72011988611988751198885+ 9011988761198882 + 36011988761198884 + 72011988761198886 + 151198876

(28)




(29)



2 (11988831 minus 311988811198882 + 31198883)(1 minus 11988821 + 21198882)32


(30)


E119883119895 = E119884119895 119895 = 1 119870 (31)




119886 = E119883 1198872 = Var119883 lArrrArr 1198881198831 = 1198881198832 = 0 (32)




119892 (119910) = 119890minus1199102radic21205871199101198992sum119895=0

120572119895119910119895

119908ℎ119890119903119890 1198992sum119895=0

120572119895119910119895 = 1198992sum119895=0

11988821198951198671198902119895 (radic119910) (33)


119873sum119896=1

119867119890119896 (119909) (34)



1 + 120598 119873sum119896=0

119867119890119896 (119909) gt 0 119909 isin R 0 le 120598 le 1205980 (35)



119905 997891997888rarr log(119890119886119905+(119887211990522) 119873sum119896=0

119887119896119888119896119905119896) (36)


only Write

120595 (119905) = 11989011990522119902119883 (119905) = 119873sum

119896=0

119888119883119896 119905119896119902119884 (119905) = 119873sum

119896=0

119888119884119896 119905119896(37)


1205951015840 (0) 119902119883 (0) + 120595 (0) 1199021015840119883 (0) = 1205951015840 (0) 119902119884 (0) + 120595 (0) 1199021015840119884 (0) 119870sum119895=0

(119870119895)120595(119895) (0) 119902(119870minus119895)119883 (0) = 119870sum119895=0

(119870119895)120595(119895) (0) 119902(119870minus119895)119884 (0) (38)



119886 = E119883 lArrrArr 1198881198831 = 0 (39)


E1198832 = 12059510158401015840 (0) 119902119883 (0) + 21205951015840 (0) 1199021015840119883 (0) + 120595 (0) 11990210158401015840119883 (0) E1198842 = 12059510158401015840 (0) 119902119884 (0) + 21205951015840 (0) 1199021015840119884 (0) + 120595 (0) 11990210158401015840119884 (0) (40)


E(119909 minus 119886119887 )2 = 1 lArrrArr 1198881198832 = 0 (41)


119890119886119902119904+(12)119887211990221199042 119873sum119896=0

119888119896119887119896119902119896119904119896

= 119890119886119902119904+(12)119887211990221199042 119873sum119896=0

119888119896 ( 11990210038161003816100381610038161199021003816100381610038161003816)119896 (119887 10038161003816100381610038161199021003816100381610038161003816)119896 119904119896

(42)



119873sum119896=0





119896=1


119896=1

119888119896119867119890119896minus1 (119909) (44)




(45)







119884 = 119883 minus E119883radicVar119883 (46)




119896=0

119888119896119867119890119896 (47)

where

119888119896 = 1119896 intinfin

minusinfin119867119890119896 (119909) 119891 (119909) 119889119909 = 1119896E119867119890119896 (119883) (48)


12 (119891 (119909minus) + 119891 (119909+)) (49)


intinfinminusinfin

11989011990924119891 (119909) 119889119909 lt infin (50)



ℎ119902119896 = E119867119890119896 (119883) 119883 sim N (0 1199022) (51)


ℎ119902119896 = 0 if 119896 is odd

11989621198962 (1198962) (1199022 minus 1)1198962 if 119896 is even (52)


120601 (119909) infinsum119896=0

ℎ119902119896119896 119867119890119896 (119909)= 120601 (119909) infinsum

119899=0

12119899119899 (1199022 minus 1)1198991198671198902119899 (119909) (53)


infinsum119899=0

(minus1)119899 (2119899)22119899 (119899)2 (1199022 minus 1)119899 (54)


(2119899)22119899 (119899)2 sim 1radic120587119899 (55)



119892119901 (119909) = 119901120601 (119909) + (1 minus 119901) 1119901120601(119909119901) (56)



119884119901 = 119883119901


= 119883119901120590119901 1205902119901 = Var119883119901 = 119901 + 1199012 minus 1199013

(57)


119888119901119896 = 1119896E119867119890119896 (119884119901)= 1119896 (119901ℎ1120590119901 119896 + (1 minus 119901) ℎ119901120590119901 119896)

(58)





1Var119883119901

= 103 + 032 minus 033 = 275482 gt 2 (60)


119902 = 112059003 997904rArr 1198882 = 0119902 = 14 997904rArr 1198882 = minus014426

(61)

minus2 minus1 1 2

minus4

minus2

2

4

6


minus2 minus1 1 2

02

04

06

08

10



E119890119902119883119891 (119883) = 119890120583119902+(12)12059021199022E119891 (119883 + 1205902119902) (62)


1198751015840 = 119890119902119883EP119890119902119883 sdot 119875 (63)


119873sum119896=0

1198871198961198881015840119896119904119896 = sum119873119896=0 119887119896119888119896 (119902 + 119904)119896sum119873

119895=0 119887119895119888119895119902119895 119904 isin R (64)


or more precisely

1198881015840119896 = 1sum119873119895=0 119887119895119888119895119902119895

119873sumℓ=119896

(ℓ119896) 119887ℓminus119896119888ℓ119902ℓminus119896 (65)


E1198751015840119890119904119883 = 1


119896=0

119887119896119904119896 1sum119873119895=0 119887119895119888119895119902119895

sdot 119873sumℓ=119896

(ℓ119896) 119887ℓminus119896119888ℓ119902ℓminus119896(66)


11988810158401 = 311988721198883 + 4119887311988841 + 11988731198883 + 11988741198884 11988810158402 = 31198871198883 + 6119887211988841 + 11988731198883 + 11988741198884 11988810158403 = 1198883 + 411988711988841 + 11988731198883 + 11988741198884 11988810158404 = 11988841 + 11988731198883 + 11988741198884

(67)



(119883(119895)1 119883(119895)

119904 ) 119895 = 1 119899 (68)


119891119883 (119909) = 120601 (119886 119887 119909) 119901 (119909) (69)


E119892 (1198831 119883119904) = intR119904119892 (1199091 119909119904)

sdot 119904prod119896=1


119896=1

119901 (119883119896)] (70)


1 119883(119895)119904 ) 119895 = 1 119899 and then

using the estimator

119899 = 1119899119899sum119895=1

[119892 (119883(119895)1 119883(119895)

119904 ) 119904prod119896=1

119901 (119883(119895)

119896)] (71)




E119890119911(119883+119884)= 119890(1198861+1198862)119905+(11988721+11988722 )(11990522)(119873(1)sum

119896=0

119888(1)119896 119905119896)(119873(2)sum119896=0

119888(2)119896 119905119896) (72)


119883 + 119884 sim GC(1198861 + 1198862 radic11988721 + 11988722 1198881 119888119873(1)+119873(2)) 119888119896 = 119896sum

119895=0

119888(1)119895 119888(2)119896minus119895(73)


119885119899 = 1198831 + sdot sdot sdot + 119883119899 (74)


( 119873sum119896=0

119888119896119905119896)119899 = 119899119873sum

119896=0


(75)


and thus






119872(119911) = E119890119911119883 = 119890119886119911+1198872(11991122)120587 (119911) (77)


119872119899 (119911) = E1198901199111198831198991 (78)


119872119899 (119911)119899 = 119872(119911) 119911 isin C (79)





119891119871 (119911) = 1119911119891119884 (log 119911) 119911 gt 0 (80)





All the functions



119892120572 (119911) = 119891119871 (119911) + 120572120601 (119911) sin (2120587 log 119911) (83)


119901 (119910) = 119873sum119896=0

119888119896119867119890119896 (119910) (84)


119904 (119910) = sin (2120587119910) 1119910notin119869 (85)


119904 (119910 + 2119896120587) = 119904 (119910) 119904 (minus119910) = minus119904 (119910) (86)


intinfinminusinfin

119890119896119910120601 (119910) 119904 (119910) 119889119910 = 0 (87)



minusintinfin0

log119891 (1199112)1 + 1199112 119889119911 lt infin (88)







119890119886+(11988722) 119873sum119896=0

119887119896119888119896 = 119890(119903minus120575)119879 (89)










(92)


119896=0

119887119896119888119896 = 119890(119903minus120575)1198791198780 (93)



(94)





119896=1

119888119896119867119890119896minus1 (minus1198892) (96)

where

1198892 = 1119887 (log(1198780119870) + 119886) (97)


Q1015840 = 119878119879EQ119878119879 sdotQ = 119890119883119879

EQ119890119883119879 sdotQ (98)


Then

EQ (1198781198791119878119879gt119870) = (EQ119878119879)EQ1015840 (1119878119879gt119870) (99)

Since119883119879


119896=1

1198881015840119896119867119890119896minus1 (minus1198891) (100)

where

1198881015840119896 = 1sum119873119895=0 119887119895119888119895

119873sumℓ=119896

(ℓ119896) 119887ℓminus119896119888ℓ 119896 = 1 1198731198891 = 1119887 [log(1198780119870) + 119886 + 1198872] = 1198892 + 119887

(101)

Hence


119896=1


119896=1

119888119896119867119890119896minus1 (minus1198892) (102)

Using




where

119888lowast119896 = 1198881015840119896 119873sum119896=0

119887119896119888119896 = 119873sumℓ=119896

(ℓ119896) 119887ℓminus119896119888ℓ 119896 = 1 119873 (105)



1198750 = 1198620 minus 1198780 + 119870119890minus119903119879 (106)


4sum119896=1



(107)

This ends the proof



Δ = 12059711986201205971198780= 119890minus120575119879(Φ(1198891) + 120601 (1198891) 119873sum

119896=1

1198881015840119896119867119890119896minus1 (minus1198891))

120574 = 1205972119862012059711987820 = 119890119886+(11988722)minus1199031198791198871198780 120601 (1198891) 119873sum119896=0

119888119896119867119890119896 (minus1198892) 120588 = 1205971198620120597119903= 119870119879119890minus119903119879(Φ(1198892) + 120601 (1198892) 119873sum

119896=1





1198881015840119896119867119890119896minus1 (minus1198891)) 119895 ge 1

(108)





120597120597119904E (119904119880 minus 119870)+ = E (1198801119904119880gt119870) (109)


2

1199043 119891119880 (119870119904 ) (110)

From 119878119879 = 1198780119890119883119879 and (100)


119896=1

1198881015840119896119867119890119896minus1 (minus1198891)) (111)

From the lemma

120574 = 1205972120597211987801198620 = 119890minus119903119879120597212059721198780EQ (1198780119880 minus 119870)+

= 1198702119890minus11990311987911987830 119891119880 (1198701198780) 119880 = 119890119883(112)

where 119883119879


119891119880 (119906) = 1119887119906119891119866 (1119887 (log 119906 minus 119886))= 1119887119906120601 (119910)

119873sum119896=0

119888119896119867119890119896 (119910)1003816100381610038161003816100381610038161003816100381610038161003816119910=(1119887)(log 119906minus119886)

(113)


120574 = 119870119890minus11990311987911988711987820 120601 (1198892) 119873sum119896=0

119888119896119867119890119896 (minus1198892)= 119890119886+(11988722)minus1199031198791198871198780 120601 (1198891) 119873sum

119896=0

119888119896119867119890119896 (minus1198892) (114)

Next




Now from (89)



119896=1

119888119896119867119890119896minus1 (minus1198892)] (118)


Define

119898(119887) = E119890119887119866minus(11988722) = sum119896=0

1198871198961198881198961198981015840 (119887) = sum

119896=1

119896119887119896minus1119888119896 = 119888lowast1 (120)


119890119886 = 119890(119903minus120575)119879minus(11988722)119898(119887) (121)


1198780119890minus120575119879EQ [( 120597120597119887 119890119887119866minus(11988722)

119898(119887) ) 1119866gtminus1198892]


(122)


EQ (1198901198871198661119866gtminus119910) = 11989011988722 [119898 (119887)Φ (119910 + 119887)

+ 120601 (119910 + 119887) 119873sum119896=1


To calculate


we need (see (105))

120597119888lowast119896120597119887 = 120597120597119887119873sumℓ=119896


120597119888lowast119873120597119887 = 0(125)




1198881015840119896119867119890119896 (minus119910 minus 119887)]

(126)



120597119886120597119888119895 119890119886+(11988722)

119873sum119896=0

119887119896119888119896 + 119887119895119890119886+(11988722) = 0or 120597119886120597119888119895 = minus

119887119895sum119873119896=0 119887119896119888119896 = minus119887

119895119890119886+(11988722)+(120575minus119903)119879(127)

Next


infin

minus1198892

(1198780119890119886+119887119910 minus 119870)120601 (119910) 119873sum119896=0

119888119896119867119890119896 (119910) 119889119910= 119890minus119903119879intinfin

minus1198892

1198780 120597119886120597119888119895 119890119886+119887119910120601 (119910)119873sum119896=0

119888119896119867119890119896 (119910) 119889119910+ 119890minus119903119879intinfin

minus1198892

(1198780119890119886+119887119910 minus 119870)120601 (119910)119867119890119895 (119910) 119889119910

(128)



119896=0


119910119892(119895) (119909) 120601 (119909) 119889119909

(129)




minus1198892


119896=1


(130)




119861119899 = 119875 119899prod119905=1

[1 +max(120572( 119878119905119878119905minus1 minus 1) 0)] (131)





119877119905 Qsim GC (119886 119887 ) 119905 = 1 119899 (133)


119905=1





1198810 = 119875 (119890minus119903 + 1205721198620)119899 (135)


1205971198810120597120585 = 120572119875 (119890minus119903 + 1205721198620)119899minus1 1205971198620120597120585 (136)


1205972119862012059711987820 = 1205722119875 (119890minus119903 + 1205721198620)119899minus2 (12059711986201205971198780 )2

+ 120572119875 (119890minus119903 + 1205721198620)119899minus1 1205972119862012059711987820 (137)



119886 = 00687119887 = 016851198883 = minus011501198884 = 003598

(138)


119904 = 61198883 = minus06898119896 = 241198884 = 08634 (139)



0

1

2

3

4

k


s


minus1

0

1

0

2

4 104

106

108

110

Option

k s







(1119887)(log(1198701198780)minus1198861015840)(11987801198901198861015840+119887119909 minus 119870)120601 (119909)

sdot ( 119873sum119896=0

119888119896119867119890119896 (119909))119889119909(140)


(1119887)(log(1198701198780)minus1198861015840)(11987801198901198861015840+119887119909 minus 119870)120601 (119909)

sdot 119867119890119895 (119909) 119889119909(141)



119886 = 01174119887 = 015951198881 = minus030536756952010661198882 = 0095420793734891531198883 = minus0123839711263352431198884 = 006120331530131559

(142)





119901 (119909) = 2119898sum119895=0

120575119895119867119890119895 (119909) (143)


119886 = minus02445119887 = 019401198881 = 16140




1198882 = 117331198883 = 042531198884 = 007320

(144)




(145)


k

s

01234

105

110

Option

0500

minus05









Competing Interests


Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














120601 (0 1 119909) = 120601 (119909) = 119890minus11990922radic2120587 (7)



120601 (119910) 119889119910 (8)



(9)



(10)



119891 (119909) = 120601 (119909) 119873sum119896=0

119888119896119867119890119896 (119909) 119909 isin R (11)



120601 (119909) 119873sum119896=0

119888119896119867119890119896 (119909) (12)



1119887120601 (119910 minus 119886119887 )119901 (119910) (13)



119867119890119896 (119910 minus 119886119887 ) 119896 = 0 1 119873 (14)


intinfinminusinfin


119889119896119889119909119896 120601 (119909) 119889119909 = 0119896 = 1 2

(15)



1199081 (119905 119909) = infinsum119899=0

119905119899119899119867119890119899 (119909) = 119890119905119909minus(11990522) (16)

Another one is

1199082 (119905 119906 119909 119910)= infinsum

119896=0

infinsum119899=0

119905119896119896 119906119899

119899 intinfin


= radic2120587119890119906(119905+119910)(17)


infin


119899

119906119899 119890119905119906100381610038161003816100381610038161003816100381610038161003816119905=119906=0

= 0 if 119896 = 119899119899 if 119896 = 119899

(18)





intinfinminusinfin

120601 (119909)119867119890119896 (119909) 119889119909 = 0 119896 = 1 2 (20)


intinfinminusinfin

119890119905119909120601 (119909)119867119890119896 (119909) 119889119909 = 11990511989611989011990522 119896 = 0 1 (21)


intinfinminusinfin

119909119899120601 (119909)119867119890119896 (119909) 119889119909= (minus1)119896 intinfin

minusinfin119909119899 ( 119889119896119889119909119896120601 (119909)) 119889119909

(22)


intinfinminusinfin

119909119899120601 (119909)119867119890119896 (119909) 119889119909=




119873and119899sum119896=0



intinfinminusinfin

119890119905119909119891 (119909) 119889119909 = 11989011990522 119873sum119896=0

119888119896119905119896 (25)



(a)

E (119884 minus 119886)119899 = 119887119899 119873and119899sum119896=0


(b)

E119890119905119884 = 119890119886119905+(119887211990522) 119873sum119896=0

119887119896119888119896119905119896 119905 isin R (27)






+ 2411988611988731198883 + 1211988741198882 + 2411988741198884 + 311988741198985 = 1198865 + 511988641198871198881 + 20119886311988721198882 + 1011988631198872 + 30119886211988731198881

+ 60119886211988731198883 + 6011988611988741198882 + 12011988611988741198884 + 151198861198874+ 1511988751198881 + 6011988751198883 + 12011988751198885

1198986 = 1198866 + 611988651198871198881 + 30119886411988721198882 + 1511988641198872 + 60119886311988731198881+ 120119886311988731198883 + 180119886211988741198882 + 360119886211988741198884+ 4511988621198874 + 9011988611988751198881 + 36011988611988751198883 + 72011988611988751198885+ 9011988761198882 + 36011988761198884 + 72011988761198886 + 151198876

(28)




(29)



2 (11988831 minus 311988811198882 + 31198883)(1 minus 11988821 + 21198882)32


(30)


E119883119895 = E119884119895 119895 = 1 119870 (31)




119886 = E119883 1198872 = Var119883 lArrrArr 1198881198831 = 1198881198832 = 0 (32)




119892 (119910) = 119890minus1199102radic21205871199101198992sum119895=0

120572119895119910119895

119908ℎ119890119903119890 1198992sum119895=0

120572119895119910119895 = 1198992sum119895=0

11988821198951198671198902119895 (radic119910) (33)


119873sum119896=1

119867119890119896 (119909) (34)



1 + 120598 119873sum119896=0

119867119890119896 (119909) gt 0 119909 isin R 0 le 120598 le 1205980 (35)



119905 997891997888rarr log(119890119886119905+(119887211990522) 119873sum119896=0

119887119896119888119896119905119896) (36)


only Write

120595 (119905) = 11989011990522119902119883 (119905) = 119873sum

119896=0

119888119883119896 119905119896119902119884 (119905) = 119873sum

119896=0

119888119884119896 119905119896(37)


1205951015840 (0) 119902119883 (0) + 120595 (0) 1199021015840119883 (0) = 1205951015840 (0) 119902119884 (0) + 120595 (0) 1199021015840119884 (0) 119870sum119895=0

(119870119895)120595(119895) (0) 119902(119870minus119895)119883 (0) = 119870sum119895=0

(119870119895)120595(119895) (0) 119902(119870minus119895)119884 (0) (38)



119886 = E119883 lArrrArr 1198881198831 = 0 (39)


E1198832 = 12059510158401015840 (0) 119902119883 (0) + 21205951015840 (0) 1199021015840119883 (0) + 120595 (0) 11990210158401015840119883 (0) E1198842 = 12059510158401015840 (0) 119902119884 (0) + 21205951015840 (0) 1199021015840119884 (0) + 120595 (0) 11990210158401015840119884 (0) (40)


E(119909 minus 119886119887 )2 = 1 lArrrArr 1198881198832 = 0 (41)


119890119886119902119904+(12)119887211990221199042 119873sum119896=0

119888119896119887119896119902119896119904119896

= 119890119886119902119904+(12)119887211990221199042 119873sum119896=0

119888119896 ( 11990210038161003816100381610038161199021003816100381610038161003816)119896 (119887 10038161003816100381610038161199021003816100381610038161003816)119896 119904119896

(42)



119873sum119896=0





119896=1


119896=1

119888119896119867119890119896minus1 (119909) (44)




(45)







119884 = 119883 minus E119883radicVar119883 (46)




119896=0

119888119896119867119890119896 (47)

where

119888119896 = 1119896 intinfin

minusinfin119867119890119896 (119909) 119891 (119909) 119889119909 = 1119896E119867119890119896 (119883) (48)


12 (119891 (119909minus) + 119891 (119909+)) (49)


intinfinminusinfin

11989011990924119891 (119909) 119889119909 lt infin (50)



ℎ119902119896 = E119867119890119896 (119883) 119883 sim N (0 1199022) (51)


ℎ119902119896 = 0 if 119896 is odd

11989621198962 (1198962) (1199022 minus 1)1198962 if 119896 is even (52)


120601 (119909) infinsum119896=0

ℎ119902119896119896 119867119890119896 (119909)= 120601 (119909) infinsum

119899=0

12119899119899 (1199022 minus 1)1198991198671198902119899 (119909) (53)


infinsum119899=0

(minus1)119899 (2119899)22119899 (119899)2 (1199022 minus 1)119899 (54)


(2119899)22119899 (119899)2 sim 1radic120587119899 (55)



119892119901 (119909) = 119901120601 (119909) + (1 minus 119901) 1119901120601(119909119901) (56)



119884119901 = 119883119901


= 119883119901120590119901 1205902119901 = Var119883119901 = 119901 + 1199012 minus 1199013

(57)


119888119901119896 = 1119896E119867119890119896 (119884119901)= 1119896 (119901ℎ1120590119901 119896 + (1 minus 119901) ℎ119901120590119901 119896)

(58)





1Var119883119901

= 103 + 032 minus 033 = 275482 gt 2 (60)


119902 = 112059003 997904rArr 1198882 = 0119902 = 14 997904rArr 1198882 = minus014426

(61)

minus2 minus1 1 2

minus4

minus2

2

4

6


minus2 minus1 1 2

02

04

06

08

10



E119890119902119883119891 (119883) = 119890120583119902+(12)12059021199022E119891 (119883 + 1205902119902) (62)


1198751015840 = 119890119902119883EP119890119902119883 sdot 119875 (63)


119873sum119896=0

1198871198961198881015840119896119904119896 = sum119873119896=0 119887119896119888119896 (119902 + 119904)119896sum119873

119895=0 119887119895119888119895119902119895 119904 isin R (64)


or more precisely

1198881015840119896 = 1sum119873119895=0 119887119895119888119895119902119895

119873sumℓ=119896

(ℓ119896) 119887ℓminus119896119888ℓ119902ℓminus119896 (65)


E1198751015840119890119904119883 = 1


119896=0

119887119896119904119896 1sum119873119895=0 119887119895119888119895119902119895

sdot 119873sumℓ=119896

(ℓ119896) 119887ℓminus119896119888ℓ119902ℓminus119896(66)


11988810158401 = 311988721198883 + 4119887311988841 + 11988731198883 + 11988741198884 11988810158402 = 31198871198883 + 6119887211988841 + 11988731198883 + 11988741198884 11988810158403 = 1198883 + 411988711988841 + 11988731198883 + 11988741198884 11988810158404 = 11988841 + 11988731198883 + 11988741198884

(67)



(119883(119895)1 119883(119895)

119904 ) 119895 = 1 119899 (68)


119891119883 (119909) = 120601 (119886 119887 119909) 119901 (119909) (69)


E119892 (1198831 119883119904) = intR119904119892 (1199091 119909119904)

sdot 119904prod119896=1


119896=1

119901 (119883119896)] (70)


1 119883(119895)119904 ) 119895 = 1 119899 and then

using the estimator

119899 = 1119899119899sum119895=1

[119892 (119883(119895)1 119883(119895)

119904 ) 119904prod119896=1

119901 (119883(119895)

119896)] (71)




E119890119911(119883+119884)= 119890(1198861+1198862)119905+(11988721+11988722 )(11990522)(119873(1)sum

119896=0

119888(1)119896 119905119896)(119873(2)sum119896=0

119888(2)119896 119905119896) (72)


119883 + 119884 sim GC(1198861 + 1198862 radic11988721 + 11988722 1198881 119888119873(1)+119873(2)) 119888119896 = 119896sum

119895=0

119888(1)119895 119888(2)119896minus119895(73)


119885119899 = 1198831 + sdot sdot sdot + 119883119899 (74)


( 119873sum119896=0

119888119896119905119896)119899 = 119899119873sum

119896=0


(75)


and thus






119872(119911) = E119890119911119883 = 119890119886119911+1198872(11991122)120587 (119911) (77)


119872119899 (119911) = E1198901199111198831198991 (78)


119872119899 (119911)119899 = 119872(119911) 119911 isin C (79)





119891119871 (119911) = 1119911119891119884 (log 119911) 119911 gt 0 (80)





All the functions



119892120572 (119911) = 119891119871 (119911) + 120572120601 (119911) sin (2120587 log 119911) (83)


119901 (119910) = 119873sum119896=0

119888119896119867119890119896 (119910) (84)


119904 (119910) = sin (2120587119910) 1119910notin119869 (85)


119904 (119910 + 2119896120587) = 119904 (119910) 119904 (minus119910) = minus119904 (119910) (86)


intinfinminusinfin

119890119896119910120601 (119910) 119904 (119910) 119889119910 = 0 (87)



minusintinfin0

log119891 (1199112)1 + 1199112 119889119911 lt infin (88)







119890119886+(11988722) 119873sum119896=0

119887119896119888119896 = 119890(119903minus120575)119879 (89)










(92)


119896=0

119887119896119888119896 = 119890(119903minus120575)1198791198780 (93)



(94)





119896=1

119888119896119867119890119896minus1 (minus1198892) (96)

where

1198892 = 1119887 (log(1198780119870) + 119886) (97)


Q1015840 = 119878119879EQ119878119879 sdotQ = 119890119883119879

EQ119890119883119879 sdotQ (98)


Then

EQ (1198781198791119878119879gt119870) = (EQ119878119879)EQ1015840 (1119878119879gt119870) (99)

Since119883119879


119896=1

1198881015840119896119867119890119896minus1 (minus1198891) (100)

where

1198881015840119896 = 1sum119873119895=0 119887119895119888119895

119873sumℓ=119896

(ℓ119896) 119887ℓminus119896119888ℓ 119896 = 1 1198731198891 = 1119887 [log(1198780119870) + 119886 + 1198872] = 1198892 + 119887

(101)

Hence


119896=1


119896=1

119888119896119867119890119896minus1 (minus1198892) (102)

Using




where

119888lowast119896 = 1198881015840119896 119873sum119896=0

119887119896119888119896 = 119873sumℓ=119896

(ℓ119896) 119887ℓminus119896119888ℓ 119896 = 1 119873 (105)



1198750 = 1198620 minus 1198780 + 119870119890minus119903119879 (106)


4sum119896=1



(107)

This ends the proof



Δ = 12059711986201205971198780= 119890minus120575119879(Φ(1198891) + 120601 (1198891) 119873sum

119896=1

1198881015840119896119867119890119896minus1 (minus1198891))

120574 = 1205972119862012059711987820 = 119890119886+(11988722)minus1199031198791198871198780 120601 (1198891) 119873sum119896=0

119888119896119867119890119896 (minus1198892) 120588 = 1205971198620120597119903= 119870119879119890minus119903119879(Φ(1198892) + 120601 (1198892) 119873sum

119896=1





1198881015840119896119867119890119896minus1 (minus1198891)) 119895 ge 1

(108)





120597120597119904E (119904119880 minus 119870)+ = E (1198801119904119880gt119870) (109)


2

1199043 119891119880 (119870119904 ) (110)

From 119878119879 = 1198780119890119883119879 and (100)


119896=1

1198881015840119896119867119890119896minus1 (minus1198891)) (111)

From the lemma

120574 = 1205972120597211987801198620 = 119890minus119903119879120597212059721198780EQ (1198780119880 minus 119870)+

= 1198702119890minus11990311987911987830 119891119880 (1198701198780) 119880 = 119890119883(112)

where 119883119879


119891119880 (119906) = 1119887119906119891119866 (1119887 (log 119906 minus 119886))= 1119887119906120601 (119910)

119873sum119896=0

119888119896119867119890119896 (119910)1003816100381610038161003816100381610038161003816100381610038161003816119910=(1119887)(log 119906minus119886)

(113)


120574 = 119870119890minus11990311987911988711987820 120601 (1198892) 119873sum119896=0

119888119896119867119890119896 (minus1198892)= 119890119886+(11988722)minus1199031198791198871198780 120601 (1198891) 119873sum

119896=0

119888119896119867119890119896 (minus1198892) (114)

Next




Now from (89)



119896=1

119888119896119867119890119896minus1 (minus1198892)] (118)


Define

119898(119887) = E119890119887119866minus(11988722) = sum119896=0

1198871198961198881198961198981015840 (119887) = sum

119896=1

119896119887119896minus1119888119896 = 119888lowast1 (120)


119890119886 = 119890(119903minus120575)119879minus(11988722)119898(119887) (121)


1198780119890minus120575119879EQ [( 120597120597119887 119890119887119866minus(11988722)

119898(119887) ) 1119866gtminus1198892]


(122)


EQ (1198901198871198661119866gtminus119910) = 11989011988722 [119898 (119887)Φ (119910 + 119887)

+ 120601 (119910 + 119887) 119873sum119896=1


To calculate


we need (see (105))

120597119888lowast119896120597119887 = 120597120597119887119873sumℓ=119896


120597119888lowast119873120597119887 = 0(125)




1198881015840119896119867119890119896 (minus119910 minus 119887)]

(126)



120597119886120597119888119895 119890119886+(11988722)

119873sum119896=0

119887119896119888119896 + 119887119895119890119886+(11988722) = 0or 120597119886120597119888119895 = minus

119887119895sum119873119896=0 119887119896119888119896 = minus119887

119895119890119886+(11988722)+(120575minus119903)119879(127)

Next


infin

minus1198892

(1198780119890119886+119887119910 minus 119870)120601 (119910) 119873sum119896=0

119888119896119867119890119896 (119910) 119889119910= 119890minus119903119879intinfin

minus1198892

1198780 120597119886120597119888119895 119890119886+119887119910120601 (119910)119873sum119896=0

119888119896119867119890119896 (119910) 119889119910+ 119890minus119903119879intinfin

minus1198892

(1198780119890119886+119887119910 minus 119870)120601 (119910)119867119890119895 (119910) 119889119910

(128)



119896=0


119910119892(119895) (119909) 120601 (119909) 119889119909

(129)




minus1198892


119896=1


(130)




119861119899 = 119875 119899prod119905=1

[1 +max(120572( 119878119905119878119905minus1 minus 1) 0)] (131)





119877119905 Qsim GC (119886 119887 ) 119905 = 1 119899 (133)


119905=1





1198810 = 119875 (119890minus119903 + 1205721198620)119899 (135)


1205971198810120597120585 = 120572119875 (119890minus119903 + 1205721198620)119899minus1 1205971198620120597120585 (136)


1205972119862012059711987820 = 1205722119875 (119890minus119903 + 1205721198620)119899minus2 (12059711986201205971198780 )2

+ 120572119875 (119890minus119903 + 1205721198620)119899minus1 1205972119862012059711987820 (137)



119886 = 00687119887 = 016851198883 = minus011501198884 = 003598

(138)


119904 = 61198883 = minus06898119896 = 241198884 = 08634 (139)



0

1

2

3

4

k


s


minus1

0

1

0

2

4 104

106

108

110

Option

k s







(1119887)(log(1198701198780)minus1198861015840)(11987801198901198861015840+119887119909 minus 119870)120601 (119909)

sdot ( 119873sum119896=0

119888119896119867119890119896 (119909))119889119909(140)


(1119887)(log(1198701198780)minus1198861015840)(11987801198901198861015840+119887119909 minus 119870)120601 (119909)

sdot 119867119890119895 (119909) 119889119909(141)



119886 = 01174119887 = 015951198881 = minus030536756952010661198882 = 0095420793734891531198883 = minus0123839711263352431198884 = 006120331530131559

(142)





119901 (119909) = 2119898sum119895=0

120575119895119867119890119895 (119909) (143)


119886 = minus02445119887 = 019401198881 = 16140




1198882 = 117331198883 = 042531198884 = 007320

(144)




(145)


k

s

01234

105

110

Option

0500

minus05









Competing Interests


Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119867119890119896 (119910 minus 119886119887 ) 119896 = 0 1 119873 (14)


intinfinminusinfin


119889119896119889119909119896 120601 (119909) 119889119909 = 0119896 = 1 2

(15)



1199081 (119905 119909) = infinsum119899=0

119905119899119899119867119890119899 (119909) = 119890119905119909minus(11990522) (16)

Another one is

1199082 (119905 119906 119909 119910)= infinsum

119896=0

infinsum119899=0

119905119896119896 119906119899

119899 intinfin


= radic2120587119890119906(119905+119910)(17)


infin


119899

119906119899 119890119905119906100381610038161003816100381610038161003816100381610038161003816119905=119906=0

= 0 if 119896 = 119899119899 if 119896 = 119899

(18)





intinfinminusinfin

120601 (119909)119867119890119896 (119909) 119889119909 = 0 119896 = 1 2 (20)


intinfinminusinfin

119890119905119909120601 (119909)119867119890119896 (119909) 119889119909 = 11990511989611989011990522 119896 = 0 1 (21)


intinfinminusinfin

119909119899120601 (119909)119867119890119896 (119909) 119889119909= (minus1)119896 intinfin

minusinfin119909119899 ( 119889119896119889119909119896120601 (119909)) 119889119909

(22)


intinfinminusinfin

119909119899120601 (119909)119867119890119896 (119909) 119889119909=




119873and119899sum119896=0



intinfinminusinfin

119890119905119909119891 (119909) 119889119909 = 11989011990522 119873sum119896=0

119888119896119905119896 (25)



(a)

E (119884 minus 119886)119899 = 119887119899 119873and119899sum119896=0


(b)

E119890119905119884 = 119890119886119905+(119887211990522) 119873sum119896=0

119887119896119888119896119905119896 119905 isin R (27)






+ 2411988611988731198883 + 1211988741198882 + 2411988741198884 + 311988741198985 = 1198865 + 511988641198871198881 + 20119886311988721198882 + 1011988631198872 + 30119886211988731198881

+ 60119886211988731198883 + 6011988611988741198882 + 12011988611988741198884 + 151198861198874+ 1511988751198881 + 6011988751198883 + 12011988751198885

1198986 = 1198866 + 611988651198871198881 + 30119886411988721198882 + 1511988641198872 + 60119886311988731198881+ 120119886311988731198883 + 180119886211988741198882 + 360119886211988741198884+ 4511988621198874 + 9011988611988751198881 + 36011988611988751198883 + 72011988611988751198885+ 9011988761198882 + 36011988761198884 + 72011988761198886 + 151198876

(28)




(29)



2 (11988831 minus 311988811198882 + 31198883)(1 minus 11988821 + 21198882)32


(30)


E119883119895 = E119884119895 119895 = 1 119870 (31)




119886 = E119883 1198872 = Var119883 lArrrArr 1198881198831 = 1198881198832 = 0 (32)




119892 (119910) = 119890minus1199102radic21205871199101198992sum119895=0

120572119895119910119895

119908ℎ119890119903119890 1198992sum119895=0

120572119895119910119895 = 1198992sum119895=0

11988821198951198671198902119895 (radic119910) (33)


119873sum119896=1

119867119890119896 (119909) (34)



1 + 120598 119873sum119896=0

119867119890119896 (119909) gt 0 119909 isin R 0 le 120598 le 1205980 (35)



119905 997891997888rarr log(119890119886119905+(119887211990522) 119873sum119896=0

119887119896119888119896119905119896) (36)


only Write

120595 (119905) = 11989011990522119902119883 (119905) = 119873sum

119896=0

119888119883119896 119905119896119902119884 (119905) = 119873sum

119896=0

119888119884119896 119905119896(37)


1205951015840 (0) 119902119883 (0) + 120595 (0) 1199021015840119883 (0) = 1205951015840 (0) 119902119884 (0) + 120595 (0) 1199021015840119884 (0) 119870sum119895=0

(119870119895)120595(119895) (0) 119902(119870minus119895)119883 (0) = 119870sum119895=0

(119870119895)120595(119895) (0) 119902(119870minus119895)119884 (0) (38)



119886 = E119883 lArrrArr 1198881198831 = 0 (39)


E1198832 = 12059510158401015840 (0) 119902119883 (0) + 21205951015840 (0) 1199021015840119883 (0) + 120595 (0) 11990210158401015840119883 (0) E1198842 = 12059510158401015840 (0) 119902119884 (0) + 21205951015840 (0) 1199021015840119884 (0) + 120595 (0) 11990210158401015840119884 (0) (40)


E(119909 minus 119886119887 )2 = 1 lArrrArr 1198881198832 = 0 (41)


119890119886119902119904+(12)119887211990221199042 119873sum119896=0

119888119896119887119896119902119896119904119896

= 119890119886119902119904+(12)119887211990221199042 119873sum119896=0

119888119896 ( 11990210038161003816100381610038161199021003816100381610038161003816)119896 (119887 10038161003816100381610038161199021003816100381610038161003816)119896 119904119896

(42)



119873sum119896=0





119896=1


119896=1

119888119896119867119890119896minus1 (119909) (44)




(45)







119884 = 119883 minus E119883radicVar119883 (46)




119896=0

119888119896119867119890119896 (47)

where

119888119896 = 1119896 intinfin

minusinfin119867119890119896 (119909) 119891 (119909) 119889119909 = 1119896E119867119890119896 (119883) (48)


12 (119891 (119909minus) + 119891 (119909+)) (49)


intinfinminusinfin

11989011990924119891 (119909) 119889119909 lt infin (50)



ℎ119902119896 = E119867119890119896 (119883) 119883 sim N (0 1199022) (51)


ℎ119902119896 = 0 if 119896 is odd

11989621198962 (1198962) (1199022 minus 1)1198962 if 119896 is even (52)


120601 (119909) infinsum119896=0

ℎ119902119896119896 119867119890119896 (119909)= 120601 (119909) infinsum

119899=0

12119899119899 (1199022 minus 1)1198991198671198902119899 (119909) (53)


infinsum119899=0

(minus1)119899 (2119899)22119899 (119899)2 (1199022 minus 1)119899 (54)


(2119899)22119899 (119899)2 sim 1radic120587119899 (55)



119892119901 (119909) = 119901120601 (119909) + (1 minus 119901) 1119901120601(119909119901) (56)



119884119901 = 119883119901


= 119883119901120590119901 1205902119901 = Var119883119901 = 119901 + 1199012 minus 1199013

(57)


119888119901119896 = 1119896E119867119890119896 (119884119901)= 1119896 (119901ℎ1120590119901 119896 + (1 minus 119901) ℎ119901120590119901 119896)

(58)





1Var119883119901

= 103 + 032 minus 033 = 275482 gt 2 (60)


119902 = 112059003 997904rArr 1198882 = 0119902 = 14 997904rArr 1198882 = minus014426

(61)

minus2 minus1 1 2

minus4

minus2

2

4

6


minus2 minus1 1 2

02

04

06

08

10



E119890119902119883119891 (119883) = 119890120583119902+(12)12059021199022E119891 (119883 + 1205902119902) (62)


1198751015840 = 119890119902119883EP119890119902119883 sdot 119875 (63)


119873sum119896=0

1198871198961198881015840119896119904119896 = sum119873119896=0 119887119896119888119896 (119902 + 119904)119896sum119873

119895=0 119887119895119888119895119902119895 119904 isin R (64)


or more precisely

1198881015840119896 = 1sum119873119895=0 119887119895119888119895119902119895

119873sumℓ=119896

(ℓ119896) 119887ℓminus119896119888ℓ119902ℓminus119896 (65)


E1198751015840119890119904119883 = 1


119896=0

119887119896119904119896 1sum119873119895=0 119887119895119888119895119902119895

sdot 119873sumℓ=119896

(ℓ119896) 119887ℓminus119896119888ℓ119902ℓminus119896(66)


11988810158401 = 311988721198883 + 4119887311988841 + 11988731198883 + 11988741198884 11988810158402 = 31198871198883 + 6119887211988841 + 11988731198883 + 11988741198884 11988810158403 = 1198883 + 411988711988841 + 11988731198883 + 11988741198884 11988810158404 = 11988841 + 11988731198883 + 11988741198884

(67)



(119883(119895)1 119883(119895)

119904 ) 119895 = 1 119899 (68)


119891119883 (119909) = 120601 (119886 119887 119909) 119901 (119909) (69)


E119892 (1198831 119883119904) = intR119904119892 (1199091 119909119904)

sdot 119904prod119896=1


119896=1

119901 (119883119896)] (70)


1 119883(119895)119904 ) 119895 = 1 119899 and then

using the estimator

119899 = 1119899119899sum119895=1

[119892 (119883(119895)1 119883(119895)

119904 ) 119904prod119896=1

119901 (119883(119895)

119896)] (71)




E119890119911(119883+119884)= 119890(1198861+1198862)119905+(11988721+11988722 )(11990522)(119873(1)sum

119896=0

119888(1)119896 119905119896)(119873(2)sum119896=0

119888(2)119896 119905119896) (72)


119883 + 119884 sim GC(1198861 + 1198862 radic11988721 + 11988722 1198881 119888119873(1)+119873(2)) 119888119896 = 119896sum

119895=0

119888(1)119895 119888(2)119896minus119895(73)


119885119899 = 1198831 + sdot sdot sdot + 119883119899 (74)


( 119873sum119896=0

119888119896119905119896)119899 = 119899119873sum

119896=0


(75)


and thus






119872(119911) = E119890119911119883 = 119890119886119911+1198872(11991122)120587 (119911) (77)


119872119899 (119911) = E1198901199111198831198991 (78)


119872119899 (119911)119899 = 119872(119911) 119911 isin C (79)





119891119871 (119911) = 1119911119891119884 (log 119911) 119911 gt 0 (80)





All the functions



119892120572 (119911) = 119891119871 (119911) + 120572120601 (119911) sin (2120587 log 119911) (83)


119901 (119910) = 119873sum119896=0

119888119896119867119890119896 (119910) (84)


119904 (119910) = sin (2120587119910) 1119910notin119869 (85)


119904 (119910 + 2119896120587) = 119904 (119910) 119904 (minus119910) = minus119904 (119910) (86)


intinfinminusinfin

119890119896119910120601 (119910) 119904 (119910) 119889119910 = 0 (87)



minusintinfin0

log119891 (1199112)1 + 1199112 119889119911 lt infin (88)







119890119886+(11988722) 119873sum119896=0

119887119896119888119896 = 119890(119903minus120575)119879 (89)










(92)


119896=0

119887119896119888119896 = 119890(119903minus120575)1198791198780 (93)



(94)





119896=1

119888119896119867119890119896minus1 (minus1198892) (96)

where

1198892 = 1119887 (log(1198780119870) + 119886) (97)


Q1015840 = 119878119879EQ119878119879 sdotQ = 119890119883119879

EQ119890119883119879 sdotQ (98)


Then

EQ (1198781198791119878119879gt119870) = (EQ119878119879)EQ1015840 (1119878119879gt119870) (99)

Since119883119879


119896=1

1198881015840119896119867119890119896minus1 (minus1198891) (100)

where

1198881015840119896 = 1sum119873119895=0 119887119895119888119895

119873sumℓ=119896

(ℓ119896) 119887ℓminus119896119888ℓ 119896 = 1 1198731198891 = 1119887 [log(1198780119870) + 119886 + 1198872] = 1198892 + 119887

(101)

Hence


119896=1


119896=1

119888119896119867119890119896minus1 (minus1198892) (102)

Using




where

119888lowast119896 = 1198881015840119896 119873sum119896=0

119887119896119888119896 = 119873sumℓ=119896

(ℓ119896) 119887ℓminus119896119888ℓ 119896 = 1 119873 (105)



1198750 = 1198620 minus 1198780 + 119870119890minus119903119879 (106)


4sum119896=1



(107)

This ends the proof



Δ = 12059711986201205971198780= 119890minus120575119879(Φ(1198891) + 120601 (1198891) 119873sum

119896=1

1198881015840119896119867119890119896minus1 (minus1198891))

120574 = 1205972119862012059711987820 = 119890119886+(11988722)minus1199031198791198871198780 120601 (1198891) 119873sum119896=0

119888119896119867119890119896 (minus1198892) 120588 = 1205971198620120597119903= 119870119879119890minus119903119879(Φ(1198892) + 120601 (1198892) 119873sum

119896=1





1198881015840119896119867119890119896minus1 (minus1198891)) 119895 ge 1

(108)





120597120597119904E (119904119880 minus 119870)+ = E (1198801119904119880gt119870) (109)


2

1199043 119891119880 (119870119904 ) (110)

From 119878119879 = 1198780119890119883119879 and (100)


119896=1

1198881015840119896119867119890119896minus1 (minus1198891)) (111)

From the lemma

120574 = 1205972120597211987801198620 = 119890minus119903119879120597212059721198780EQ (1198780119880 minus 119870)+

= 1198702119890minus11990311987911987830 119891119880 (1198701198780) 119880 = 119890119883(112)

where 119883119879


119891119880 (119906) = 1119887119906119891119866 (1119887 (log 119906 minus 119886))= 1119887119906120601 (119910)

119873sum119896=0

119888119896119867119890119896 (119910)1003816100381610038161003816100381610038161003816100381610038161003816119910=(1119887)(log 119906minus119886)

(113)


120574 = 119870119890minus11990311987911988711987820 120601 (1198892) 119873sum119896=0

119888119896119867119890119896 (minus1198892)= 119890119886+(11988722)minus1199031198791198871198780 120601 (1198891) 119873sum

119896=0

119888119896119867119890119896 (minus1198892) (114)

Next




Now from (89)



119896=1

119888119896119867119890119896minus1 (minus1198892)] (118)


Define

119898(119887) = E119890119887119866minus(11988722) = sum119896=0

1198871198961198881198961198981015840 (119887) = sum

119896=1

119896119887119896minus1119888119896 = 119888lowast1 (120)


119890119886 = 119890(119903minus120575)119879minus(11988722)119898(119887) (121)


1198780119890minus120575119879EQ [( 120597120597119887 119890119887119866minus(11988722)

119898(119887) ) 1119866gtminus1198892]


(122)


EQ (1198901198871198661119866gtminus119910) = 11989011988722 [119898 (119887)Φ (119910 + 119887)

+ 120601 (119910 + 119887) 119873sum119896=1


To calculate


we need (see (105))

120597119888lowast119896120597119887 = 120597120597119887119873sumℓ=119896


120597119888lowast119873120597119887 = 0(125)




1198881015840119896119867119890119896 (minus119910 minus 119887)]

(126)



120597119886120597119888119895 119890119886+(11988722)

119873sum119896=0

119887119896119888119896 + 119887119895119890119886+(11988722) = 0or 120597119886120597119888119895 = minus

119887119895sum119873119896=0 119887119896119888119896 = minus119887

119895119890119886+(11988722)+(120575minus119903)119879(127)

Next


infin

minus1198892

(1198780119890119886+119887119910 minus 119870)120601 (119910) 119873sum119896=0

119888119896119867119890119896 (119910) 119889119910= 119890minus119903119879intinfin

minus1198892

1198780 120597119886120597119888119895 119890119886+119887119910120601 (119910)119873sum119896=0

119888119896119867119890119896 (119910) 119889119910+ 119890minus119903119879intinfin

minus1198892

(1198780119890119886+119887119910 minus 119870)120601 (119910)119867119890119895 (119910) 119889119910

(128)



119896=0


119910119892(119895) (119909) 120601 (119909) 119889119909

(129)




minus1198892


119896=1


(130)




119861119899 = 119875 119899prod119905=1

[1 +max(120572( 119878119905119878119905minus1 minus 1) 0)] (131)





119877119905 Qsim GC (119886 119887 ) 119905 = 1 119899 (133)


119905=1





1198810 = 119875 (119890minus119903 + 1205721198620)119899 (135)


1205971198810120597120585 = 120572119875 (119890minus119903 + 1205721198620)119899minus1 1205971198620120597120585 (136)


1205972119862012059711987820 = 1205722119875 (119890minus119903 + 1205721198620)119899minus2 (12059711986201205971198780 )2

+ 120572119875 (119890minus119903 + 1205721198620)119899minus1 1205972119862012059711987820 (137)



119886 = 00687119887 = 016851198883 = minus011501198884 = 003598

(138)


119904 = 61198883 = minus06898119896 = 241198884 = 08634 (139)



0

1

2

3

4

k


s


minus1

0

1

0

2

4 104

106

108

110

Option

k s







(1119887)(log(1198701198780)minus1198861015840)(11987801198901198861015840+119887119909 minus 119870)120601 (119909)

sdot ( 119873sum119896=0

119888119896119867119890119896 (119909))119889119909(140)


(1119887)(log(1198701198780)minus1198861015840)(11987801198901198861015840+119887119909 minus 119870)120601 (119909)

sdot 119867119890119895 (119909) 119889119909(141)



119886 = 01174119887 = 015951198881 = minus030536756952010661198882 = 0095420793734891531198883 = minus0123839711263352431198884 = 006120331530131559

(142)





119901 (119909) = 2119898sum119895=0

120575119895119867119890119895 (119909) (143)


119886 = minus02445119887 = 019401198881 = 16140




1198882 = 117331198883 = 042531198884 = 007320

(144)




(145)


k

s

01234

105

110

Option

0500

minus05









Competing Interests


Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











+ 2411988611988731198883 + 1211988741198882 + 2411988741198884 + 311988741198985 = 1198865 + 511988641198871198881 + 20119886311988721198882 + 1011988631198872 + 30119886211988731198881

+ 60119886211988731198883 + 6011988611988741198882 + 12011988611988741198884 + 151198861198874+ 1511988751198881 + 6011988751198883 + 12011988751198885

1198986 = 1198866 + 611988651198871198881 + 30119886411988721198882 + 1511988641198872 + 60119886311988731198881+ 120119886311988731198883 + 180119886211988741198882 + 360119886211988741198884+ 4511988621198874 + 9011988611988751198881 + 36011988611988751198883 + 72011988611988751198885+ 9011988761198882 + 36011988761198884 + 72011988761198886 + 151198876

(28)




(29)



2 (11988831 minus 311988811198882 + 31198883)(1 minus 11988821 + 21198882)32


(30)


E119883119895 = E119884119895 119895 = 1 119870 (31)




119886 = E119883 1198872 = Var119883 lArrrArr 1198881198831 = 1198881198832 = 0 (32)




119892 (119910) = 119890minus1199102radic21205871199101198992sum119895=0

120572119895119910119895

119908ℎ119890119903119890 1198992sum119895=0

120572119895119910119895 = 1198992sum119895=0

11988821198951198671198902119895 (radic119910) (33)


119873sum119896=1

119867119890119896 (119909) (34)



1 + 120598 119873sum119896=0

119867119890119896 (119909) gt 0 119909 isin R 0 le 120598 le 1205980 (35)



119905 997891997888rarr log(119890119886119905+(119887211990522) 119873sum119896=0

119887119896119888119896119905119896) (36)


only Write

120595 (119905) = 11989011990522119902119883 (119905) = 119873sum

119896=0

119888119883119896 119905119896119902119884 (119905) = 119873sum

119896=0

119888119884119896 119905119896(37)


1205951015840 (0) 119902119883 (0) + 120595 (0) 1199021015840119883 (0) = 1205951015840 (0) 119902119884 (0) + 120595 (0) 1199021015840119884 (0) 119870sum119895=0

(119870119895)120595(119895) (0) 119902(119870minus119895)119883 (0) = 119870sum119895=0

(119870119895)120595(119895) (0) 119902(119870minus119895)119884 (0) (38)



119886 = E119883 lArrrArr 1198881198831 = 0 (39)


E1198832 = 12059510158401015840 (0) 119902119883 (0) + 21205951015840 (0) 1199021015840119883 (0) + 120595 (0) 11990210158401015840119883 (0) E1198842 = 12059510158401015840 (0) 119902119884 (0) + 21205951015840 (0) 1199021015840119884 (0) + 120595 (0) 11990210158401015840119884 (0) (40)


E(119909 minus 119886119887 )2 = 1 lArrrArr 1198881198832 = 0 (41)


119890119886119902119904+(12)119887211990221199042 119873sum119896=0

119888119896119887119896119902119896119904119896

= 119890119886119902119904+(12)119887211990221199042 119873sum119896=0

119888119896 ( 11990210038161003816100381610038161199021003816100381610038161003816)119896 (119887 10038161003816100381610038161199021003816100381610038161003816)119896 119904119896

(42)



119873sum119896=0





119896=1


119896=1

119888119896119867119890119896minus1 (119909) (44)




(45)







119884 = 119883 minus E119883radicVar119883 (46)




119896=0

119888119896119867119890119896 (47)

where

119888119896 = 1119896 intinfin

minusinfin119867119890119896 (119909) 119891 (119909) 119889119909 = 1119896E119867119890119896 (119883) (48)


12 (119891 (119909minus) + 119891 (119909+)) (49)


intinfinminusinfin

11989011990924119891 (119909) 119889119909 lt infin (50)



ℎ119902119896 = E119867119890119896 (119883) 119883 sim N (0 1199022) (51)


ℎ119902119896 = 0 if 119896 is odd

11989621198962 (1198962) (1199022 minus 1)1198962 if 119896 is even (52)


120601 (119909) infinsum119896=0

ℎ119902119896119896 119867119890119896 (119909)= 120601 (119909) infinsum

119899=0

12119899119899 (1199022 minus 1)1198991198671198902119899 (119909) (53)


infinsum119899=0

(minus1)119899 (2119899)22119899 (119899)2 (1199022 minus 1)119899 (54)


(2119899)22119899 (119899)2 sim 1radic120587119899 (55)



119892119901 (119909) = 119901120601 (119909) + (1 minus 119901) 1119901120601(119909119901) (56)



119884119901 = 119883119901


= 119883119901120590119901 1205902119901 = Var119883119901 = 119901 + 1199012 minus 1199013

(57)


119888119901119896 = 1119896E119867119890119896 (119884119901)= 1119896 (119901ℎ1120590119901 119896 + (1 minus 119901) ℎ119901120590119901 119896)

(58)





1Var119883119901

= 103 + 032 minus 033 = 275482 gt 2 (60)


119902 = 112059003 997904rArr 1198882 = 0119902 = 14 997904rArr 1198882 = minus014426

(61)

minus2 minus1 1 2

minus4

minus2

2

4

6


minus2 minus1 1 2

02

04

06

08

10



E119890119902119883119891 (119883) = 119890120583119902+(12)12059021199022E119891 (119883 + 1205902119902) (62)


1198751015840 = 119890119902119883EP119890119902119883 sdot 119875 (63)


119873sum119896=0

1198871198961198881015840119896119904119896 = sum119873119896=0 119887119896119888119896 (119902 + 119904)119896sum119873

119895=0 119887119895119888119895119902119895 119904 isin R (64)


or more precisely

1198881015840119896 = 1sum119873119895=0 119887119895119888119895119902119895

119873sumℓ=119896

(ℓ119896) 119887ℓminus119896119888ℓ119902ℓminus119896 (65)


E1198751015840119890119904119883 = 1


119896=0

119887119896119904119896 1sum119873119895=0 119887119895119888119895119902119895

sdot 119873sumℓ=119896

(ℓ119896) 119887ℓminus119896119888ℓ119902ℓminus119896(66)


11988810158401 = 311988721198883 + 4119887311988841 + 11988731198883 + 11988741198884 11988810158402 = 31198871198883 + 6119887211988841 + 11988731198883 + 11988741198884 11988810158403 = 1198883 + 411988711988841 + 11988731198883 + 11988741198884 11988810158404 = 11988841 + 11988731198883 + 11988741198884

(67)



(119883(119895)1 119883(119895)

119904 ) 119895 = 1 119899 (68)


119891119883 (119909) = 120601 (119886 119887 119909) 119901 (119909) (69)


E119892 (1198831 119883119904) = intR119904119892 (1199091 119909119904)

sdot 119904prod119896=1


119896=1

119901 (119883119896)] (70)


1 119883(119895)119904 ) 119895 = 1 119899 and then

using the estimator

119899 = 1119899119899sum119895=1

[119892 (119883(119895)1 119883(119895)

119904 ) 119904prod119896=1

119901 (119883(119895)

119896)] (71)




E119890119911(119883+119884)= 119890(1198861+1198862)119905+(11988721+11988722 )(11990522)(119873(1)sum

119896=0

119888(1)119896 119905119896)(119873(2)sum119896=0

119888(2)119896 119905119896) (72)


119883 + 119884 sim GC(1198861 + 1198862 radic11988721 + 11988722 1198881 119888119873(1)+119873(2)) 119888119896 = 119896sum

119895=0

119888(1)119895 119888(2)119896minus119895(73)


119885119899 = 1198831 + sdot sdot sdot + 119883119899 (74)


( 119873sum119896=0

119888119896119905119896)119899 = 119899119873sum

119896=0


(75)


and thus






119872(119911) = E119890119911119883 = 119890119886119911+1198872(11991122)120587 (119911) (77)


119872119899 (119911) = E1198901199111198831198991 (78)


119872119899 (119911)119899 = 119872(119911) 119911 isin C (79)





119891119871 (119911) = 1119911119891119884 (log 119911) 119911 gt 0 (80)





All the functions



119892120572 (119911) = 119891119871 (119911) + 120572120601 (119911) sin (2120587 log 119911) (83)


119901 (119910) = 119873sum119896=0

119888119896119867119890119896 (119910) (84)


119904 (119910) = sin (2120587119910) 1119910notin119869 (85)


119904 (119910 + 2119896120587) = 119904 (119910) 119904 (minus119910) = minus119904 (119910) (86)


intinfinminusinfin

119890119896119910120601 (119910) 119904 (119910) 119889119910 = 0 (87)



minusintinfin0

log119891 (1199112)1 + 1199112 119889119911 lt infin (88)







119890119886+(11988722) 119873sum119896=0

119887119896119888119896 = 119890(119903minus120575)119879 (89)










(92)


119896=0

119887119896119888119896 = 119890(119903minus120575)1198791198780 (93)



(94)





119896=1

119888119896119867119890119896minus1 (minus1198892) (96)

where

1198892 = 1119887 (log(1198780119870) + 119886) (97)


Q1015840 = 119878119879EQ119878119879 sdotQ = 119890119883119879

EQ119890119883119879 sdotQ (98)


Then

EQ (1198781198791119878119879gt119870) = (EQ119878119879)EQ1015840 (1119878119879gt119870) (99)

Since119883119879


119896=1

1198881015840119896119867119890119896minus1 (minus1198891) (100)

where

1198881015840119896 = 1sum119873119895=0 119887119895119888119895

119873sumℓ=119896

(ℓ119896) 119887ℓminus119896119888ℓ 119896 = 1 1198731198891 = 1119887 [log(1198780119870) + 119886 + 1198872] = 1198892 + 119887

(101)

Hence


119896=1


119896=1

119888119896119867119890119896minus1 (minus1198892) (102)

Using




where

119888lowast119896 = 1198881015840119896 119873sum119896=0

119887119896119888119896 = 119873sumℓ=119896

(ℓ119896) 119887ℓminus119896119888ℓ 119896 = 1 119873 (105)



1198750 = 1198620 minus 1198780 + 119870119890minus119903119879 (106)


4sum119896=1



(107)

This ends the proof



Δ = 12059711986201205971198780= 119890minus120575119879(Φ(1198891) + 120601 (1198891) 119873sum

119896=1

1198881015840119896119867119890119896minus1 (minus1198891))

120574 = 1205972119862012059711987820 = 119890119886+(11988722)minus1199031198791198871198780 120601 (1198891) 119873sum119896=0

119888119896119867119890119896 (minus1198892) 120588 = 1205971198620120597119903= 119870119879119890minus119903119879(Φ(1198892) + 120601 (1198892) 119873sum

119896=1





1198881015840119896119867119890119896minus1 (minus1198891)) 119895 ge 1

(108)





120597120597119904E (119904119880 minus 119870)+ = E (1198801119904119880gt119870) (109)


2

1199043 119891119880 (119870119904 ) (110)

From 119878119879 = 1198780119890119883119879 and (100)


119896=1

1198881015840119896119867119890119896minus1 (minus1198891)) (111)

From the lemma

120574 = 1205972120597211987801198620 = 119890minus119903119879120597212059721198780EQ (1198780119880 minus 119870)+

= 1198702119890minus11990311987911987830 119891119880 (1198701198780) 119880 = 119890119883(112)

where 119883119879


119891119880 (119906) = 1119887119906119891119866 (1119887 (log 119906 minus 119886))= 1119887119906120601 (119910)

119873sum119896=0

119888119896119867119890119896 (119910)1003816100381610038161003816100381610038161003816100381610038161003816119910=(1119887)(log 119906minus119886)

(113)


120574 = 119870119890minus11990311987911988711987820 120601 (1198892) 119873sum119896=0

119888119896119867119890119896 (minus1198892)= 119890119886+(11988722)minus1199031198791198871198780 120601 (1198891) 119873sum

119896=0

119888119896119867119890119896 (minus1198892) (114)

Next




Now from (89)



119896=1

119888119896119867119890119896minus1 (minus1198892)] (118)


Define

119898(119887) = E119890119887119866minus(11988722) = sum119896=0

1198871198961198881198961198981015840 (119887) = sum

119896=1

119896119887119896minus1119888119896 = 119888lowast1 (120)


119890119886 = 119890(119903minus120575)119879minus(11988722)119898(119887) (121)


1198780119890minus120575119879EQ [( 120597120597119887 119890119887119866minus(11988722)

119898(119887) ) 1119866gtminus1198892]


(122)


EQ (1198901198871198661119866gtminus119910) = 11989011988722 [119898 (119887)Φ (119910 + 119887)

+ 120601 (119910 + 119887) 119873sum119896=1


To calculate


we need (see (105))

120597119888lowast119896120597119887 = 120597120597119887119873sumℓ=119896


120597119888lowast119873120597119887 = 0(125)




1198881015840119896119867119890119896 (minus119910 minus 119887)]

(126)



120597119886120597119888119895 119890119886+(11988722)

119873sum119896=0

119887119896119888119896 + 119887119895119890119886+(11988722) = 0or 120597119886120597119888119895 = minus

119887119895sum119873119896=0 119887119896119888119896 = minus119887

119895119890119886+(11988722)+(120575minus119903)119879(127)

Next


infin

minus1198892

(1198780119890119886+119887119910 minus 119870)120601 (119910) 119873sum119896=0

119888119896119867119890119896 (119910) 119889119910= 119890minus119903119879intinfin

minus1198892

1198780 120597119886120597119888119895 119890119886+119887119910120601 (119910)119873sum119896=0

119888119896119867119890119896 (119910) 119889119910+ 119890minus119903119879intinfin

minus1198892

(1198780119890119886+119887119910 minus 119870)120601 (119910)119867119890119895 (119910) 119889119910

(128)



119896=0


119910119892(119895) (119909) 120601 (119909) 119889119909

(129)




minus1198892


119896=1


(130)




119861119899 = 119875 119899prod119905=1

[1 +max(120572( 119878119905119878119905minus1 minus 1) 0)] (131)





119877119905 Qsim GC (119886 119887 ) 119905 = 1 119899 (133)


119905=1





1198810 = 119875 (119890minus119903 + 1205721198620)119899 (135)


1205971198810120597120585 = 120572119875 (119890minus119903 + 1205721198620)119899minus1 1205971198620120597120585 (136)


1205972119862012059711987820 = 1205722119875 (119890minus119903 + 1205721198620)119899minus2 (12059711986201205971198780 )2

+ 120572119875 (119890minus119903 + 1205721198620)119899minus1 1205972119862012059711987820 (137)



119886 = 00687119887 = 016851198883 = minus011501198884 = 003598

(138)


119904 = 61198883 = minus06898119896 = 241198884 = 08634 (139)



0

1

2

3

4

k


s


minus1

0

1

0

2

4 104

106

108

110

Option

k s







(1119887)(log(1198701198780)minus1198861015840)(11987801198901198861015840+119887119909 minus 119870)120601 (119909)

sdot ( 119873sum119896=0

119888119896119867119890119896 (119909))119889119909(140)


(1119887)(log(1198701198780)minus1198861015840)(11987801198901198861015840+119887119909 minus 119870)120601 (119909)

sdot 119867119890119895 (119909) 119889119909(141)



119886 = 01174119887 = 015951198881 = minus030536756952010661198882 = 0095420793734891531198883 = minus0123839711263352431198884 = 006120331530131559

(142)





119901 (119909) = 2119898sum119895=0

120575119895119867119890119895 (119909) (143)


119886 = minus02445119887 = 019401198881 = 16140




1198882 = 117331198883 = 042531198884 = 007320

(144)




(145)


k

s

01234

105

110

Option

0500

minus05









Competing Interests


Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1 + 120598 119873sum119896=0

119867119890119896 (119909) gt 0 119909 isin R 0 le 120598 le 1205980 (35)



119905 997891997888rarr log(119890119886119905+(119887211990522) 119873sum119896=0

119887119896119888119896119905119896) (36)


only Write

120595 (119905) = 11989011990522119902119883 (119905) = 119873sum

119896=0

119888119883119896 119905119896119902119884 (119905) = 119873sum

119896=0

119888119884119896 119905119896(37)


1205951015840 (0) 119902119883 (0) + 120595 (0) 1199021015840119883 (0) = 1205951015840 (0) 119902119884 (0) + 120595 (0) 1199021015840119884 (0) 119870sum119895=0

(119870119895)120595(119895) (0) 119902(119870minus119895)119883 (0) = 119870sum119895=0

(119870119895)120595(119895) (0) 119902(119870minus119895)119884 (0) (38)



119886 = E119883 lArrrArr 1198881198831 = 0 (39)


E1198832 = 12059510158401015840 (0) 119902119883 (0) + 21205951015840 (0) 1199021015840119883 (0) + 120595 (0) 11990210158401015840119883 (0) E1198842 = 12059510158401015840 (0) 119902119884 (0) + 21205951015840 (0) 1199021015840119884 (0) + 120595 (0) 11990210158401015840119884 (0) (40)


E(119909 minus 119886119887 )2 = 1 lArrrArr 1198881198832 = 0 (41)


119890119886119902119904+(12)119887211990221199042 119873sum119896=0

119888119896119887119896119902119896119904119896

= 119890119886119902119904+(12)119887211990221199042 119873sum119896=0

119888119896 ( 11990210038161003816100381610038161199021003816100381610038161003816)119896 (119887 10038161003816100381610038161199021003816100381610038161003816)119896 119904119896

(42)



119873sum119896=0





119896=1


119896=1

119888119896119867119890119896minus1 (119909) (44)




(45)







119884 = 119883 minus E119883radicVar119883 (46)




119896=0

119888119896119867119890119896 (47)

where

119888119896 = 1119896 intinfin

minusinfin119867119890119896 (119909) 119891 (119909) 119889119909 = 1119896E119867119890119896 (119883) (48)


12 (119891 (119909minus) + 119891 (119909+)) (49)


intinfinminusinfin

11989011990924119891 (119909) 119889119909 lt infin (50)



ℎ119902119896 = E119867119890119896 (119883) 119883 sim N (0 1199022) (51)


ℎ119902119896 = 0 if 119896 is odd

11989621198962 (1198962) (1199022 minus 1)1198962 if 119896 is even (52)


120601 (119909) infinsum119896=0

ℎ119902119896119896 119867119890119896 (119909)= 120601 (119909) infinsum

119899=0

12119899119899 (1199022 minus 1)1198991198671198902119899 (119909) (53)


infinsum119899=0

(minus1)119899 (2119899)22119899 (119899)2 (1199022 minus 1)119899 (54)


(2119899)22119899 (119899)2 sim 1radic120587119899 (55)



119892119901 (119909) = 119901120601 (119909) + (1 minus 119901) 1119901120601(119909119901) (56)



119884119901 = 119883119901


= 119883119901120590119901 1205902119901 = Var119883119901 = 119901 + 1199012 minus 1199013

(57)


119888119901119896 = 1119896E119867119890119896 (119884119901)= 1119896 (119901ℎ1120590119901 119896 + (1 minus 119901) ℎ119901120590119901 119896)

(58)





1Var119883119901

= 103 + 032 minus 033 = 275482 gt 2 (60)


119902 = 112059003 997904rArr 1198882 = 0119902 = 14 997904rArr 1198882 = minus014426

(61)

minus2 minus1 1 2

minus4

minus2

2

4

6


minus2 minus1 1 2

02

04

06

08

10



E119890119902119883119891 (119883) = 119890120583119902+(12)12059021199022E119891 (119883 + 1205902119902) (62)


1198751015840 = 119890119902119883EP119890119902119883 sdot 119875 (63)


119873sum119896=0

1198871198961198881015840119896119904119896 = sum119873119896=0 119887119896119888119896 (119902 + 119904)119896sum119873

119895=0 119887119895119888119895119902119895 119904 isin R (64)


or more precisely

1198881015840119896 = 1sum119873119895=0 119887119895119888119895119902119895

119873sumℓ=119896

(ℓ119896) 119887ℓminus119896119888ℓ119902ℓminus119896 (65)


E1198751015840119890119904119883 = 1


119896=0

119887119896119904119896 1sum119873119895=0 119887119895119888119895119902119895

sdot 119873sumℓ=119896

(ℓ119896) 119887ℓminus119896119888ℓ119902ℓminus119896(66)


11988810158401 = 311988721198883 + 4119887311988841 + 11988731198883 + 11988741198884 11988810158402 = 31198871198883 + 6119887211988841 + 11988731198883 + 11988741198884 11988810158403 = 1198883 + 411988711988841 + 11988731198883 + 11988741198884 11988810158404 = 11988841 + 11988731198883 + 11988741198884

(67)



(119883(119895)1 119883(119895)

119904 ) 119895 = 1 119899 (68)


119891119883 (119909) = 120601 (119886 119887 119909) 119901 (119909) (69)


E119892 (1198831 119883119904) = intR119904119892 (1199091 119909119904)

sdot 119904prod119896=1


119896=1

119901 (119883119896)] (70)


1 119883(119895)119904 ) 119895 = 1 119899 and then

using the estimator

119899 = 1119899119899sum119895=1

[119892 (119883(119895)1 119883(119895)

119904 ) 119904prod119896=1

119901 (119883(119895)

119896)] (71)




E119890119911(119883+119884)= 119890(1198861+1198862)119905+(11988721+11988722 )(11990522)(119873(1)sum

119896=0

119888(1)119896 119905119896)(119873(2)sum119896=0

119888(2)119896 119905119896) (72)


119883 + 119884 sim GC(1198861 + 1198862 radic11988721 + 11988722 1198881 119888119873(1)+119873(2)) 119888119896 = 119896sum

119895=0

119888(1)119895 119888(2)119896minus119895(73)


119885119899 = 1198831 + sdot sdot sdot + 119883119899 (74)


( 119873sum119896=0

119888119896119905119896)119899 = 119899119873sum

119896=0


(75)


and thus






119872(119911) = E119890119911119883 = 119890119886119911+1198872(11991122)120587 (119911) (77)


119872119899 (119911) = E1198901199111198831198991 (78)


119872119899 (119911)119899 = 119872(119911) 119911 isin C (79)





119891119871 (119911) = 1119911119891119884 (log 119911) 119911 gt 0 (80)





All the functions



119892120572 (119911) = 119891119871 (119911) + 120572120601 (119911) sin (2120587 log 119911) (83)


119901 (119910) = 119873sum119896=0

119888119896119867119890119896 (119910) (84)


119904 (119910) = sin (2120587119910) 1119910notin119869 (85)


119904 (119910 + 2119896120587) = 119904 (119910) 119904 (minus119910) = minus119904 (119910) (86)


intinfinminusinfin

119890119896119910120601 (119910) 119904 (119910) 119889119910 = 0 (87)



minusintinfin0

log119891 (1199112)1 + 1199112 119889119911 lt infin (88)







119890119886+(11988722) 119873sum119896=0

119887119896119888119896 = 119890(119903minus120575)119879 (89)










(92)


119896=0

119887119896119888119896 = 119890(119903minus120575)1198791198780 (93)



(94)





119896=1

119888119896119867119890119896minus1 (minus1198892) (96)

where

1198892 = 1119887 (log(1198780119870) + 119886) (97)


Q1015840 = 119878119879EQ119878119879 sdotQ = 119890119883119879

EQ119890119883119879 sdotQ (98)


Then

EQ (1198781198791119878119879gt119870) = (EQ119878119879)EQ1015840 (1119878119879gt119870) (99)

Since119883119879


119896=1

1198881015840119896119867119890119896minus1 (minus1198891) (100)

where

1198881015840119896 = 1sum119873119895=0 119887119895119888119895

119873sumℓ=119896

(ℓ119896) 119887ℓminus119896119888ℓ 119896 = 1 1198731198891 = 1119887 [log(1198780119870) + 119886 + 1198872] = 1198892 + 119887

(101)

Hence


119896=1


119896=1

119888119896119867119890119896minus1 (minus1198892) (102)

Using




where

119888lowast119896 = 1198881015840119896 119873sum119896=0

119887119896119888119896 = 119873sumℓ=119896

(ℓ119896) 119887ℓminus119896119888ℓ 119896 = 1 119873 (105)



1198750 = 1198620 minus 1198780 + 119870119890minus119903119879 (106)


4sum119896=1



(107)

This ends the proof



Δ = 12059711986201205971198780= 119890minus120575119879(Φ(1198891) + 120601 (1198891) 119873sum

119896=1

1198881015840119896119867119890119896minus1 (minus1198891))

120574 = 1205972119862012059711987820 = 119890119886+(11988722)minus1199031198791198871198780 120601 (1198891) 119873sum119896=0

119888119896119867119890119896 (minus1198892) 120588 = 1205971198620120597119903= 119870119879119890minus119903119879(Φ(1198892) + 120601 (1198892) 119873sum

119896=1





1198881015840119896119867119890119896minus1 (minus1198891)) 119895 ge 1

(108)





120597120597119904E (119904119880 minus 119870)+ = E (1198801119904119880gt119870) (109)


2

1199043 119891119880 (119870119904 ) (110)

From 119878119879 = 1198780119890119883119879 and (100)


119896=1

1198881015840119896119867119890119896minus1 (minus1198891)) (111)

From the lemma

120574 = 1205972120597211987801198620 = 119890minus119903119879120597212059721198780EQ (1198780119880 minus 119870)+

= 1198702119890minus11990311987911987830 119891119880 (1198701198780) 119880 = 119890119883(112)

where 119883119879


119891119880 (119906) = 1119887119906119891119866 (1119887 (log 119906 minus 119886))= 1119887119906120601 (119910)

119873sum119896=0

119888119896119867119890119896 (119910)1003816100381610038161003816100381610038161003816100381610038161003816119910=(1119887)(log 119906minus119886)

(113)


120574 = 119870119890minus11990311987911988711987820 120601 (1198892) 119873sum119896=0

119888119896119867119890119896 (minus1198892)= 119890119886+(11988722)minus1199031198791198871198780 120601 (1198891) 119873sum

119896=0

119888119896119867119890119896 (minus1198892) (114)

Next




Now from (89)



119896=1

119888119896119867119890119896minus1 (minus1198892)] (118)


Define

119898(119887) = E119890119887119866minus(11988722) = sum119896=0

1198871198961198881198961198981015840 (119887) = sum

119896=1

119896119887119896minus1119888119896 = 119888lowast1 (120)


119890119886 = 119890(119903minus120575)119879minus(11988722)119898(119887) (121)


1198780119890minus120575119879EQ [( 120597120597119887 119890119887119866minus(11988722)

119898(119887) ) 1119866gtminus1198892]


(122)


EQ (1198901198871198661119866gtminus119910) = 11989011988722 [119898 (119887)Φ (119910 + 119887)

+ 120601 (119910 + 119887) 119873sum119896=1


To calculate


we need (see (105))

120597119888lowast119896120597119887 = 120597120597119887119873sumℓ=119896


120597119888lowast119873120597119887 = 0(125)




1198881015840119896119867119890119896 (minus119910 minus 119887)]

(126)



120597119886120597119888119895 119890119886+(11988722)

119873sum119896=0

119887119896119888119896 + 119887119895119890119886+(11988722) = 0or 120597119886120597119888119895 = minus

119887119895sum119873119896=0 119887119896119888119896 = minus119887

119895119890119886+(11988722)+(120575minus119903)119879(127)

Next


infin

minus1198892

(1198780119890119886+119887119910 minus 119870)120601 (119910) 119873sum119896=0

119888119896119867119890119896 (119910) 119889119910= 119890minus119903119879intinfin

minus1198892

1198780 120597119886120597119888119895 119890119886+119887119910120601 (119910)119873sum119896=0

119888119896119867119890119896 (119910) 119889119910+ 119890minus119903119879intinfin

minus1198892

(1198780119890119886+119887119910 minus 119870)120601 (119910)119867119890119895 (119910) 119889119910

(128)



119896=0


119910119892(119895) (119909) 120601 (119909) 119889119909

(129)




minus1198892


119896=1


(130)




119861119899 = 119875 119899prod119905=1

[1 +max(120572( 119878119905119878119905minus1 minus 1) 0)] (131)





119877119905 Qsim GC (119886 119887 ) 119905 = 1 119899 (133)


119905=1





1198810 = 119875 (119890minus119903 + 1205721198620)119899 (135)


1205971198810120597120585 = 120572119875 (119890minus119903 + 1205721198620)119899minus1 1205971198620120597120585 (136)


1205972119862012059711987820 = 1205722119875 (119890minus119903 + 1205721198620)119899minus2 (12059711986201205971198780 )2

+ 120572119875 (119890minus119903 + 1205721198620)119899minus1 1205972119862012059711987820 (137)



119886 = 00687119887 = 016851198883 = minus011501198884 = 003598

(138)


119904 = 61198883 = minus06898119896 = 241198884 = 08634 (139)



0

1

2

3

4

k


s


minus1

0

1

0

2

4 104

106

108

110

Option

k s







(1119887)(log(1198701198780)minus1198861015840)(11987801198901198861015840+119887119909 minus 119870)120601 (119909)

sdot ( 119873sum119896=0

119888119896119867119890119896 (119909))119889119909(140)


(1119887)(log(1198701198780)minus1198861015840)(11987801198901198861015840+119887119909 minus 119870)120601 (119909)

sdot 119867119890119895 (119909) 119889119909(141)



119886 = 01174119887 = 015951198881 = minus030536756952010661198882 = 0095420793734891531198883 = minus0123839711263352431198884 = 006120331530131559

(142)





119901 (119909) = 2119898sum119895=0

120575119895119867119890119895 (119909) (143)


119886 = minus02445119887 = 019401198881 = 16140




1198882 = 117331198883 = 042531198884 = 007320

(144)




(145)


k

s

01234

105

110

Option

0500

minus05









Competing Interests


Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119884 = 119883 minus E119883radicVar119883 (46)




119896=0

119888119896119867119890119896 (47)

where

119888119896 = 1119896 intinfin

minusinfin119867119890119896 (119909) 119891 (119909) 119889119909 = 1119896E119867119890119896 (119883) (48)


12 (119891 (119909minus) + 119891 (119909+)) (49)


intinfinminusinfin

11989011990924119891 (119909) 119889119909 lt infin (50)



ℎ119902119896 = E119867119890119896 (119883) 119883 sim N (0 1199022) (51)


ℎ119902119896 = 0 if 119896 is odd

11989621198962 (1198962) (1199022 minus 1)1198962 if 119896 is even (52)


120601 (119909) infinsum119896=0

ℎ119902119896119896 119867119890119896 (119909)= 120601 (119909) infinsum

119899=0

12119899119899 (1199022 minus 1)1198991198671198902119899 (119909) (53)


infinsum119899=0

(minus1)119899 (2119899)22119899 (119899)2 (1199022 minus 1)119899 (54)


(2119899)22119899 (119899)2 sim 1radic120587119899 (55)



119892119901 (119909) = 119901120601 (119909) + (1 minus 119901) 1119901120601(119909119901) (56)



119884119901 = 119883119901


= 119883119901120590119901 1205902119901 = Var119883119901 = 119901 + 1199012 minus 1199013

(57)


119888119901119896 = 1119896E119867119890119896 (119884119901)= 1119896 (119901ℎ1120590119901 119896 + (1 minus 119901) ℎ119901120590119901 119896)

(58)





1Var119883119901

= 103 + 032 minus 033 = 275482 gt 2 (60)


119902 = 112059003 997904rArr 1198882 = 0119902 = 14 997904rArr 1198882 = minus014426

(61)

minus2 minus1 1 2

minus4

minus2

2

4

6


minus2 minus1 1 2

02

04

06

08

10



E119890119902119883119891 (119883) = 119890120583119902+(12)12059021199022E119891 (119883 + 1205902119902) (62)


1198751015840 = 119890119902119883EP119890119902119883 sdot 119875 (63)


119873sum119896=0

1198871198961198881015840119896119904119896 = sum119873119896=0 119887119896119888119896 (119902 + 119904)119896sum119873

119895=0 119887119895119888119895119902119895 119904 isin R (64)


or more precisely

1198881015840119896 = 1sum119873119895=0 119887119895119888119895119902119895

119873sumℓ=119896

(ℓ119896) 119887ℓminus119896119888ℓ119902ℓminus119896 (65)


E1198751015840119890119904119883 = 1


119896=0

119887119896119904119896 1sum119873119895=0 119887119895119888119895119902119895

sdot 119873sumℓ=119896

(ℓ119896) 119887ℓminus119896119888ℓ119902ℓminus119896(66)


11988810158401 = 311988721198883 + 4119887311988841 + 11988731198883 + 11988741198884 11988810158402 = 31198871198883 + 6119887211988841 + 11988731198883 + 11988741198884 11988810158403 = 1198883 + 411988711988841 + 11988731198883 + 11988741198884 11988810158404 = 11988841 + 11988731198883 + 11988741198884

(67)



(119883(119895)1 119883(119895)

119904 ) 119895 = 1 119899 (68)


119891119883 (119909) = 120601 (119886 119887 119909) 119901 (119909) (69)


E119892 (1198831 119883119904) = intR119904119892 (1199091 119909119904)

sdot 119904prod119896=1


119896=1

119901 (119883119896)] (70)


1 119883(119895)119904 ) 119895 = 1 119899 and then

using the estimator

119899 = 1119899119899sum119895=1

[119892 (119883(119895)1 119883(119895)

119904 ) 119904prod119896=1

119901 (119883(119895)

119896)] (71)




E119890119911(119883+119884)= 119890(1198861+1198862)119905+(11988721+11988722 )(11990522)(119873(1)sum

119896=0

119888(1)119896 119905119896)(119873(2)sum119896=0

119888(2)119896 119905119896) (72)


119883 + 119884 sim GC(1198861 + 1198862 radic11988721 + 11988722 1198881 119888119873(1)+119873(2)) 119888119896 = 119896sum

119895=0

119888(1)119895 119888(2)119896minus119895(73)


119885119899 = 1198831 + sdot sdot sdot + 119883119899 (74)


( 119873sum119896=0

119888119896119905119896)119899 = 119899119873sum

119896=0


(75)


and thus






119872(119911) = E119890119911119883 = 119890119886119911+1198872(11991122)120587 (119911) (77)


119872119899 (119911) = E1198901199111198831198991 (78)


119872119899 (119911)119899 = 119872(119911) 119911 isin C (79)





119891119871 (119911) = 1119911119891119884 (log 119911) 119911 gt 0 (80)





All the functions



119892120572 (119911) = 119891119871 (119911) + 120572120601 (119911) sin (2120587 log 119911) (83)


119901 (119910) = 119873sum119896=0

119888119896119867119890119896 (119910) (84)


119904 (119910) = sin (2120587119910) 1119910notin119869 (85)


119904 (119910 + 2119896120587) = 119904 (119910) 119904 (minus119910) = minus119904 (119910) (86)


intinfinminusinfin

119890119896119910120601 (119910) 119904 (119910) 119889119910 = 0 (87)



minusintinfin0

log119891 (1199112)1 + 1199112 119889119911 lt infin (88)







119890119886+(11988722) 119873sum119896=0

119887119896119888119896 = 119890(119903minus120575)119879 (89)










(92)


119896=0

119887119896119888119896 = 119890(119903minus120575)1198791198780 (93)



(94)





119896=1

119888119896119867119890119896minus1 (minus1198892) (96)

where

1198892 = 1119887 (log(1198780119870) + 119886) (97)


Q1015840 = 119878119879EQ119878119879 sdotQ = 119890119883119879

EQ119890119883119879 sdotQ (98)


Then

EQ (1198781198791119878119879gt119870) = (EQ119878119879)EQ1015840 (1119878119879gt119870) (99)

Since119883119879


119896=1

1198881015840119896119867119890119896minus1 (minus1198891) (100)

where

1198881015840119896 = 1sum119873119895=0 119887119895119888119895

119873sumℓ=119896

(ℓ119896) 119887ℓminus119896119888ℓ 119896 = 1 1198731198891 = 1119887 [log(1198780119870) + 119886 + 1198872] = 1198892 + 119887

(101)

Hence


119896=1


119896=1

119888119896119867119890119896minus1 (minus1198892) (102)

Using




where

119888lowast119896 = 1198881015840119896 119873sum119896=0

119887119896119888119896 = 119873sumℓ=119896

(ℓ119896) 119887ℓminus119896119888ℓ 119896 = 1 119873 (105)



1198750 = 1198620 minus 1198780 + 119870119890minus119903119879 (106)


4sum119896=1



(107)

This ends the proof



Δ = 12059711986201205971198780= 119890minus120575119879(Φ(1198891) + 120601 (1198891) 119873sum

119896=1

1198881015840119896119867119890119896minus1 (minus1198891))

120574 = 1205972119862012059711987820 = 119890119886+(11988722)minus1199031198791198871198780 120601 (1198891) 119873sum119896=0

119888119896119867119890119896 (minus1198892) 120588 = 1205971198620120597119903= 119870119879119890minus119903119879(Φ(1198892) + 120601 (1198892) 119873sum

119896=1





1198881015840119896119867119890119896minus1 (minus1198891)) 119895 ge 1

(108)





120597120597119904E (119904119880 minus 119870)+ = E (1198801119904119880gt119870) (109)


2

1199043 119891119880 (119870119904 ) (110)

From 119878119879 = 1198780119890119883119879 and (100)


119896=1

1198881015840119896119867119890119896minus1 (minus1198891)) (111)

From the lemma

120574 = 1205972120597211987801198620 = 119890minus119903119879120597212059721198780EQ (1198780119880 minus 119870)+

= 1198702119890minus11990311987911987830 119891119880 (1198701198780) 119880 = 119890119883(112)

where 119883119879


119891119880 (119906) = 1119887119906119891119866 (1119887 (log 119906 minus 119886))= 1119887119906120601 (119910)

119873sum119896=0

119888119896119867119890119896 (119910)1003816100381610038161003816100381610038161003816100381610038161003816119910=(1119887)(log 119906minus119886)

(113)


120574 = 119870119890minus11990311987911988711987820 120601 (1198892) 119873sum119896=0

119888119896119867119890119896 (minus1198892)= 119890119886+(11988722)minus1199031198791198871198780 120601 (1198891) 119873sum

119896=0

119888119896119867119890119896 (minus1198892) (114)

Next




Now from (89)



119896=1

119888119896119867119890119896minus1 (minus1198892)] (118)


Define

119898(119887) = E119890119887119866minus(11988722) = sum119896=0

1198871198961198881198961198981015840 (119887) = sum

119896=1

119896119887119896minus1119888119896 = 119888lowast1 (120)


119890119886 = 119890(119903minus120575)119879minus(11988722)119898(119887) (121)


1198780119890minus120575119879EQ [( 120597120597119887 119890119887119866minus(11988722)

119898(119887) ) 1119866gtminus1198892]


(122)


EQ (1198901198871198661119866gtminus119910) = 11989011988722 [119898 (119887)Φ (119910 + 119887)

+ 120601 (119910 + 119887) 119873sum119896=1


To calculate


we need (see (105))

120597119888lowast119896120597119887 = 120597120597119887119873sumℓ=119896


120597119888lowast119873120597119887 = 0(125)




1198881015840119896119867119890119896 (minus119910 minus 119887)]

(126)



120597119886120597119888119895 119890119886+(11988722)

119873sum119896=0

119887119896119888119896 + 119887119895119890119886+(11988722) = 0or 120597119886120597119888119895 = minus

119887119895sum119873119896=0 119887119896119888119896 = minus119887

119895119890119886+(11988722)+(120575minus119903)119879(127)

Next


infin

minus1198892

(1198780119890119886+119887119910 minus 119870)120601 (119910) 119873sum119896=0

119888119896119867119890119896 (119910) 119889119910= 119890minus119903119879intinfin

minus1198892

1198780 120597119886120597119888119895 119890119886+119887119910120601 (119910)119873sum119896=0

119888119896119867119890119896 (119910) 119889119910+ 119890minus119903119879intinfin

minus1198892

(1198780119890119886+119887119910 minus 119870)120601 (119910)119867119890119895 (119910) 119889119910

(128)



119896=0


119910119892(119895) (119909) 120601 (119909) 119889119909

(129)




minus1198892


119896=1


(130)




119861119899 = 119875 119899prod119905=1

[1 +max(120572( 119878119905119878119905minus1 minus 1) 0)] (131)





119877119905 Qsim GC (119886 119887 ) 119905 = 1 119899 (133)


119905=1





1198810 = 119875 (119890minus119903 + 1205721198620)119899 (135)


1205971198810120597120585 = 120572119875 (119890minus119903 + 1205721198620)119899minus1 1205971198620120597120585 (136)


1205972119862012059711987820 = 1205722119875 (119890minus119903 + 1205721198620)119899minus2 (12059711986201205971198780 )2

+ 120572119875 (119890minus119903 + 1205721198620)119899minus1 1205972119862012059711987820 (137)



119886 = 00687119887 = 016851198883 = minus011501198884 = 003598

(138)


119904 = 61198883 = minus06898119896 = 241198884 = 08634 (139)



0

1

2

3

4

k


s


minus1

0

1

0

2

4 104

106

108

110

Option

k s







(1119887)(log(1198701198780)minus1198861015840)(11987801198901198861015840+119887119909 minus 119870)120601 (119909)

sdot ( 119873sum119896=0

119888119896119867119890119896 (119909))119889119909(140)


(1119887)(log(1198701198780)minus1198861015840)(11987801198901198861015840+119887119909 minus 119870)120601 (119909)

sdot 119867119890119895 (119909) 119889119909(141)



119886 = 01174119887 = 015951198881 = minus030536756952010661198882 = 0095420793734891531198883 = minus0123839711263352431198884 = 006120331530131559

(142)





119901 (119909) = 2119898sum119895=0

120575119895119867119890119895 (119909) (143)


119886 = minus02445119887 = 019401198881 = 16140




1198882 = 117331198883 = 042531198884 = 007320

(144)




(145)


k

s

01234

105

110

Option

0500

minus05









Competing Interests


Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119884119901 = 119883119901


= 119883119901120590119901 1205902119901 = Var119883119901 = 119901 + 1199012 minus 1199013

(57)


119888119901119896 = 1119896E119867119890119896 (119884119901)= 1119896 (119901ℎ1120590119901 119896 + (1 minus 119901) ℎ119901120590119901 119896)

(58)





1Var119883119901

= 103 + 032 minus 033 = 275482 gt 2 (60)


119902 = 112059003 997904rArr 1198882 = 0119902 = 14 997904rArr 1198882 = minus014426

(61)

minus2 minus1 1 2

minus4

minus2

2

4

6


minus2 minus1 1 2

02

04

06

08

10



E119890119902119883119891 (119883) = 119890120583119902+(12)12059021199022E119891 (119883 + 1205902119902) (62)


1198751015840 = 119890119902119883EP119890119902119883 sdot 119875 (63)


119873sum119896=0

1198871198961198881015840119896119904119896 = sum119873119896=0 119887119896119888119896 (119902 + 119904)119896sum119873

119895=0 119887119895119888119895119902119895 119904 isin R (64)


or more precisely

1198881015840119896 = 1sum119873119895=0 119887119895119888119895119902119895

119873sumℓ=119896

(ℓ119896) 119887ℓminus119896119888ℓ119902ℓminus119896 (65)


E1198751015840119890119904119883 = 1


119896=0

119887119896119904119896 1sum119873119895=0 119887119895119888119895119902119895

sdot 119873sumℓ=119896

(ℓ119896) 119887ℓminus119896119888ℓ119902ℓminus119896(66)


11988810158401 = 311988721198883 + 4119887311988841 + 11988731198883 + 11988741198884 11988810158402 = 31198871198883 + 6119887211988841 + 11988731198883 + 11988741198884 11988810158403 = 1198883 + 411988711988841 + 11988731198883 + 11988741198884 11988810158404 = 11988841 + 11988731198883 + 11988741198884

(67)



(119883(119895)1 119883(119895)

119904 ) 119895 = 1 119899 (68)


119891119883 (119909) = 120601 (119886 119887 119909) 119901 (119909) (69)


E119892 (1198831 119883119904) = intR119904119892 (1199091 119909119904)

sdot 119904prod119896=1


119896=1

119901 (119883119896)] (70)


1 119883(119895)119904 ) 119895 = 1 119899 and then

using the estimator

119899 = 1119899119899sum119895=1

[119892 (119883(119895)1 119883(119895)

119904 ) 119904prod119896=1

119901 (119883(119895)

119896)] (71)




E119890119911(119883+119884)= 119890(1198861+1198862)119905+(11988721+11988722 )(11990522)(119873(1)sum

119896=0

119888(1)119896 119905119896)(119873(2)sum119896=0

119888(2)119896 119905119896) (72)


119883 + 119884 sim GC(1198861 + 1198862 radic11988721 + 11988722 1198881 119888119873(1)+119873(2)) 119888119896 = 119896sum

119895=0

119888(1)119895 119888(2)119896minus119895(73)


119885119899 = 1198831 + sdot sdot sdot + 119883119899 (74)


( 119873sum119896=0

119888119896119905119896)119899 = 119899119873sum

119896=0


(75)


and thus






119872(119911) = E119890119911119883 = 119890119886119911+1198872(11991122)120587 (119911) (77)


119872119899 (119911) = E1198901199111198831198991 (78)


119872119899 (119911)119899 = 119872(119911) 119911 isin C (79)





119891119871 (119911) = 1119911119891119884 (log 119911) 119911 gt 0 (80)





All the functions



119892120572 (119911) = 119891119871 (119911) + 120572120601 (119911) sin (2120587 log 119911) (83)


119901 (119910) = 119873sum119896=0

119888119896119867119890119896 (119910) (84)


119904 (119910) = sin (2120587119910) 1119910notin119869 (85)


119904 (119910 + 2119896120587) = 119904 (119910) 119904 (minus119910) = minus119904 (119910) (86)


intinfinminusinfin

119890119896119910120601 (119910) 119904 (119910) 119889119910 = 0 (87)



minusintinfin0

log119891 (1199112)1 + 1199112 119889119911 lt infin (88)







119890119886+(11988722) 119873sum119896=0

119887119896119888119896 = 119890(119903minus120575)119879 (89)










(92)


119896=0

119887119896119888119896 = 119890(119903minus120575)1198791198780 (93)



(94)





119896=1

119888119896119867119890119896minus1 (minus1198892) (96)

where

1198892 = 1119887 (log(1198780119870) + 119886) (97)


Q1015840 = 119878119879EQ119878119879 sdotQ = 119890119883119879

EQ119890119883119879 sdotQ (98)


Then

EQ (1198781198791119878119879gt119870) = (EQ119878119879)EQ1015840 (1119878119879gt119870) (99)

Since119883119879


119896=1

1198881015840119896119867119890119896minus1 (minus1198891) (100)

where

1198881015840119896 = 1sum119873119895=0 119887119895119888119895

119873sumℓ=119896

(ℓ119896) 119887ℓminus119896119888ℓ 119896 = 1 1198731198891 = 1119887 [log(1198780119870) + 119886 + 1198872] = 1198892 + 119887

(101)

Hence


119896=1


119896=1

119888119896119867119890119896minus1 (minus1198892) (102)

Using




where

119888lowast119896 = 1198881015840119896 119873sum119896=0

119887119896119888119896 = 119873sumℓ=119896

(ℓ119896) 119887ℓminus119896119888ℓ 119896 = 1 119873 (105)



1198750 = 1198620 minus 1198780 + 119870119890minus119903119879 (106)


4sum119896=1



(107)

This ends the proof



Δ = 12059711986201205971198780= 119890minus120575119879(Φ(1198891) + 120601 (1198891) 119873sum

119896=1

1198881015840119896119867119890119896minus1 (minus1198891))

120574 = 1205972119862012059711987820 = 119890119886+(11988722)minus1199031198791198871198780 120601 (1198891) 119873sum119896=0

119888119896119867119890119896 (minus1198892) 120588 = 1205971198620120597119903= 119870119879119890minus119903119879(Φ(1198892) + 120601 (1198892) 119873sum

119896=1





1198881015840119896119867119890119896minus1 (minus1198891)) 119895 ge 1

(108)





120597120597119904E (119904119880 minus 119870)+ = E (1198801119904119880gt119870) (109)


2

1199043 119891119880 (119870119904 ) (110)

From 119878119879 = 1198780119890119883119879 and (100)


119896=1

1198881015840119896119867119890119896minus1 (minus1198891)) (111)

From the lemma

120574 = 1205972120597211987801198620 = 119890minus119903119879120597212059721198780EQ (1198780119880 minus 119870)+

= 1198702119890minus11990311987911987830 119891119880 (1198701198780) 119880 = 119890119883(112)

where 119883119879


119891119880 (119906) = 1119887119906119891119866 (1119887 (log 119906 minus 119886))= 1119887119906120601 (119910)

119873sum119896=0

119888119896119867119890119896 (119910)1003816100381610038161003816100381610038161003816100381610038161003816119910=(1119887)(log 119906minus119886)

(113)


120574 = 119870119890minus11990311987911988711987820 120601 (1198892) 119873sum119896=0

119888119896119867119890119896 (minus1198892)= 119890119886+(11988722)minus1199031198791198871198780 120601 (1198891) 119873sum

119896=0

119888119896119867119890119896 (minus1198892) (114)

Next




Now from (89)



119896=1

119888119896119867119890119896minus1 (minus1198892)] (118)


Define

119898(119887) = E119890119887119866minus(11988722) = sum119896=0

1198871198961198881198961198981015840 (119887) = sum

119896=1

119896119887119896minus1119888119896 = 119888lowast1 (120)


119890119886 = 119890(119903minus120575)119879minus(11988722)119898(119887) (121)


1198780119890minus120575119879EQ [( 120597120597119887 119890119887119866minus(11988722)

119898(119887) ) 1119866gtminus1198892]


(122)


EQ (1198901198871198661119866gtminus119910) = 11989011988722 [119898 (119887)Φ (119910 + 119887)

+ 120601 (119910 + 119887) 119873sum119896=1


To calculate


we need (see (105))

120597119888lowast119896120597119887 = 120597120597119887119873sumℓ=119896


120597119888lowast119873120597119887 = 0(125)




1198881015840119896119867119890119896 (minus119910 minus 119887)]

(126)



120597119886120597119888119895 119890119886+(11988722)

119873sum119896=0

119887119896119888119896 + 119887119895119890119886+(11988722) = 0or 120597119886120597119888119895 = minus

119887119895sum119873119896=0 119887119896119888119896 = minus119887

119895119890119886+(11988722)+(120575minus119903)119879(127)

Next


infin

minus1198892

(1198780119890119886+119887119910 minus 119870)120601 (119910) 119873sum119896=0

119888119896119867119890119896 (119910) 119889119910= 119890minus119903119879intinfin

minus1198892

1198780 120597119886120597119888119895 119890119886+119887119910120601 (119910)119873sum119896=0

119888119896119867119890119896 (119910) 119889119910+ 119890minus119903119879intinfin

minus1198892

(1198780119890119886+119887119910 minus 119870)120601 (119910)119867119890119895 (119910) 119889119910

(128)



119896=0


119910119892(119895) (119909) 120601 (119909) 119889119909

(129)




minus1198892


119896=1


(130)




119861119899 = 119875 119899prod119905=1

[1 +max(120572( 119878119905119878119905minus1 minus 1) 0)] (131)





119877119905 Qsim GC (119886 119887 ) 119905 = 1 119899 (133)


119905=1





1198810 = 119875 (119890minus119903 + 1205721198620)119899 (135)


1205971198810120597120585 = 120572119875 (119890minus119903 + 1205721198620)119899minus1 1205971198620120597120585 (136)


1205972119862012059711987820 = 1205722119875 (119890minus119903 + 1205721198620)119899minus2 (12059711986201205971198780 )2

+ 120572119875 (119890minus119903 + 1205721198620)119899minus1 1205972119862012059711987820 (137)



119886 = 00687119887 = 016851198883 = minus011501198884 = 003598

(138)


119904 = 61198883 = minus06898119896 = 241198884 = 08634 (139)



0

1

2

3

4

k


s


minus1

0

1

0

2

4 104

106

108

110

Option

k s







(1119887)(log(1198701198780)minus1198861015840)(11987801198901198861015840+119887119909 minus 119870)120601 (119909)

sdot ( 119873sum119896=0

119888119896119867119890119896 (119909))119889119909(140)


(1119887)(log(1198701198780)minus1198861015840)(11987801198901198861015840+119887119909 minus 119870)120601 (119909)

sdot 119867119890119895 (119909) 119889119909(141)



119886 = 01174119887 = 015951198881 = minus030536756952010661198882 = 0095420793734891531198883 = minus0123839711263352431198884 = 006120331530131559

(142)





119901 (119909) = 2119898sum119895=0

120575119895119867119890119895 (119909) (143)


119886 = minus02445119887 = 019401198881 = 16140




1198882 = 117331198883 = 042531198884 = 007320

(144)




(145)


k

s

01234

105

110

Option

0500

minus05









Competing Interests


Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










or more precisely

1198881015840119896 = 1sum119873119895=0 119887119895119888119895119902119895

119873sumℓ=119896

(ℓ119896) 119887ℓminus119896119888ℓ119902ℓminus119896 (65)


E1198751015840119890119904119883 = 1


119896=0

119887119896119904119896 1sum119873119895=0 119887119895119888119895119902119895

sdot 119873sumℓ=119896

(ℓ119896) 119887ℓminus119896119888ℓ119902ℓminus119896(66)


11988810158401 = 311988721198883 + 4119887311988841 + 11988731198883 + 11988741198884 11988810158402 = 31198871198883 + 6119887211988841 + 11988731198883 + 11988741198884 11988810158403 = 1198883 + 411988711988841 + 11988731198883 + 11988741198884 11988810158404 = 11988841 + 11988731198883 + 11988741198884

(67)



(119883(119895)1 119883(119895)

119904 ) 119895 = 1 119899 (68)


119891119883 (119909) = 120601 (119886 119887 119909) 119901 (119909) (69)


E119892 (1198831 119883119904) = intR119904119892 (1199091 119909119904)

sdot 119904prod119896=1


119896=1

119901 (119883119896)] (70)


1 119883(119895)119904 ) 119895 = 1 119899 and then

using the estimator

119899 = 1119899119899sum119895=1

[119892 (119883(119895)1 119883(119895)

119904 ) 119904prod119896=1

119901 (119883(119895)

119896)] (71)




E119890119911(119883+119884)= 119890(1198861+1198862)119905+(11988721+11988722 )(11990522)(119873(1)sum

119896=0

119888(1)119896 119905119896)(119873(2)sum119896=0

119888(2)119896 119905119896) (72)


119883 + 119884 sim GC(1198861 + 1198862 radic11988721 + 11988722 1198881 119888119873(1)+119873(2)) 119888119896 = 119896sum

119895=0

119888(1)119895 119888(2)119896minus119895(73)


119885119899 = 1198831 + sdot sdot sdot + 119883119899 (74)


( 119873sum119896=0

119888119896119905119896)119899 = 119899119873sum

119896=0


(75)


and thus






119872(119911) = E119890119911119883 = 119890119886119911+1198872(11991122)120587 (119911) (77)


119872119899 (119911) = E1198901199111198831198991 (78)


119872119899 (119911)119899 = 119872(119911) 119911 isin C (79)





119891119871 (119911) = 1119911119891119884 (log 119911) 119911 gt 0 (80)





All the functions



119892120572 (119911) = 119891119871 (119911) + 120572120601 (119911) sin (2120587 log 119911) (83)


119901 (119910) = 119873sum119896=0

119888119896119867119890119896 (119910) (84)


119904 (119910) = sin (2120587119910) 1119910notin119869 (85)


119904 (119910 + 2119896120587) = 119904 (119910) 119904 (minus119910) = minus119904 (119910) (86)


intinfinminusinfin

119890119896119910120601 (119910) 119904 (119910) 119889119910 = 0 (87)



minusintinfin0

log119891 (1199112)1 + 1199112 119889119911 lt infin (88)







119890119886+(11988722) 119873sum119896=0

119887119896119888119896 = 119890(119903minus120575)119879 (89)










(92)


119896=0

119887119896119888119896 = 119890(119903minus120575)1198791198780 (93)



(94)





119896=1

119888119896119867119890119896minus1 (minus1198892) (96)

where

1198892 = 1119887 (log(1198780119870) + 119886) (97)


Q1015840 = 119878119879EQ119878119879 sdotQ = 119890119883119879

EQ119890119883119879 sdotQ (98)


Then

EQ (1198781198791119878119879gt119870) = (EQ119878119879)EQ1015840 (1119878119879gt119870) (99)

Since119883119879


119896=1

1198881015840119896119867119890119896minus1 (minus1198891) (100)

where

1198881015840119896 = 1sum119873119895=0 119887119895119888119895

119873sumℓ=119896

(ℓ119896) 119887ℓminus119896119888ℓ 119896 = 1 1198731198891 = 1119887 [log(1198780119870) + 119886 + 1198872] = 1198892 + 119887

(101)

Hence


119896=1


119896=1

119888119896119867119890119896minus1 (minus1198892) (102)

Using




where

119888lowast119896 = 1198881015840119896 119873sum119896=0

119887119896119888119896 = 119873sumℓ=119896

(ℓ119896) 119887ℓminus119896119888ℓ 119896 = 1 119873 (105)



1198750 = 1198620 minus 1198780 + 119870119890minus119903119879 (106)


4sum119896=1



(107)

This ends the proof



Δ = 12059711986201205971198780= 119890minus120575119879(Φ(1198891) + 120601 (1198891) 119873sum

119896=1

1198881015840119896119867119890119896minus1 (minus1198891))

120574 = 1205972119862012059711987820 = 119890119886+(11988722)minus1199031198791198871198780 120601 (1198891) 119873sum119896=0

119888119896119867119890119896 (minus1198892) 120588 = 1205971198620120597119903= 119870119879119890minus119903119879(Φ(1198892) + 120601 (1198892) 119873sum

119896=1





1198881015840119896119867119890119896minus1 (minus1198891)) 119895 ge 1

(108)





120597120597119904E (119904119880 minus 119870)+ = E (1198801119904119880gt119870) (109)


2

1199043 119891119880 (119870119904 ) (110)

From 119878119879 = 1198780119890119883119879 and (100)


119896=1

1198881015840119896119867119890119896minus1 (minus1198891)) (111)

From the lemma

120574 = 1205972120597211987801198620 = 119890minus119903119879120597212059721198780EQ (1198780119880 minus 119870)+

= 1198702119890minus11990311987911987830 119891119880 (1198701198780) 119880 = 119890119883(112)

where 119883119879


119891119880 (119906) = 1119887119906119891119866 (1119887 (log 119906 minus 119886))= 1119887119906120601 (119910)

119873sum119896=0

119888119896119867119890119896 (119910)1003816100381610038161003816100381610038161003816100381610038161003816119910=(1119887)(log 119906minus119886)

(113)


120574 = 119870119890minus11990311987911988711987820 120601 (1198892) 119873sum119896=0

119888119896119867119890119896 (minus1198892)= 119890119886+(11988722)minus1199031198791198871198780 120601 (1198891) 119873sum

119896=0

119888119896119867119890119896 (minus1198892) (114)

Next




Now from (89)



119896=1

119888119896119867119890119896minus1 (minus1198892)] (118)


Define

119898(119887) = E119890119887119866minus(11988722) = sum119896=0

1198871198961198881198961198981015840 (119887) = sum

119896=1

119896119887119896minus1119888119896 = 119888lowast1 (120)


119890119886 = 119890(119903minus120575)119879minus(11988722)119898(119887) (121)


1198780119890minus120575119879EQ [( 120597120597119887 119890119887119866minus(11988722)

119898(119887) ) 1119866gtminus1198892]


(122)


EQ (1198901198871198661119866gtminus119910) = 11989011988722 [119898 (119887)Φ (119910 + 119887)

+ 120601 (119910 + 119887) 119873sum119896=1


To calculate


we need (see (105))

120597119888lowast119896120597119887 = 120597120597119887119873sumℓ=119896


120597119888lowast119873120597119887 = 0(125)




1198881015840119896119867119890119896 (minus119910 minus 119887)]

(126)



120597119886120597119888119895 119890119886+(11988722)

119873sum119896=0

119887119896119888119896 + 119887119895119890119886+(11988722) = 0or 120597119886120597119888119895 = minus

119887119895sum119873119896=0 119887119896119888119896 = minus119887

119895119890119886+(11988722)+(120575minus119903)119879(127)

Next


infin

minus1198892

(1198780119890119886+119887119910 minus 119870)120601 (119910) 119873sum119896=0

119888119896119867119890119896 (119910) 119889119910= 119890minus119903119879intinfin

minus1198892

1198780 120597119886120597119888119895 119890119886+119887119910120601 (119910)119873sum119896=0

119888119896119867119890119896 (119910) 119889119910+ 119890minus119903119879intinfin

minus1198892

(1198780119890119886+119887119910 minus 119870)120601 (119910)119867119890119895 (119910) 119889119910

(128)



119896=0


119910119892(119895) (119909) 120601 (119909) 119889119909

(129)




minus1198892


119896=1


(130)




119861119899 = 119875 119899prod119905=1

[1 +max(120572( 119878119905119878119905minus1 minus 1) 0)] (131)





119877119905 Qsim GC (119886 119887 ) 119905 = 1 119899 (133)


119905=1





1198810 = 119875 (119890minus119903 + 1205721198620)119899 (135)


1205971198810120597120585 = 120572119875 (119890minus119903 + 1205721198620)119899minus1 1205971198620120597120585 (136)


1205972119862012059711987820 = 1205722119875 (119890minus119903 + 1205721198620)119899minus2 (12059711986201205971198780 )2

+ 120572119875 (119890minus119903 + 1205721198620)119899minus1 1205972119862012059711987820 (137)



119886 = 00687119887 = 016851198883 = minus011501198884 = 003598

(138)


119904 = 61198883 = minus06898119896 = 241198884 = 08634 (139)



0

1

2

3

4

k


s


minus1

0

1

0

2

4 104

106

108

110

Option

k s







(1119887)(log(1198701198780)minus1198861015840)(11987801198901198861015840+119887119909 minus 119870)120601 (119909)

sdot ( 119873sum119896=0

119888119896119867119890119896 (119909))119889119909(140)


(1119887)(log(1198701198780)minus1198861015840)(11987801198901198861015840+119887119909 minus 119870)120601 (119909)

sdot 119867119890119895 (119909) 119889119909(141)



119886 = 01174119887 = 015951198881 = minus030536756952010661198882 = 0095420793734891531198883 = minus0123839711263352431198884 = 006120331530131559

(142)





119901 (119909) = 2119898sum119895=0

120575119895119867119890119895 (119909) (143)


119886 = minus02445119887 = 019401198881 = 16140




1198882 = 117331198883 = 042531198884 = 007320

(144)




(145)


k

s

01234

105

110

Option

0500

minus05









Competing Interests


Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










and thus






119872(119911) = E119890119911119883 = 119890119886119911+1198872(11991122)120587 (119911) (77)


119872119899 (119911) = E1198901199111198831198991 (78)


119872119899 (119911)119899 = 119872(119911) 119911 isin C (79)





119891119871 (119911) = 1119911119891119884 (log 119911) 119911 gt 0 (80)





All the functions



119892120572 (119911) = 119891119871 (119911) + 120572120601 (119911) sin (2120587 log 119911) (83)


119901 (119910) = 119873sum119896=0

119888119896119867119890119896 (119910) (84)


119904 (119910) = sin (2120587119910) 1119910notin119869 (85)


119904 (119910 + 2119896120587) = 119904 (119910) 119904 (minus119910) = minus119904 (119910) (86)


intinfinminusinfin

119890119896119910120601 (119910) 119904 (119910) 119889119910 = 0 (87)



minusintinfin0

log119891 (1199112)1 + 1199112 119889119911 lt infin (88)







119890119886+(11988722) 119873sum119896=0

119887119896119888119896 = 119890(119903minus120575)119879 (89)










(92)


119896=0

119887119896119888119896 = 119890(119903minus120575)1198791198780 (93)



(94)





119896=1

119888119896119867119890119896minus1 (minus1198892) (96)

where

1198892 = 1119887 (log(1198780119870) + 119886) (97)


Q1015840 = 119878119879EQ119878119879 sdotQ = 119890119883119879

EQ119890119883119879 sdotQ (98)


Then

EQ (1198781198791119878119879gt119870) = (EQ119878119879)EQ1015840 (1119878119879gt119870) (99)

Since119883119879


119896=1

1198881015840119896119867119890119896minus1 (minus1198891) (100)

where

1198881015840119896 = 1sum119873119895=0 119887119895119888119895

119873sumℓ=119896

(ℓ119896) 119887ℓminus119896119888ℓ 119896 = 1 1198731198891 = 1119887 [log(1198780119870) + 119886 + 1198872] = 1198892 + 119887

(101)

Hence


119896=1


119896=1

119888119896119867119890119896minus1 (minus1198892) (102)

Using




where

119888lowast119896 = 1198881015840119896 119873sum119896=0

119887119896119888119896 = 119873sumℓ=119896

(ℓ119896) 119887ℓminus119896119888ℓ 119896 = 1 119873 (105)



1198750 = 1198620 minus 1198780 + 119870119890minus119903119879 (106)


4sum119896=1



(107)

This ends the proof



Δ = 12059711986201205971198780= 119890minus120575119879(Φ(1198891) + 120601 (1198891) 119873sum

119896=1

1198881015840119896119867119890119896minus1 (minus1198891))

120574 = 1205972119862012059711987820 = 119890119886+(11988722)minus1199031198791198871198780 120601 (1198891) 119873sum119896=0

119888119896119867119890119896 (minus1198892) 120588 = 1205971198620120597119903= 119870119879119890minus119903119879(Φ(1198892) + 120601 (1198892) 119873sum

119896=1





1198881015840119896119867119890119896minus1 (minus1198891)) 119895 ge 1

(108)





120597120597119904E (119904119880 minus 119870)+ = E (1198801119904119880gt119870) (109)


2

1199043 119891119880 (119870119904 ) (110)

From 119878119879 = 1198780119890119883119879 and (100)


119896=1

1198881015840119896119867119890119896minus1 (minus1198891)) (111)

From the lemma

120574 = 1205972120597211987801198620 = 119890minus119903119879120597212059721198780EQ (1198780119880 minus 119870)+

= 1198702119890minus11990311987911987830 119891119880 (1198701198780) 119880 = 119890119883(112)

where 119883119879


119891119880 (119906) = 1119887119906119891119866 (1119887 (log 119906 minus 119886))= 1119887119906120601 (119910)

119873sum119896=0

119888119896119867119890119896 (119910)1003816100381610038161003816100381610038161003816100381610038161003816119910=(1119887)(log 119906minus119886)

(113)


120574 = 119870119890minus11990311987911988711987820 120601 (1198892) 119873sum119896=0

119888119896119867119890119896 (minus1198892)= 119890119886+(11988722)minus1199031198791198871198780 120601 (1198891) 119873sum

119896=0

119888119896119867119890119896 (minus1198892) (114)

Next




Now from (89)



119896=1

119888119896119867119890119896minus1 (minus1198892)] (118)


Define

119898(119887) = E119890119887119866minus(11988722) = sum119896=0

1198871198961198881198961198981015840 (119887) = sum

119896=1

119896119887119896minus1119888119896 = 119888lowast1 (120)


119890119886 = 119890(119903minus120575)119879minus(11988722)119898(119887) (121)


1198780119890minus120575119879EQ [( 120597120597119887 119890119887119866minus(11988722)

119898(119887) ) 1119866gtminus1198892]


(122)


EQ (1198901198871198661119866gtminus119910) = 11989011988722 [119898 (119887)Φ (119910 + 119887)

+ 120601 (119910 + 119887) 119873sum119896=1


To calculate


we need (see (105))

120597119888lowast119896120597119887 = 120597120597119887119873sumℓ=119896


120597119888lowast119873120597119887 = 0(125)




1198881015840119896119867119890119896 (minus119910 minus 119887)]

(126)



120597119886120597119888119895 119890119886+(11988722)

119873sum119896=0

119887119896119888119896 + 119887119895119890119886+(11988722) = 0or 120597119886120597119888119895 = minus

119887119895sum119873119896=0 119887119896119888119896 = minus119887

119895119890119886+(11988722)+(120575minus119903)119879(127)

Next


infin

minus1198892

(1198780119890119886+119887119910 minus 119870)120601 (119910) 119873sum119896=0

119888119896119867119890119896 (119910) 119889119910= 119890minus119903119879intinfin

minus1198892

1198780 120597119886120597119888119895 119890119886+119887119910120601 (119910)119873sum119896=0

119888119896119867119890119896 (119910) 119889119910+ 119890minus119903119879intinfin

minus1198892

(1198780119890119886+119887119910 minus 119870)120601 (119910)119867119890119895 (119910) 119889119910

(128)



119896=0


119910119892(119895) (119909) 120601 (119909) 119889119909

(129)




minus1198892


119896=1


(130)




119861119899 = 119875 119899prod119905=1

[1 +max(120572( 119878119905119878119905minus1 minus 1) 0)] (131)





119877119905 Qsim GC (119886 119887 ) 119905 = 1 119899 (133)


119905=1





1198810 = 119875 (119890minus119903 + 1205721198620)119899 (135)


1205971198810120597120585 = 120572119875 (119890minus119903 + 1205721198620)119899minus1 1205971198620120597120585 (136)


1205972119862012059711987820 = 1205722119875 (119890minus119903 + 1205721198620)119899minus2 (12059711986201205971198780 )2

+ 120572119875 (119890minus119903 + 1205721198620)119899minus1 1205972119862012059711987820 (137)



119886 = 00687119887 = 016851198883 = minus011501198884 = 003598

(138)


119904 = 61198883 = minus06898119896 = 241198884 = 08634 (139)



0

1

2

3

4

k


s


minus1

0

1

0

2

4 104

106

108

110

Option

k s







(1119887)(log(1198701198780)minus1198861015840)(11987801198901198861015840+119887119909 minus 119870)120601 (119909)

sdot ( 119873sum119896=0

119888119896119867119890119896 (119909))119889119909(140)


(1119887)(log(1198701198780)minus1198861015840)(11987801198901198861015840+119887119909 minus 119870)120601 (119909)

sdot 119867119890119895 (119909) 119889119909(141)



119886 = 01174119887 = 015951198881 = minus030536756952010661198882 = 0095420793734891531198883 = minus0123839711263352431198884 = 006120331530131559

(142)





119901 (119909) = 2119898sum119895=0

120575119895119867119890119895 (119909) (143)


119886 = minus02445119887 = 019401198881 = 16140




1198882 = 117331198883 = 042531198884 = 007320

(144)




(145)


k

s

01234

105

110

Option

0500

minus05









Competing Interests


Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











minusintinfin0

log119891 (1199112)1 + 1199112 119889119911 lt infin (88)







119890119886+(11988722) 119873sum119896=0

119887119896119888119896 = 119890(119903minus120575)119879 (89)










(92)


119896=0

119887119896119888119896 = 119890(119903minus120575)1198791198780 (93)



(94)





119896=1

119888119896119867119890119896minus1 (minus1198892) (96)

where

1198892 = 1119887 (log(1198780119870) + 119886) (97)


Q1015840 = 119878119879EQ119878119879 sdotQ = 119890119883119879

EQ119890119883119879 sdotQ (98)


Then

EQ (1198781198791119878119879gt119870) = (EQ119878119879)EQ1015840 (1119878119879gt119870) (99)

Since119883119879


119896=1

1198881015840119896119867119890119896minus1 (minus1198891) (100)

where

1198881015840119896 = 1sum119873119895=0 119887119895119888119895

119873sumℓ=119896

(ℓ119896) 119887ℓminus119896119888ℓ 119896 = 1 1198731198891 = 1119887 [log(1198780119870) + 119886 + 1198872] = 1198892 + 119887

(101)

Hence


119896=1


119896=1

119888119896119867119890119896minus1 (minus1198892) (102)

Using




where

119888lowast119896 = 1198881015840119896 119873sum119896=0

119887119896119888119896 = 119873sumℓ=119896

(ℓ119896) 119887ℓminus119896119888ℓ 119896 = 1 119873 (105)



1198750 = 1198620 minus 1198780 + 119870119890minus119903119879 (106)


4sum119896=1



(107)

This ends the proof



Δ = 12059711986201205971198780= 119890minus120575119879(Φ(1198891) + 120601 (1198891) 119873sum

119896=1

1198881015840119896119867119890119896minus1 (minus1198891))

120574 = 1205972119862012059711987820 = 119890119886+(11988722)minus1199031198791198871198780 120601 (1198891) 119873sum119896=0

119888119896119867119890119896 (minus1198892) 120588 = 1205971198620120597119903= 119870119879119890minus119903119879(Φ(1198892) + 120601 (1198892) 119873sum

119896=1





1198881015840119896119867119890119896minus1 (minus1198891)) 119895 ge 1

(108)





120597120597119904E (119904119880 minus 119870)+ = E (1198801119904119880gt119870) (109)


2

1199043 119891119880 (119870119904 ) (110)

From 119878119879 = 1198780119890119883119879 and (100)


119896=1

1198881015840119896119867119890119896minus1 (minus1198891)) (111)

From the lemma

120574 = 1205972120597211987801198620 = 119890minus119903119879120597212059721198780EQ (1198780119880 minus 119870)+

= 1198702119890minus11990311987911987830 119891119880 (1198701198780) 119880 = 119890119883(112)

where 119883119879


119891119880 (119906) = 1119887119906119891119866 (1119887 (log 119906 minus 119886))= 1119887119906120601 (119910)

119873sum119896=0

119888119896119867119890119896 (119910)1003816100381610038161003816100381610038161003816100381610038161003816119910=(1119887)(log 119906minus119886)

(113)


120574 = 119870119890minus11990311987911988711987820 120601 (1198892) 119873sum119896=0

119888119896119867119890119896 (minus1198892)= 119890119886+(11988722)minus1199031198791198871198780 120601 (1198891) 119873sum

119896=0

119888119896119867119890119896 (minus1198892) (114)

Next




Now from (89)



119896=1

119888119896119867119890119896minus1 (minus1198892)] (118)


Define

119898(119887) = E119890119887119866minus(11988722) = sum119896=0

1198871198961198881198961198981015840 (119887) = sum

119896=1

119896119887119896minus1119888119896 = 119888lowast1 (120)


119890119886 = 119890(119903minus120575)119879minus(11988722)119898(119887) (121)


1198780119890minus120575119879EQ [( 120597120597119887 119890119887119866minus(11988722)

119898(119887) ) 1119866gtminus1198892]


(122)


EQ (1198901198871198661119866gtminus119910) = 11989011988722 [119898 (119887)Φ (119910 + 119887)

+ 120601 (119910 + 119887) 119873sum119896=1


To calculate


we need (see (105))

120597119888lowast119896120597119887 = 120597120597119887119873sumℓ=119896


120597119888lowast119873120597119887 = 0(125)




1198881015840119896119867119890119896 (minus119910 minus 119887)]

(126)



120597119886120597119888119895 119890119886+(11988722)

119873sum119896=0

119887119896119888119896 + 119887119895119890119886+(11988722) = 0or 120597119886120597119888119895 = minus

119887119895sum119873119896=0 119887119896119888119896 = minus119887

119895119890119886+(11988722)+(120575minus119903)119879(127)

Next


infin

minus1198892

(1198780119890119886+119887119910 minus 119870)120601 (119910) 119873sum119896=0

119888119896119867119890119896 (119910) 119889119910= 119890minus119903119879intinfin

minus1198892

1198780 120597119886120597119888119895 119890119886+119887119910120601 (119910)119873sum119896=0

119888119896119867119890119896 (119910) 119889119910+ 119890minus119903119879intinfin

minus1198892

(1198780119890119886+119887119910 minus 119870)120601 (119910)119867119890119895 (119910) 119889119910

(128)



119896=0


119910119892(119895) (119909) 120601 (119909) 119889119909

(129)




minus1198892


119896=1


(130)




119861119899 = 119875 119899prod119905=1

[1 +max(120572( 119878119905119878119905minus1 minus 1) 0)] (131)





119877119905 Qsim GC (119886 119887 ) 119905 = 1 119899 (133)


119905=1





1198810 = 119875 (119890minus119903 + 1205721198620)119899 (135)


1205971198810120597120585 = 120572119875 (119890minus119903 + 1205721198620)119899minus1 1205971198620120597120585 (136)


1205972119862012059711987820 = 1205722119875 (119890minus119903 + 1205721198620)119899minus2 (12059711986201205971198780 )2

+ 120572119875 (119890minus119903 + 1205721198620)119899minus1 1205972119862012059711987820 (137)



119886 = 00687119887 = 016851198883 = minus011501198884 = 003598

(138)


119904 = 61198883 = minus06898119896 = 241198884 = 08634 (139)



0

1

2

3

4

k


s


minus1

0

1

0

2

4 104

106

108

110

Option

k s







(1119887)(log(1198701198780)minus1198861015840)(11987801198901198861015840+119887119909 minus 119870)120601 (119909)

sdot ( 119873sum119896=0

119888119896119867119890119896 (119909))119889119909(140)


(1119887)(log(1198701198780)minus1198861015840)(11987801198901198861015840+119887119909 minus 119870)120601 (119909)

sdot 119867119890119895 (119909) 119889119909(141)



119886 = 01174119887 = 015951198881 = minus030536756952010661198882 = 0095420793734891531198883 = minus0123839711263352431198884 = 006120331530131559

(142)





119901 (119909) = 2119898sum119895=0

120575119895119867119890119895 (119909) (143)


119886 = minus02445119887 = 019401198881 = 16140




1198882 = 117331198883 = 042531198884 = 007320

(144)




(145)


k

s

01234

105

110

Option

0500

minus05









Competing Interests


Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Then

EQ (1198781198791119878119879gt119870) = (EQ119878119879)EQ1015840 (1119878119879gt119870) (99)

Since119883119879


119896=1

1198881015840119896119867119890119896minus1 (minus1198891) (100)

where

1198881015840119896 = 1sum119873119895=0 119887119895119888119895

119873sumℓ=119896

(ℓ119896) 119887ℓminus119896119888ℓ 119896 = 1 1198731198891 = 1119887 [log(1198780119870) + 119886 + 1198872] = 1198892 + 119887

(101)

Hence


119896=1


119896=1

119888119896119867119890119896minus1 (minus1198892) (102)

Using




where

119888lowast119896 = 1198881015840119896 119873sum119896=0

119887119896119888119896 = 119873sumℓ=119896

(ℓ119896) 119887ℓminus119896119888ℓ 119896 = 1 119873 (105)



1198750 = 1198620 minus 1198780 + 119870119890minus119903119879 (106)


4sum119896=1



(107)

This ends the proof



Δ = 12059711986201205971198780= 119890minus120575119879(Φ(1198891) + 120601 (1198891) 119873sum

119896=1

1198881015840119896119867119890119896minus1 (minus1198891))

120574 = 1205972119862012059711987820 = 119890119886+(11988722)minus1199031198791198871198780 120601 (1198891) 119873sum119896=0

119888119896119867119890119896 (minus1198892) 120588 = 1205971198620120597119903= 119870119879119890minus119903119879(Φ(1198892) + 120601 (1198892) 119873sum

119896=1





1198881015840119896119867119890119896minus1 (minus1198891)) 119895 ge 1

(108)





120597120597119904E (119904119880 minus 119870)+ = E (1198801119904119880gt119870) (109)


2

1199043 119891119880 (119870119904 ) (110)

From 119878119879 = 1198780119890119883119879 and (100)


119896=1

1198881015840119896119867119890119896minus1 (minus1198891)) (111)

From the lemma

120574 = 1205972120597211987801198620 = 119890minus119903119879120597212059721198780EQ (1198780119880 minus 119870)+

= 1198702119890minus11990311987911987830 119891119880 (1198701198780) 119880 = 119890119883(112)

where 119883119879


119891119880 (119906) = 1119887119906119891119866 (1119887 (log 119906 minus 119886))= 1119887119906120601 (119910)

119873sum119896=0

119888119896119867119890119896 (119910)1003816100381610038161003816100381610038161003816100381610038161003816119910=(1119887)(log 119906minus119886)

(113)


120574 = 119870119890minus11990311987911988711987820 120601 (1198892) 119873sum119896=0

119888119896119867119890119896 (minus1198892)= 119890119886+(11988722)minus1199031198791198871198780 120601 (1198891) 119873sum

119896=0

119888119896119867119890119896 (minus1198892) (114)

Next




Now from (89)



119896=1

119888119896119867119890119896minus1 (minus1198892)] (118)


Define

119898(119887) = E119890119887119866minus(11988722) = sum119896=0

1198871198961198881198961198981015840 (119887) = sum

119896=1

119896119887119896minus1119888119896 = 119888lowast1 (120)


119890119886 = 119890(119903minus120575)119879minus(11988722)119898(119887) (121)


1198780119890minus120575119879EQ [( 120597120597119887 119890119887119866minus(11988722)

119898(119887) ) 1119866gtminus1198892]


(122)


EQ (1198901198871198661119866gtminus119910) = 11989011988722 [119898 (119887)Φ (119910 + 119887)

+ 120601 (119910 + 119887) 119873sum119896=1


To calculate


we need (see (105))

120597119888lowast119896120597119887 = 120597120597119887119873sumℓ=119896


120597119888lowast119873120597119887 = 0(125)




1198881015840119896119867119890119896 (minus119910 minus 119887)]

(126)



120597119886120597119888119895 119890119886+(11988722)

119873sum119896=0

119887119896119888119896 + 119887119895119890119886+(11988722) = 0or 120597119886120597119888119895 = minus

119887119895sum119873119896=0 119887119896119888119896 = minus119887

119895119890119886+(11988722)+(120575minus119903)119879(127)

Next


infin

minus1198892

(1198780119890119886+119887119910 minus 119870)120601 (119910) 119873sum119896=0

119888119896119867119890119896 (119910) 119889119910= 119890minus119903119879intinfin

minus1198892

1198780 120597119886120597119888119895 119890119886+119887119910120601 (119910)119873sum119896=0

119888119896119867119890119896 (119910) 119889119910+ 119890minus119903119879intinfin

minus1198892

(1198780119890119886+119887119910 minus 119870)120601 (119910)119867119890119895 (119910) 119889119910

(128)



119896=0


119910119892(119895) (119909) 120601 (119909) 119889119909

(129)




minus1198892


119896=1


(130)




119861119899 = 119875 119899prod119905=1

[1 +max(120572( 119878119905119878119905minus1 minus 1) 0)] (131)





119877119905 Qsim GC (119886 119887 ) 119905 = 1 119899 (133)


119905=1





1198810 = 119875 (119890minus119903 + 1205721198620)119899 (135)


1205971198810120597120585 = 120572119875 (119890minus119903 + 1205721198620)119899minus1 1205971198620120597120585 (136)


1205972119862012059711987820 = 1205722119875 (119890minus119903 + 1205721198620)119899minus2 (12059711986201205971198780 )2

+ 120572119875 (119890minus119903 + 1205721198620)119899minus1 1205972119862012059711987820 (137)



119886 = 00687119887 = 016851198883 = minus011501198884 = 003598

(138)


119904 = 61198883 = minus06898119896 = 241198884 = 08634 (139)



0

1

2

3

4

k


s


minus1

0

1

0

2

4 104

106

108

110

Option

k s







(1119887)(log(1198701198780)minus1198861015840)(11987801198901198861015840+119887119909 minus 119870)120601 (119909)

sdot ( 119873sum119896=0

119888119896119867119890119896 (119909))119889119909(140)


(1119887)(log(1198701198780)minus1198861015840)(11987801198901198861015840+119887119909 minus 119870)120601 (119909)

sdot 119867119890119895 (119909) 119889119909(141)



119886 = 01174119887 = 015951198881 = minus030536756952010661198882 = 0095420793734891531198883 = minus0123839711263352431198884 = 006120331530131559

(142)





119901 (119909) = 2119898sum119895=0

120575119895119867119890119895 (119909) (143)


119886 = minus02445119887 = 019401198881 = 16140




1198882 = 117331198883 = 042531198884 = 007320

(144)




(145)


k

s

01234

105

110

Option

0500

minus05









Competing Interests


Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












120597120597119904E (119904119880 minus 119870)+ = E (1198801119904119880gt119870) (109)


2

1199043 119891119880 (119870119904 ) (110)

From 119878119879 = 1198780119890119883119879 and (100)


119896=1

1198881015840119896119867119890119896minus1 (minus1198891)) (111)

From the lemma

120574 = 1205972120597211987801198620 = 119890minus119903119879120597212059721198780EQ (1198780119880 minus 119870)+

= 1198702119890minus11990311987911987830 119891119880 (1198701198780) 119880 = 119890119883(112)

where 119883119879


119891119880 (119906) = 1119887119906119891119866 (1119887 (log 119906 minus 119886))= 1119887119906120601 (119910)

119873sum119896=0

119888119896119867119890119896 (119910)1003816100381610038161003816100381610038161003816100381610038161003816119910=(1119887)(log 119906minus119886)

(113)


120574 = 119870119890minus11990311987911988711987820 120601 (1198892) 119873sum119896=0

119888119896119867119890119896 (minus1198892)= 119890119886+(11988722)minus1199031198791198871198780 120601 (1198891) 119873sum

119896=0

119888119896119867119890119896 (minus1198892) (114)

Next




Now from (89)



119896=1

119888119896119867119890119896minus1 (minus1198892)] (118)


Define

119898(119887) = E119890119887119866minus(11988722) = sum119896=0

1198871198961198881198961198981015840 (119887) = sum

119896=1

119896119887119896minus1119888119896 = 119888lowast1 (120)


119890119886 = 119890(119903minus120575)119879minus(11988722)119898(119887) (121)


1198780119890minus120575119879EQ [( 120597120597119887 119890119887119866minus(11988722)

119898(119887) ) 1119866gtminus1198892]


(122)


EQ (1198901198871198661119866gtminus119910) = 11989011988722 [119898 (119887)Φ (119910 + 119887)

+ 120601 (119910 + 119887) 119873sum119896=1


To calculate


we need (see (105))

120597119888lowast119896120597119887 = 120597120597119887119873sumℓ=119896


120597119888lowast119873120597119887 = 0(125)




1198881015840119896119867119890119896 (minus119910 minus 119887)]

(126)



120597119886120597119888119895 119890119886+(11988722)

119873sum119896=0

119887119896119888119896 + 119887119895119890119886+(11988722) = 0or 120597119886120597119888119895 = minus

119887119895sum119873119896=0 119887119896119888119896 = minus119887

119895119890119886+(11988722)+(120575minus119903)119879(127)

Next


infin

minus1198892

(1198780119890119886+119887119910 minus 119870)120601 (119910) 119873sum119896=0

119888119896119867119890119896 (119910) 119889119910= 119890minus119903119879intinfin

minus1198892

1198780 120597119886120597119888119895 119890119886+119887119910120601 (119910)119873sum119896=0

119888119896119867119890119896 (119910) 119889119910+ 119890minus119903119879intinfin

minus1198892

(1198780119890119886+119887119910 minus 119870)120601 (119910)119867119890119895 (119910) 119889119910

(128)



119896=0


119910119892(119895) (119909) 120601 (119909) 119889119909

(129)




minus1198892


119896=1


(130)




119861119899 = 119875 119899prod119905=1

[1 +max(120572( 119878119905119878119905minus1 minus 1) 0)] (131)





119877119905 Qsim GC (119886 119887 ) 119905 = 1 119899 (133)


119905=1





1198810 = 119875 (119890minus119903 + 1205721198620)119899 (135)


1205971198810120597120585 = 120572119875 (119890minus119903 + 1205721198620)119899minus1 1205971198620120597120585 (136)


1205972119862012059711987820 = 1205722119875 (119890minus119903 + 1205721198620)119899minus2 (12059711986201205971198780 )2

+ 120572119875 (119890minus119903 + 1205721198620)119899minus1 1205972119862012059711987820 (137)



119886 = 00687119887 = 016851198883 = minus011501198884 = 003598

(138)


119904 = 61198883 = minus06898119896 = 241198884 = 08634 (139)



0

1

2

3

4

k


s


minus1

0

1

0

2

4 104

106

108

110

Option

k s







(1119887)(log(1198701198780)minus1198861015840)(11987801198901198861015840+119887119909 minus 119870)120601 (119909)

sdot ( 119873sum119896=0

119888119896119867119890119896 (119909))119889119909(140)


(1119887)(log(1198701198780)minus1198861015840)(11987801198901198861015840+119887119909 minus 119870)120601 (119909)

sdot 119867119890119895 (119909) 119889119909(141)



119886 = 01174119887 = 015951198881 = minus030536756952010661198882 = 0095420793734891531198883 = minus0123839711263352431198884 = 006120331530131559

(142)





119901 (119909) = 2119898sum119895=0

120575119895119867119890119895 (119909) (143)


119886 = minus02445119887 = 019401198881 = 16140




1198882 = 117331198883 = 042531198884 = 007320

(144)




(145)


k

s

01234

105

110

Option

0500

minus05









Competing Interests


Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1198881015840119896119867119890119896 (minus119910 minus 119887)]

(126)



120597119886120597119888119895 119890119886+(11988722)

119873sum119896=0

119887119896119888119896 + 119887119895119890119886+(11988722) = 0or 120597119886120597119888119895 = minus

119887119895sum119873119896=0 119887119896119888119896 = minus119887

119895119890119886+(11988722)+(120575minus119903)119879(127)

Next


infin

minus1198892

(1198780119890119886+119887119910 minus 119870)120601 (119910) 119873sum119896=0

119888119896119867119890119896 (119910) 119889119910= 119890minus119903119879intinfin

minus1198892

1198780 120597119886120597119888119895 119890119886+119887119910120601 (119910)119873sum119896=0

119888119896119867119890119896 (119910) 119889119910+ 119890minus119903119879intinfin

minus1198892

(1198780119890119886+119887119910 minus 119870)120601 (119910)119867119890119895 (119910) 119889119910

(128)



119896=0


119910119892(119895) (119909) 120601 (119909) 119889119909

(129)




minus1198892


119896=1


(130)




119861119899 = 119875 119899prod119905=1

[1 +max(120572( 119878119905119878119905minus1 minus 1) 0)] (131)





119877119905 Qsim GC (119886 119887 ) 119905 = 1 119899 (133)


119905=1





1198810 = 119875 (119890minus119903 + 1205721198620)119899 (135)


1205971198810120597120585 = 120572119875 (119890minus119903 + 1205721198620)119899minus1 1205971198620120597120585 (136)


1205972119862012059711987820 = 1205722119875 (119890minus119903 + 1205721198620)119899minus2 (12059711986201205971198780 )2

+ 120572119875 (119890minus119903 + 1205721198620)119899minus1 1205972119862012059711987820 (137)



119886 = 00687119887 = 016851198883 = minus011501198884 = 003598

(138)


119904 = 61198883 = minus06898119896 = 241198884 = 08634 (139)



0

1

2

3

4

k


s


minus1

0

1

0

2

4 104

106

108

110

Option

k s







(1119887)(log(1198701198780)minus1198861015840)(11987801198901198861015840+119887119909 minus 119870)120601 (119909)

sdot ( 119873sum119896=0

119888119896119867119890119896 (119909))119889119909(140)


(1119887)(log(1198701198780)minus1198861015840)(11987801198901198861015840+119887119909 minus 119870)120601 (119909)

sdot 119867119890119895 (119909) 119889119909(141)



119886 = 01174119887 = 015951198881 = minus030536756952010661198882 = 0095420793734891531198883 = minus0123839711263352431198884 = 006120331530131559

(142)





119901 (119909) = 2119898sum119895=0

120575119895119867119890119895 (119909) (143)


119886 = minus02445119887 = 019401198881 = 16140




1198882 = 117331198883 = 042531198884 = 007320

(144)




(145)


k

s

01234

105

110

Option

0500

minus05









Competing Interests


Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198810 = 119875 (119890minus119903 + 1205721198620)119899 (135)


1205971198810120597120585 = 120572119875 (119890minus119903 + 1205721198620)119899minus1 1205971198620120597120585 (136)


1205972119862012059711987820 = 1205722119875 (119890minus119903 + 1205721198620)119899minus2 (12059711986201205971198780 )2

+ 120572119875 (119890minus119903 + 1205721198620)119899minus1 1205972119862012059711987820 (137)



119886 = 00687119887 = 016851198883 = minus011501198884 = 003598

(138)


119904 = 61198883 = minus06898119896 = 241198884 = 08634 (139)



0

1

2

3

4

k


s


minus1

0

1

0

2

4 104

106

108

110

Option

k s







(1119887)(log(1198701198780)minus1198861015840)(11987801198901198861015840+119887119909 minus 119870)120601 (119909)

sdot ( 119873sum119896=0

119888119896119867119890119896 (119909))119889119909(140)


(1119887)(log(1198701198780)minus1198861015840)(11987801198901198861015840+119887119909 minus 119870)120601 (119909)

sdot 119867119890119895 (119909) 119889119909(141)



119886 = 01174119887 = 015951198881 = minus030536756952010661198882 = 0095420793734891531198883 = minus0123839711263352431198884 = 006120331530131559

(142)





119901 (119909) = 2119898sum119895=0

120575119895119867119890119895 (119909) (143)


119886 = minus02445119887 = 019401198881 = 16140




1198882 = 117331198883 = 042531198884 = 007320

(144)




(145)


k

s

01234

105

110

Option

0500

minus05









Competing Interests


Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












(1119887)(log(1198701198780)minus1198861015840)(11987801198901198861015840+119887119909 minus 119870)120601 (119909)

sdot ( 119873sum119896=0

119888119896119867119890119896 (119909))119889119909(140)


(1119887)(log(1198701198780)minus1198861015840)(11987801198901198861015840+119887119909 minus 119870)120601 (119909)

sdot 119867119890119895 (119909) 119889119909(141)



119886 = 01174119887 = 015951198881 = minus030536756952010661198882 = 0095420793734891531198883 = minus0123839711263352431198884 = 006120331530131559

(142)





119901 (119909) = 2119898sum119895=0

120575119895119867119890119895 (119909) (143)


119886 = minus02445119887 = 019401198881 = 16140




1198882 = 117331198883 = 042531198884 = 007320

(144)




(145)


k

s

01234

105

110

Option

0500

minus05









Competing Interests


Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1198882 = 117331198883 = 042531198884 = 007320

(144)




(145)


k

s

01234

105

110

Option

0500

minus05









Competing Interests


Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














Competing Interests


Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









researcharticle gram-charlier processes and applications

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