VOLTERRA EQUATIONS WITH FRACTIONALSTOCHASTIC INTEGRALS

MAHMOUD M EL-BORAI KHAIRIA EL-SAID EL-NADI

OSAMA L MOSTAFA AND HAMDY M AHMED

Received 24 December 2003 and in revised form 6 May 2004

Some fractional stochastic systems of integral equations are studied The fractional sto-chastic Skorohod integrals are also studied The existence and uniquness of the consid-ered stochastic fractional systems are established An application of the fractional Black-Scholes is considered

1 Introduction

We assume that a probability space (ΩηP) is given where Ω denotes the space C(R+Rk) equipped with the topology of uniform convergence on compact sets η the Borelσ-field of Ω and P a probability measure on Ω

Let Wt(ω)= ω(t) t ge 0 be a Wiener process For any t ge 0 we define ηt = σω(s)s lt torZ where Z denotes the class of the elements in ηt which have zero P-measure

Pardoux and Protter discussed the existence and uniqueness of the solution of thestochastic integral equation of the form

Xt = X0 +int t

0F(tsXs

)ds+

ksumi=1

int t0Gi(Ht tsXs

)dWi

s (11)

whose solution Xt should be Rd-valued and ηt adapted process Ht is an Rp-valued(see [16]) It is supposed that F maps ts 0le s lt ttimesRd into Rd and Gi (i= 12 k)maps Rptimests 0le s lt ttimesRd into Rd

In the present work we study the existence uniqueness and continuity of the solutionof the fractional stochastic integral equation of the form

Xt = X0 + Iβt F(tsXs

)+

ksumi=1

Wβt Gi

(Ht tsXs

) (12)

where 0 lt β lt 1 Iβt F(tsXs) the fractional integral of F(tsXs) is defined by (see [22])

Iβt F(tsXs

)= 1Γ(β)

int t0

F(tsXs

)(tminus s)1minusβ ds (13)

Copyright copy 2004 Hindawi Publishing CorporationMathematical Problems in Engineering 20045 (2004) 453ndash4682000 Mathematics Subject Classification 60H05 60G57URL httpdxdoiorg101155S1024123X04312020

454 Volterra equations with fractional stochastic integrals

The fractional Wiener process Wβt Gi(Ht tsXs) of Gi(Ht tsXs) is defined by (see [7])

Wβt Gi

(Ht tsXs

)= 1Γ((β+ 1)2

) int t0

Gi(Ht tsXs

)(tminus s)(1minusβ)2 dWi

s (14)

Stochastic Volterra equations have been studied in several papers (see [2 3 4 6 16 1718]) In this work we will use the Skorohod integral (see [9 10 11 12 13 14 15 19 20])to interpret (12) as in [16] since the integrands in the stochastic integrals are not adaptedtherefore we cannot use as usual the Ito integral to interpret the equation In Section 2we state some results concerning the Skorohod integral which will be used later togetherwith the precise interpretation of (12) In Sections 3 and 4 we prove the existence anduniqueness of a solution to (12) in two steps In Section 5 we establish under additionalassumptions the existence of almost surely (as) continuous modification of the solutionprocess In Section 6 we show the continuity of the solution of (12)

Equation (12) has many important financial applications These systems arise if weconsider the fractional analog of a portfolio (see [1]) The fractional Black-Scholes marketconsists of a bank account or a bond and a stock The price process At of the bond at timet is given by

At = expint t

0r(s)ds

(15)

where r(s)ge 0 sisin [0 t] is the interest rate A portfolio is a pair (utvt) of random vari-ables for fixed t isin [0T] The price Xt of the stock could be governed by a fractionalVolterra equation of the form

Xt = X0 +1

Γ(β)

int t0

micro(s)Xs(tminus s)1minusβ ds+

1Γ((β+ 1)2

) int t0

σ(s)Xs(tminus s)(1minusβ)2 dWs (16)

Here the drift microge 0 and volatility σ gt 0 are continuous functions on [0T] The numbersut and vt are the bond and stock units respectively (held by an investor) Hence thecorresponding value process is

Vt = utAt + vtXt (17)

The process Vt could be governed by the equation

Vt =V0 +1

Γ(β)

int t0

r(s)Asus(tminus s)1minusβ ds+

1Γ(β)

int t0

micro(s)Xsvs(tminus s)1minusβ ds

+1

Γ((1minusβ)2

) int t0

σ(s)vsXs(tminus s)(1minusβ)2 dWs

(18)

Mahmoud M El-Borai et al 455

2 The Skorohod integral

We will now define the Skorohod integral Most of this section is a review of some basicnotations and a few results from [4 16]

Let again Ω= C(R+Rk) let η be its Borel field and let P denote Wiener measure on(Ωη)

Wt(ω)= ω(t) (21)

Let η0t = σWs 0 le s lt t and ηt = η0

t orZ where Z denotes the class of sets which havezero P-measure of η

For hisin L2(R+Rk) we denote the Wiener integral by

W(h)=int T

0

(h(t)dWt

) (22)

LetA denote the dense subset of L2(ΩηP) consisting of those classes of random variablesof the form

F = f(W(h1) W

(hn))

(23)

where n isin N (N denotes the set of nonnegative integers) f isin Cinfinb (Rn) h1 hn isinL2(R+Rk) Cinfinb is the set of infinitely differentiable functions on [0b] whose derivativesof any order are null at b If F has the form (23) we define its derivative in the directioni as the process Di

tF t ge 0 defined by

DitF =

nsumk=1

part f

partxk

(W(h1) W

(hn))hik(t) (24)

DF will stand for the k-dimensional process DtF = (D1t F Dk

t F) t ge 0Proposition 21 The differential operators Di i= 1 k are unbounded closable opera-tors from L2(Ω) into L2(ΩtimesR+)

Let D12i be the closure of A with respect to the norm

Fi12 = F2 + middot DiF∥∥L2(R+)

∥∥2 (25)

where F2 = (E(F2))12 E(X) is the mathematical expectation of X and

g2L2(R+) =

intinfin0g2(t)dt (26)

Similarly the domain D12 =⋂ki=1D

12i is the closure of A with respect to the norm (see

[10])

F12 = F2 +ksumi=1

middot DiF∥∥L2(R+)

∥∥2 (27)

456 Volterra equations with fractional stochastic integrals

We identify Di and D with their closed extensions (D12 is the domain of D L2(Ω)rarrL2(ΩtimesR+Rk))

We denote byD12iloc the set of measurable Frsquos which are such that there exists a sequence

(ΩnFn) nisinN sub ηtimesD12i with the two properties

(i) Ωn uarrΩ as nrarrinfin(ii) Fn|Ωn = F|Ωn nisinN

For F isinD12iloc we define without ambiguity Di

tF =DitFn on ΩntimesR+ for all nisinN D12

loc isdefined similarly

For i = 1 k we define δi the Skorohod integral with respect to Wit as the adjoint

of Di that is Domδi (the set of adapted processes for the Skorohod integral) is the set ofuisin L2(ΩtimesR+) which are such that there exists a constant c with

∣∣∣∣Eintinfin

0DitFut dt

∣∣∣∣le cF2 forallF isin A (28)

If uisinDomδi δi(u) is defined as the unique element of L2(Ω) which satisfies

E(δi(u)F

)= Eintinfin

0DitFut dt forallF isinA (29)

Let L12i = L2(R+D12

i ) We have that L12i subDomδi and for uisin L12

i

E[δi(u)2]= E

intinfin0u2t dt+E

intinfin0

intinfin0DisutD

itus dsdt (210)

Note that if uisin L2loc(R+D12

i ) then u1[0T] isin L12i for any T gt 0 and we can write

int T0utdW

it = δi

(u1[0T]

) (211)

The Skorohod integral is a local operation on L2loc(R+D12

i ) in the sense that if uv isinL2

loc(R+D12i ) then

int t0 usdW

is =

int t0 vsdW

is as on ω us(ω)= vs(ω) for almost all s lt t

Let L12iloc denote the set of measurable processes u which are such that for any T gt 0

there exists a sequence

(ΩTn uTn

) nisinN

sub ηtimesL12i (212)

such that

(i) ΩTn uarrΩ as as nrarrinfin

(ii) u= uTn dPtimesdt ae on ΩTn times [0T] nisinN

Mahmoud M El-Borai et al 457

For uisin L12iloc we can define its Skorohod integral with respect to Wi

t by

int t0usdW

is =

int t0uTnsdW

is on ΩT

n times [0T] (213)

Finally L12 =⋂ki=1L

12i and L12

loc is defined similarly as L12iloc

We now introduce the particular class of integrands which we will use belowLet u R+timesΩtimesRP rarrR satisfy the following

(i) For all x isinRp (tω)rarr (tωx) is ηt progressively measurable(ii) For all (tω)isinR+timesΩ u(tωmiddot)isin C1(Rp)

(iii) For some increasing function φ R+rarrR+ |u(tωx)|+|uprime(tωx)|leφ(|x|) forall (tωx)isinR+timesΩtimesRp where uprime(tx) stands for the gradient (partupartx)(tx)

Let θ be a p-dimensional random vector such that

(iv) θ j isinD12i

⋂Linfin(Ω) j = 1 p

We fix T gt 0 and consider

Ii(x)=int T

0

u(tx)(T minus t)(1minusβ)2 dW

it 0 lt β lt 1 (214)

Define moreover vt = u(tθ)Under conditions (i) (ii) (iii) and (iv) the following proposition holdsIt is proved in [10] that

int T0u(tθ)dWi

t =int T

0u(tx)dWi

t |x=θ minusint T

0uprime(tθ)Di

tθdt (215)

The same relation is proved in [16] under slightly different conditions Equation (215) isused to define the Skorohod integral

int T0 u(tθ)dWi

t

Proposition 22 The random field Ii(x) x isinRp defined above possesses an as continu-ous modification so that the random variable Ii(θ) can be defined v isinDomδi (see [10 16])

Condition (iv) can be replaced by (ivprime) θ j isinD12iloc j = 1 p

Under conditions (i) (ii) (iii) and (ivprime) v isin (Domδi)loc in the sense that there existsa sequence (Ωnvn) nisinN sub ηtimesDomδi such that Ωn uarrΩ as and vn|Ωn = v|Ωn

Indeed let (Ωprimenθn) be a localizing sequence for θ in (D12

i )p and let ψn nisinN subCinfinc (RpRp) satisfy ψn(x)= x whenever |x| le n

Define vn(t) = u(tψn(θn)) Ωn =Ωprimen

⋂|θ| le n then (Ωnvn) n isin N satisfies theabove conditions

It is then natural to define for every εgt0 the Skorohod integralint Tminusε

0 (vt(T minus t)(1minusβ)2)

dWit by formula (215) and the latter coincides with

int Tminusε0 (vn(t)(T minus t)(1minusβ)2)dWi

t on Ωn

It is clear that limεrarr0int Tminusε

0 (vt(T minus t)(1minusβ)2)dWit exists in the mean by using the norm

middot 2

458 Volterra equations with fractional stochastic integrals

3 Statement of the problem interpretation of (12)

Our aim is to study the equation

Xt = X0 + aint t

0

F(tsXs

)(tminus s)1minusβ ds+ b

ksumi=1

int t0

Gi(Ht tsXs

)(tminus s)(1minusβ)2 dWi

s

0 lt β lt 1 a= 1Γ(β)

b = 1Γ((β+ 1)2

) (31)

We define D = (ts)isinR2+ 0le s lt t

The coefficients F and G are given as follows F DtimesRd rarrRd is measurable and foreach (sx)isinR+timesRd F(middotsx) is ηt progressively measurable on Ωtimes [s+infin)

For i= 1 k Gi RptimesDtimesRd rarrRd is measurable for each (h tx) Gi(h tmiddotx) is ηsprogressively measurable on Ωtimes [0 t] and for each (ω tsx) Gi(middot tsx) is of class C1Ht is a given progressively measurable p-dimensional process It will follow from thesehypotheses that we will be able to construct a progressively measurable solution Xt

Therefore for each t the process Gi(h tsXs) s isin [0T] is of the form vs = u(sθ)with u(sh)=Gi(h tsXs) and θ =Ht We will impose below conditions on G Ht andthe solution Xt so as to satisfy requirements (i) (ii) (iii) and (ivprime) of Section 2

In particular we will consider only nonanticipating solutions Therefore the stochasticintegrals in (31) will be interpreted according to (215) that is

int t0

Gi(Ht tsXs

)(tminus s)(1minusβ)2 dWi

s =int t

0

Gi(h tsXs

)(tminus s)(1minusβ)2 dW

is|h=Ht minus

int t0

Gprimei(Ht tsXs

)(tminus s)(1minusβ)2 Di

sHt ds (32)

In other words we can rewrite (31) as

Xt = X0 +int t

0F(tsXs

)ds+ b

ksumi=1

int t0

Gi(h tsXs

)(tminus s)(1minusβ)2 dW

is|h=Ht (33)

where

F(tsx)= a F(tsx)(tminus s)1minusβ minus b

ksumi=1

Gprimei(Ht tsx

)DisHt

(tminus s)(1minusβ)2 (34)

and the stochastic integrals are now the usual Ito integralsWe will show below that (33) makes sense for any progressively measurable process X

which satisfies X isin⋂tgt0Lq(0 t) as for some q gt p We will find such a solution to (33)

it will then follow from (32) that it is a solution to (31) Similarly uniqueness for (31)in the above class will follow from uniqueness for (33) in that class

4 Existence and uniqueness under strong hypotheses

We formulate a set of further hypotheses (those stated in Section 3 are assumed to holdthroughout the paper) under which we will establish a first result of the existence anduniqueness of a solution of (12)

Mahmoud M El-Borai et al 459

Let B be an open bounded subset of Rp K gt 0 and q gt p such that

(H1) X0 isin Lq(Ωη0PRd)(H2) P(Ht isin Bforallt ge 0)= 1(H3) H isin (L12)p |DsHt| le K as 0le s lt t(H4) |F(tsx)|+

sumki=1 |Gi(h tsx)|+

sumki=1 |Gprimei (h tsx)| le K(1 + |x|) for any 0le s lt t

hisin B x isinRd(H5) |F(tsx) minus F(ts y)| +

sumki=1 |Gi(h tsx) minus Gi(h ts y)| +

sumki=1 |Gprimei (h tsx)

minusGprimei (h ts y)| le K|xminus y| for any 0le s lt t hisin B and x y isinRd as

Note that q will be a fixed real number such that q gt p and Lqprog(Ωtimes (0 t)) will

stand for the space Lq(Ωtimes (0 t)ξtP times λ) where ξt denotes the σ-algebra of progres-sively measurable subsets of Ωtimes (0 t) and λ denotes the Lebesgue measure on (0 t) andset 1minusβ = αq 0 lt α lt 1

Lemma 41 Let X isin⋂tgt0Lqprog(Ωtimes (0 t)) where q gt p and suppose that (H4) is in force

Then for any T ge t gt 0 and iisin 1 k the random field

int t0

Gi(h tsXs

)(tminus s)α2q dWi

s hisin B

(41)

possesses an as continuous modification

Proof Using Burkholder-Gundy and Holderrsquos inequalities together with (H4) we obtain

E

(∣∣∣∣int t

0

Gi(h tsXs

)(tminus s)α2q dWi

s minusint t

0

Gi(k tsXs

)(tminus s)α2q dWi

s

∣∣∣∣q)

le c1Eint t

0

∣∣Gi(h tsXs

)minusGi(k tsXs

)∣∣q(tminus s)α2 ds

le c1Kq|hminus k|q

int t0

E(1 +

∣∣Xs∣∣q)(tminus s)α2 ds

le c2Kq|hminus k|q

int t0(tminus s)minusα2ds

le c3|hminus k|qt1minusα2

(42)

where c1 c2 and c3 are positive constants The result now follows from the multidimen-sional generalization of Kolmogrovrsquos lemma (see [1 8])

We can now assume that for fixed t the random field (41) is as continuous in hprovided X isin Lqprog(Ωtimes (0 t))

From X isin⋂tgt0Lqprog(Ωtimes (0 t)) define

It(X h)= bksumi=1

int t0

Gi(h tsXs

)(tminus s)α2q dWi

s hisinRp t gt 0

Jt(X)=int t

0F(tsXs

)ds+ It

(X Ht

)

(43)

460 Volterra equations with fractional stochastic integrals

Lemma 42 For any t gt 0 there is a constant c gt 0 such that

E(∣∣Jt(X)

∣∣q)le c(int t

0E(1 +

∣∣Xs∣∣q)

1(tminus s)α +

1(tminus s)α2

ds)

0 lt α lt 1 (44)

Proof

∣∣∣∣int t

0F(tsXs

)ds∣∣∣∣q

le cint t

0

∣∣F(tsXs)∣∣q(tminus s)α +

ksumi=1

∣∣Gprimei(h tsXs)|qh=Ht

∣∣DisHt

∣∣q(tminus s)α2 ds

le cint t

0

(1 +

∣∣Xs∣∣q)

1(tminus s)α +

1(tminus s)α2

ds

(45)

where we have used (H2) (H3) and (H4)

∣∣It(X Ht)∣∣le sup

hisinB

∣∣It(X h)∣∣ (46)

(c c are positive constants)It is easy to show using in particular (H4) and Lebesguersquos dominated convergence

theorem that the mapping

hminusrarr It(X h) (47)

from Rp into Lq(Ω) is differentiable and that

partIt(X h)parthj

= bint t

0

part

parthj

ksumi=1

(Gi(h tsXs

)(tminus s)α2q

)dWi

s (48)

Since q gt p we can infer from Sobolevrsquos embedding theorem (see [13]) that

E

(suphisinB

∣∣It(X h)∣∣q)le cE

intB

(∣∣It(X h)∣∣q +

psumj=1

∣∣∣∣ partItparthj(X h)

∣∣∣∣q)dh (49)

It then follows from the Burkholder-Gundy inequality that

E

(suphisinB

∣∣It(X h)∣∣q)le cE

intB

int t0

ksumi=1

(∣∣Gi(h tsXs

)∣∣q(tminus s)α2 +

psumj=1

∣∣∣∣ part

parthj

Gi(h tsXs

)(tminus s)α2q

∣∣∣∣q)dsdh

le c(int

Bdh)(Eint t

0

(1 +

∣∣Xs∣∣q)(tminus s)α2 ds

)

le cint t

0

E(1 +

∣∣Xs∣∣q)(tminus s)α2 ds

(410)

where we have used (H4) and B is bounded From (45) and (410) the proof is complete

A similar argument using (H5) instead of (H4) yields the following lemma

Mahmoud M El-Borai et al 461

Lemma 43 For any 0 lt t le T there exists a constant c gt 0 such that

E(∣∣Jt(X)minus Jt(Y)

∣∣q)le cint t

0E∣∣XsminusYs∣∣q

1

(tminus s)α +1

(tminus s)α2ds (411)

We are now in a position to prove the main result of this section

Theorem 44 Under conditions (H1) (H2) (H3) (H4) and (H5) there exists a uniqueelement X isin ⋂tgt0L

qprog(Ωtimes (0 t)) which solves (31) Moreover if τ is a stopping time

uniqueness holds on the random interval [0τ]

Proof Equation (31) can be rewritten as

Xt = X0 + Jt(X) t ge 0 (412)

Uniqueness Let X Y isin⋂tgt0Lqprog(Ωtimes (0 t)) and let τ be a stopping time such that

Xt = X0 + Jt(X) Yt = X0 + Jt(Y) 0le t le τ (413)

From Lemma 43

E(∣∣Xt minusYt∣∣q)= E(∣∣Jt(X)minus Jt(Y)

∣∣q)le c

int t0E∣∣XsminusYs∣∣q

1

(tminus s)α +1

(tminus s)α2ds

(414)

Set (E|Xt minusYt|q)= γt It is easy to see that

γt le 2cη(t)int t

0

γs(tminus s)α ds (415)

where η(t)=Max(1 tα) and c is a positive constant Thus (see [5 21])

γt le M2(Γ(tminusα)

)2

Γ(2(1minusα)

) int t0

γs(tminus s)2αminus1

ds (416)

and by a classical argument we get

γt le Mn(Γ(tminusα)

)nΓ(n(1minusα)

) int t0

γs(tminus s)nαminus(nminus1)

ds (417)

where M = 2cη(T) Thus for sufficiently large n n gt 1(1minusα) we get from (417)

γt le Kn

Γ(n(1minusα)

) int t0γs ds (418)

where K is a positive constantTaking the limit as nrarrinfin we find that (418) leads to γt = 0To prove the existence we define a sequence Xn

t 0le t le T n= 012 as follows

Xot = Xo Xn+1

t = Xo + Jt(Xn) T le t le 0 (419)

462 Volterra equations with fractional stochastic integrals

Using Lemma 42 we can see that

Xn isin⋂tgt0

Lqprog

(Ωtimes [0T]

) (420)

It then follows from Lemma 43 that

gn+1(t)le 2cη(t)int t

0

gn(s)(tminus s)α ds (421)

where

gn+1(t)= E(∣∣Xn+1t minusXn

t

∣∣q) (422)

By using a similar argument we can write

gn le Kn

Γ(n(1minusα)

) int t0go(s)ds (423)

The last estimation implies that Xn is a Cauchy sequence in Lqprog(Ωtimes [0T]) Then there

exists X such that Xn rarr X in⋂tgt0L

qprog(Ωtimes [0T]) and again using Lemma 43 we can

pass to the limit in (420) yielding that X solves (412)

5 An existence and uniqueness result under weaker assumptions

We formulate a new set of weaker hypotheses

(H1prime) X0 is η0 measurable(H2prime) H isin (L12

loc)pHt is a progressively measurable process which can be localized in(L12)p by a progressively measurable sequence

We assume that there exists an increasing progressively measurable process Ut t ge 0with values in R+ such that

(H3prime) |Ht|+sumk

i=1 |DisHt| leUt as 0le s lt t

Finally we suppose that for any N gt 0 there exists an increasing progressively measur-able process VN

t t ge 0 with values in R+ such that

(H4prime) |F(tsx)|+sumk

i=1 |Gi(h tsx)|+sumk

i=1 |Gprimei (h tsx)| le VNt (1+|x|) for all |h|leN

0le s lt t and x isinRd(H5prime) |F(tsx) minus F(ts y)| +

sumki=1 |Gi(h tsx) minus Gi(h ts y)| +

sumki=1 |Gprimei (h tsx)

minusGprimei (h ts y)| leVNt |xminus y| for all |h| leN 0le s lt t x y isinRd

Let again q be a fixed real number with q gt p and set 1minusβ = αq 0 lt α lt 1 We havethe following theorem

Theorem 51 Equation (31) has a unique solution in the class of progressively measurableprocesses which satisfy

X isin⋂tgt0

Lq(0 t) as (51)

Mahmoud M El-Borai et al 463

Proof (a) We first see how (31) makes sense if X isin⋂tgt0Lq(0 t) as

That is we have to show that for fixed t gt 0

int t0

Gi(h tsXs

)(tminus s)α2q dWi

s hisinRP

(52)

is a well-defined random field which possesses an as continuous versionFor that sake we define

τn = inftint t

0

∣∣Xs∣∣q(tminus s)α dsge n or VN

t ge n (53)

The argument of Lemma 41 can be used to show that

hminusrarrint tandτn

0

Gi(h tsXs

)(tminus s)α2q dWi

s (54)

possesses an as continuous modification on |h| le N Since this is true for any n andN and cupnτn ge t =Ω as the result follows

(b) Existence we want to show existence on an arbitrary interval [0T] (T will be fixedbelow)

Let Hn nisinN denote a progressively measurable localizing sequence forH in (L12)p

on [0T] Since from (H3prime) suptleT |Ht| is as finite we can and do assume without lossof generality that

∣∣Hnt (ω)

∣∣le n forall(tω)isin [0T]timesΩ (55)

Note that Ht(ω) =Hnt (ω) as on ΩT

n for all t isin [0T] where ΩTn uarrΩ as as nrarrinfin We

moreover define Xn0 = X01|X0|len Sn = inft supsltt |DsH

nt |orVn

t ge n We consider theequation

Xnt = Xn

0 +int t

0Fn(tsXn

s

)ds+ b

ksumi=1

int t0

Gni

(h tsXn

s

)(tminus s)α2q dWi

s|h=Hnt (56)

where

Fn(tsx)= 1[0Sn](s)

[aF(tsx)(tminus s)αq minus b

ksumi=1

Gprimei(Hnt tsx

)DisH

nt

(tminus s)α2q]

Gni (h tsx)= 1[0Sn](s)Gi(h tsx)

(57)

It is easy to see that theorem (415) applies to (56)Define

Sn(ω)=Sn(ω)and inf

t le T

int t0

∣∣Hs(ω)minusHns (ω)

∣∣ds gt 0

if∣∣X0(ω)

∣∣ lt n

0 otherwise(58)

464 Volterra equations with fractional stochastic integrals

Sn is a stopping time and it follows from the uniqueness part of Theorem 44 that ifmgt n

Xmt = Xn

t on[0 Sn

]as (59)

Since moreover Sn = T uarrΩ as we can define the process Xt on [0T] by Xt = Xnt

on [0 Sn] for all nisinNClearly X isin Lq(0T) as and solves (31) on [0T] Since T is arbitrary the existence

is proved(c) Uniqueness it suffices to prove uniqueness on an arbitrary interval [0T] Let

Xt t isin [0T] be a progressively measurable process such that X isin Lq(0T) as andX solves (31) It suffices to show that X coincides with the solution we have just con-structed

Let

Sn(ω)= S(ω)and inft le T

int t0

∣∣Xs(ω)∣∣q

(tminus s)α ds gt n

Xnt = XtandSn

(510)

Sn isin Lq(Ωtimes [0T]) and it solves (56) with Sn replaced by SnThen

Xnt (ω)= Xt(ω)timesP ae on

[0 Sn

] (511)

where is the Lebesgue measure on the real lineThe result follows from the fact that Sn ge T uarrΩ asNote that the above solution satisfies in fact X isin⋂qgt1

⋂tgt0L

q(0 t) as

6 Continuity of the solution

We want to give additional conditions under which the solution of (31) is an as contin-uous process

(H6) For all (sx)isinR+timesRd trarr F(tsx)(tminus s)αq is as continuous on (s+infin)(H7) Ht t ge 0 is as continuous(H8) For all iisin 1 k sisinR+ trarrDi

sHt is as continuous on (s+infin)(H9) For all (sx) isin R+ timesRd i isin 1 k (th)rarr Gprimei (h tsx)(tminus s)α2q is as con-

tinuous on (s+infin)timesRP

We also suppose that there exist δ gt 0 gt 0 such that for all N gt 0 |h| le N 0 le s lttand r x isinRd |tminus r| le 1

(H10) There exists an increasing process VNt t ge 0 such that

∣∣Gi(h tsx)minusGi(hrsx)∣∣leVN

t |tminus r|δ(1 + |x|) (61)

Theorem 61 Under conditions (H1prime) (H2prime) (H3prime) (H4prime) (H5prime) and (H6) (H7) (H8)(H9) (H10) the unique solution of (31) (which is progressively measurable and belongs asto⋂qgt1⋂tgt0L

q(0 t)) has as continuous modification

Mahmoud M El-Borai et al 465

Proof We need to show only that whenever X isin⋂qgt1⋂tgt0L

q(0 t) as Jt(X) t gt 0 hasan as continuous modification

(a) We first show that trarr int t0 F(tsXs)ds is as continuous

Note that (H6) (H7) (H8) and (H9) imply that for all (sx)isinR+timesRd trarr F(tsx)is as continuous on (s+infin)

Moreover from (H2prime) (H3prime) (H4prime) and the fact that X isin⋂qgt1⋂tgt0L

q(0 t) as forany T gt 0 there exists a process ZTs sisin [0T] such that

∣∣F(tsXs)∣∣le ZTs ∣∣Gprimei(Ht tsXs

)DisHt

∣∣le ZTs 0le s lt t le T asint t

0ZTs

1

(tminus s)α2q +1

(tminus s)αqds ltinfin as

(62)

Let first tn nisinN be a sequence such that tn lt t for any n and tnrarr t as nrarrinfin then

int t0F(tsXs

)dsminus

int tn0F(tnsXs

)ds=

int ttnF(tsXs

)ds+

int tn0

[F(tsXs

)minus F(tnsXs)]ds

∣∣∣∣int ttnF(tsXs

)ds∣∣∣∣le

int ttnZTs

1

(tminus s)α2q +1

(tminus s)αqds

(63)

and the latter tends as to 0 as nrarrinfin

∣∣∣∣int tn

0

[F(tsXs

)minus F(tnsXs)]ds∣∣∣∣le

int t0

∣∣F(tsXs)minus F(tnsXs)∣∣ds (64)

which tends to 0 as nrarrinfin A similar argument gives the same result when tn gt t tnrarr t(b) We next show that t rarr It(X Ht) possesses an as continuous modification This

follows from (H7) and

(th)minusrarr It(X h) (65)

has an as continuous modificationBy localization it suffices to prove (65) under assumptions (H2) (H3) (H4) (H5)

and (H6) (H7) (H8) (H9) (H10) with VNt (ω) in (H10) replaced by a constant K

and in case X0 isin⋂qgt1L

q(ΩRd)It then suffices to show that under the above hypotheses there exists Cq gt 0 such that

for any hk isinRP the numbers tr will be positive Suppose to fix the ideas that 0le r lt tthen

It(X h)minus Ir(X k)=int tr

Gi(h tsXs

)(tminus s)α2q dWi

s +int r

0

[Gi(h tsXs

)(tminus s)α2q minus Gi

(hrsXs

)(rminus s)α2q

]dWi

s

+int r

0

[Gi(h tsXs

)(rminus s)α2q minus Gi

(krsXs

)(rminus s)α2q

]dWi

s

(66)

466 Volterra equations with fractional stochastic integrals

It follows from the Burkholder-Gundy inequality that

E(∣∣∣∣int tr

Gi(h tsXs

)(tminus s)α2q dWi

s

∣∣∣∣q)le cq

ksumi=1

E

[(int tr

∣∣Gi(h tsXs

)∣∣2

(tminus s)αq ds)q2]

le cq(tminus r)(qminus2)2ksumi=1

Eint tr

∣∣Gi(h tsXs

)∣∣q(tminus s)α2 ds

le cq(tminus r)(qminus2)2int tr

E(1 +

∣∣Xs∣∣q)(tminus s)α2 ds

le Cq(tminus r)(qminus2)2(tminus r)1minusα2

(67)

From (H4) for Gi we deduce as in Lemma 41 that

E

(∣∣∣∣int r

0

[Gi(hrsXs

)minusGi(krsXs

)(rminus s)α2q

]dWi

s

∣∣∣∣q)

le cq(hminus k)qint r

0

E(1 +

∣∣Xs∣∣q)(rminus s)α2 dsle Cq(hminus k)qr1minusα2

(68)

From (H10) and the fact that

X isin⋂qgt1

⋂tgt0

Lq(Ωtimes (0 t)

)

E

(∣∣∣∣int r

0

[Gi(h tsXs

)(tminus s)α2q minus Gi

(hrsXs

)(rminus s)α2q

]dWi

s

∣∣∣∣q)

= E(∣∣∣∣int r

0

(1

(rminus s)α2q[Gi(h tsXs

)minusGi(hrsXs

)]

+Gi(h tsXs

)[ 1(tminus s)α2q minus

1(rminus s)α2q

])dWi

s

∣∣∣∣q)

le E(cq

int r0

(∣∣Gi(h tsXs

)minusGi(hrsXs

)∣∣q(rminus s)α

+∣∣Gi

(h tsXs

)∣∣q((rminus s)minusα2qminus (tminus s)minusα2q)q)ds

)

le cqKq(int r

0

( |tminus r|δq(rminus s)α E

(1 +

∣∣Xs∣∣q))ds

+int r

0

(E(1 +

∣∣Xs∣∣q)((rminus s)minusα2minus (tminus s)minusα2))ds)

le cq[|tminus r|δq

int r0

(rminus s)minusα2ds+int r

0

((rminus s)minusα2minus (tminus s)minusα2)ds]

le Cq(|tminus r|δqr1minusα2 +

(r1minusα2 + (tminus r)1minusα2minus t1minusα2))

(69)

Mahmoud M El-Borai et al 467

which from the above estimate yields

E(∣∣It(X h)minus Ir(X k)

∣∣q)le Cq

[|tminus r|1minusα2 + |tminus r|(qminusα)2

+ r1minusα2(|tminus r|δq + |hminus k|q)+(r1minusα2minus t1minusα2)]

(610)

this completes the proof

Acknowledgment

We are very grateful to the referees for valuable comments

References

[1] C Bender and R J Elliott Binary market models and discrete wick products preprint 2002[2] M A Berger and V J Mizel Volterra equations with Ito integrals I J Integral Equations 2

(1980) no 3 187ndash245[3] Volterra equations with Ito integrals II J Integral Equations 2 (1980) no 4 319ndash337[4] An extension of the stochastic integral Ann Probab 10 (1982) no 2 435ndash450[5] M M El-Borai Some probability densities and fundamental solutions of fractional evolution

equations Chaos Solitons Fractals 14 (2002) no 3 433ndash440[6] A M Kolodh On the existence solutions of stochastic Volterra equations Theory of Random

Processes 11 (1983) 51ndash57 (Russian)[7] B B Mandelbrot and J W Van Ness Fractional Brownian motions fractional noises and appli-

cations SIAM Rev 10 (1968) 422ndash437[8] P-A Meyer Flot drsquoune equation differentielle stochastique (drsquoapres Malliavin Bismut Kunita)

[Flow of a stochastic differential equation (following Malliavin Bismut Kunita)] Seminar onProbability XV (University of Strasbourg Strasbourg 19791980) Lecture Notes in Mathvol 850 Springer-Verlag Berlin 1981 pp 103ndash117 (French)

[9] D Nualart Noncausal stochastic integrals and calculus Stochastic Analysis and Related Topics(Silivri 1986) Lecture Notes in Math vol 1316 Springer-Verlag Berlin 1988 pp 80ndash129

[10] D Nualart and E Pardoux Stochastic calculus with anticipating integrands Probab TheoryRelated Fields 78 (1988) no 4 535ndash581

[11] D Nualart and M Zakai Generalized stochastic integrals and the Malliavin calculus ProbabTheory Related Fields 73 (1986) no 2 255ndash280

[12] D Ocone and E Pardoux A generalized Ito-Ventzell formula Application to a class of anticipat-ing stochastic differential equations Ann Inst H Poincare Probab Statist 25 (1989) no 139ndash71

[13] Linear stochastic differential equations with boundary conditions Probab Theory Re-lated Fields 82 (1989) no 4 489ndash526

[14] S Ogawa Sur la question drsquoexistence de solutions drsquoune equation differentielle stochastique dutype noncausal [On the existence of solutions of a stochastic differential equation of noncausaltype] J Math Kyoto Univ 24 (1984) no 4 699ndash704 (French)

[15] E Pardoux and P Protter A two-sided stochastic integral and its calculus Probab Theory RelatedFields 76 (1987) no 1 15ndash49

[16] Stochastic Volterra equations with anticipating coefficients Ann Probab 18 (1990)no 4 1635ndash1655

[17] P Protter Volterra equations driven by semimartingales Ann Probab 13 (1985) no 2 519ndash530

468 Volterra equations with fractional stochastic integrals

[18] A N V Rao and C P Tsokos On the existence uniqueness and stability behavior of a randomsolution to a nonlinear perturbed stochastic integro-differential equation Information andControl 27 (1975) 61ndash74

[19] Y Shiota A linear stochastic integral equation containing the extended Ito integral Math RepToyama Univ 9 (1986) 43ndash65

[20] A V Skorohod On a generalization of the stochastic integral Teor Verojatnost i Primenen 20(1975) no 2 223ndash238 (Russian)

[21] A-S Sznitman Martingales dependant drsquoun parametre une formule drsquoIto [Martingales depend-ing on a parameter an Ito formula] Z Wahrsch Verw Gebiete 60 (1982) no 1 41ndash70(French)

[22] W Wyss The fractional diffusion equation J Math Phys 27 (1986) no 11 2782ndash2785

Mahmoud M El-Borai Faculty of Science Alexandria University Alexandria EgyptE-mail address m m elboraiyahoocom

Khairia El-Said El-Nadi Faculty of Science Alexandria University Alexandria EgyptE-mail address khairia el saidhotmailcom

Osama L Mostafa Faculty of Science Alexandria University Alexandria EgyptE-mail address olmoustafayahoocom

Hamdy M Ahmed Faculty of Science Alexandria University Alexandria EgyptE-mail address hamdy 17egyahoocom

Mahmoud M El-Borai et al 455

2 The Skorohod integral

We will now define the Skorohod integral Most of this section is a review of some basicnotations and a few results from [4 16]

Let again Ω= C(R+Rk) let η be its Borel field and let P denote Wiener measure on(Ωη)

Wt(ω)= ω(t) (21)

Let η0t = σWs 0 le s lt t and ηt = η0

t orZ where Z denotes the class of sets which havezero P-measure of η

For hisin L2(R+Rk) we denote the Wiener integral by

W(h)=int T

0

(h(t)dWt

) (22)

LetA denote the dense subset of L2(ΩηP) consisting of those classes of random variablesof the form

F = f(W(h1) W

(hn))

(23)

where n isin N (N denotes the set of nonnegative integers) f isin Cinfinb (Rn) h1 hn isinL2(R+Rk) Cinfinb is the set of infinitely differentiable functions on [0b] whose derivativesof any order are null at b If F has the form (23) we define its derivative in the directioni as the process Di

tF t ge 0 defined by

DitF =

nsumk=1

part f

partxk

(W(h1) W

(hn))hik(t) (24)

DF will stand for the k-dimensional process DtF = (D1t F Dk

t F) t ge 0Proposition 21 The differential operators Di i= 1 k are unbounded closable opera-tors from L2(Ω) into L2(ΩtimesR+)

Let D12i be the closure of A with respect to the norm

Fi12 = F2 + middot DiF∥∥L2(R+)

∥∥2 (25)

where F2 = (E(F2))12 E(X) is the mathematical expectation of X and

g2L2(R+) =

intinfin0g2(t)dt (26)

Similarly the domain D12 =⋂ki=1D

12i is the closure of A with respect to the norm (see

[10])

F12 = F2 +ksumi=1

middot DiF∥∥L2(R+)

∥∥2 (27)

456 Volterra equations with fractional stochastic integrals

We identify Di and D with their closed extensions (D12 is the domain of D L2(Ω)rarrL2(ΩtimesR+Rk))

We denote byD12iloc the set of measurable Frsquos which are such that there exists a sequence

(ΩnFn) nisinN sub ηtimesD12i with the two properties

(i) Ωn uarrΩ as nrarrinfin(ii) Fn|Ωn = F|Ωn nisinN

For F isinD12iloc we define without ambiguity Di

tF =DitFn on ΩntimesR+ for all nisinN D12

loc isdefined similarly

For i = 1 k we define δi the Skorohod integral with respect to Wit as the adjoint

of Di that is Domδi (the set of adapted processes for the Skorohod integral) is the set ofuisin L2(ΩtimesR+) which are such that there exists a constant c with

∣∣∣∣Eintinfin

0DitFut dt

∣∣∣∣le cF2 forallF isin A (28)

If uisinDomδi δi(u) is defined as the unique element of L2(Ω) which satisfies

E(δi(u)F

)= Eintinfin

0DitFut dt forallF isinA (29)

Let L12i = L2(R+D12

i ) We have that L12i subDomδi and for uisin L12

i

E[δi(u)2]= E

intinfin0u2t dt+E

intinfin0

intinfin0DisutD

itus dsdt (210)

Note that if uisin L2loc(R+D12

i ) then u1[0T] isin L12i for any T gt 0 and we can write

int T0utdW

it = δi

(u1[0T]

) (211)

The Skorohod integral is a local operation on L2loc(R+D12

i ) in the sense that if uv isinL2

loc(R+D12i ) then

int t0 usdW

is =

int t0 vsdW

is as on ω us(ω)= vs(ω) for almost all s lt t

Let L12iloc denote the set of measurable processes u which are such that for any T gt 0

there exists a sequence

(ΩTn uTn

) nisinN

sub ηtimesL12i (212)

such that

(i) ΩTn uarrΩ as as nrarrinfin

(ii) u= uTn dPtimesdt ae on ΩTn times [0T] nisinN

Mahmoud M El-Borai et al 457

For uisin L12iloc we can define its Skorohod integral with respect to Wi

t by

int t0usdW

is =

int t0uTnsdW

is on ΩT

n times [0T] (213)

Finally L12 =⋂ki=1L

12i and L12

loc is defined similarly as L12iloc

We now introduce the particular class of integrands which we will use belowLet u R+timesΩtimesRP rarrR satisfy the following

(i) For all x isinRp (tω)rarr (tωx) is ηt progressively measurable(ii) For all (tω)isinR+timesΩ u(tωmiddot)isin C1(Rp)

(iii) For some increasing function φ R+rarrR+ |u(tωx)|+|uprime(tωx)|leφ(|x|) forall (tωx)isinR+timesΩtimesRp where uprime(tx) stands for the gradient (partupartx)(tx)

Let θ be a p-dimensional random vector such that

(iv) θ j isinD12i

⋂Linfin(Ω) j = 1 p

We fix T gt 0 and consider

Ii(x)=int T

0

u(tx)(T minus t)(1minusβ)2 dW

it 0 lt β lt 1 (214)

Define moreover vt = u(tθ)Under conditions (i) (ii) (iii) and (iv) the following proposition holdsIt is proved in [10] that

int T0u(tθ)dWi

t =int T

0u(tx)dWi

t |x=θ minusint T

0uprime(tθ)Di

tθdt (215)

The same relation is proved in [16] under slightly different conditions Equation (215) isused to define the Skorohod integral

int T0 u(tθ)dWi

t

Proposition 22 The random field Ii(x) x isinRp defined above possesses an as continu-ous modification so that the random variable Ii(θ) can be defined v isinDomδi (see [10 16])

Condition (iv) can be replaced by (ivprime) θ j isinD12iloc j = 1 p

Under conditions (i) (ii) (iii) and (ivprime) v isin (Domδi)loc in the sense that there existsa sequence (Ωnvn) nisinN sub ηtimesDomδi such that Ωn uarrΩ as and vn|Ωn = v|Ωn

Indeed let (Ωprimenθn) be a localizing sequence for θ in (D12

i )p and let ψn nisinN subCinfinc (RpRp) satisfy ψn(x)= x whenever |x| le n

Define vn(t) = u(tψn(θn)) Ωn =Ωprimen

⋂|θ| le n then (Ωnvn) n isin N satisfies theabove conditions

It is then natural to define for every εgt0 the Skorohod integralint Tminusε

0 (vt(T minus t)(1minusβ)2)

dWit by formula (215) and the latter coincides with

int Tminusε0 (vn(t)(T minus t)(1minusβ)2)dWi

t on Ωn

It is clear that limεrarr0int Tminusε

0 (vt(T minus t)(1minusβ)2)dWit exists in the mean by using the norm

middot 2

458 Volterra equations with fractional stochastic integrals

3 Statement of the problem interpretation of (12)

Our aim is to study the equation

Xt = X0 + aint t

0

F(tsXs

)(tminus s)1minusβ ds+ b

ksumi=1

int t0

Gi(Ht tsXs

)(tminus s)(1minusβ)2 dWi

s

0 lt β lt 1 a= 1Γ(β)

b = 1Γ((β+ 1)2

) (31)

We define D = (ts)isinR2+ 0le s lt t

The coefficients F and G are given as follows F DtimesRd rarrRd is measurable and foreach (sx)isinR+timesRd F(middotsx) is ηt progressively measurable on Ωtimes [s+infin)

For i= 1 k Gi RptimesDtimesRd rarrRd is measurable for each (h tx) Gi(h tmiddotx) is ηsprogressively measurable on Ωtimes [0 t] and for each (ω tsx) Gi(middot tsx) is of class C1Ht is a given progressively measurable p-dimensional process It will follow from thesehypotheses that we will be able to construct a progressively measurable solution Xt

Therefore for each t the process Gi(h tsXs) s isin [0T] is of the form vs = u(sθ)with u(sh)=Gi(h tsXs) and θ =Ht We will impose below conditions on G Ht andthe solution Xt so as to satisfy requirements (i) (ii) (iii) and (ivprime) of Section 2

In particular we will consider only nonanticipating solutions Therefore the stochasticintegrals in (31) will be interpreted according to (215) that is

int t0

Gi(Ht tsXs

)(tminus s)(1minusβ)2 dWi

s =int t

0

Gi(h tsXs

)(tminus s)(1minusβ)2 dW

is|h=Ht minus

int t0

Gprimei(Ht tsXs

)(tminus s)(1minusβ)2 Di

sHt ds (32)

In other words we can rewrite (31) as

Xt = X0 +int t

0F(tsXs

)ds+ b

ksumi=1

int t0

Gi(h tsXs

)(tminus s)(1minusβ)2 dW

is|h=Ht (33)

where

F(tsx)= a F(tsx)(tminus s)1minusβ minus b

ksumi=1

Gprimei(Ht tsx

)DisHt

(tminus s)(1minusβ)2 (34)

and the stochastic integrals are now the usual Ito integralsWe will show below that (33) makes sense for any progressively measurable process X

which satisfies X isin⋂tgt0Lq(0 t) as for some q gt p We will find such a solution to (33)

it will then follow from (32) that it is a solution to (31) Similarly uniqueness for (31)in the above class will follow from uniqueness for (33) in that class

4 Existence and uniqueness under strong hypotheses

We formulate a set of further hypotheses (those stated in Section 3 are assumed to holdthroughout the paper) under which we will establish a first result of the existence anduniqueness of a solution of (12)

Mahmoud M El-Borai et al 459

Let B be an open bounded subset of Rp K gt 0 and q gt p such that

(H1) X0 isin Lq(Ωη0PRd)(H2) P(Ht isin Bforallt ge 0)= 1(H3) H isin (L12)p |DsHt| le K as 0le s lt t(H4) |F(tsx)|+

sumki=1 |Gi(h tsx)|+

sumki=1 |Gprimei (h tsx)| le K(1 + |x|) for any 0le s lt t

hisin B x isinRd(H5) |F(tsx) minus F(ts y)| +

sumki=1 |Gi(h tsx) minus Gi(h ts y)| +

sumki=1 |Gprimei (h tsx)

minusGprimei (h ts y)| le K|xminus y| for any 0le s lt t hisin B and x y isinRd as

Note that q will be a fixed real number such that q gt p and Lqprog(Ωtimes (0 t)) will

stand for the space Lq(Ωtimes (0 t)ξtP times λ) where ξt denotes the σ-algebra of progres-sively measurable subsets of Ωtimes (0 t) and λ denotes the Lebesgue measure on (0 t) andset 1minusβ = αq 0 lt α lt 1

Lemma 41 Let X isin⋂tgt0Lqprog(Ωtimes (0 t)) where q gt p and suppose that (H4) is in force

Then for any T ge t gt 0 and iisin 1 k the random field

int t0

Gi(h tsXs

)(tminus s)α2q dWi

s hisin B

(41)

possesses an as continuous modification

Proof Using Burkholder-Gundy and Holderrsquos inequalities together with (H4) we obtain

E

(∣∣∣∣int t

0

Gi(h tsXs

)(tminus s)α2q dWi

s minusint t

0

Gi(k tsXs

)(tminus s)α2q dWi

s

∣∣∣∣q)

le c1Eint t

0

∣∣Gi(h tsXs

)minusGi(k tsXs

)∣∣q(tminus s)α2 ds

le c1Kq|hminus k|q

int t0

E(1 +

∣∣Xs∣∣q)(tminus s)α2 ds

le c2Kq|hminus k|q

int t0(tminus s)minusα2ds

le c3|hminus k|qt1minusα2

(42)

where c1 c2 and c3 are positive constants The result now follows from the multidimen-sional generalization of Kolmogrovrsquos lemma (see [1 8])

We can now assume that for fixed t the random field (41) is as continuous in hprovided X isin Lqprog(Ωtimes (0 t))

From X isin⋂tgt0Lqprog(Ωtimes (0 t)) define

It(X h)= bksumi=1

int t0

Gi(h tsXs

)(tminus s)α2q dWi

s hisinRp t gt 0

Jt(X)=int t

0F(tsXs

)ds+ It

(X Ht

)

(43)

460 Volterra equations with fractional stochastic integrals

Lemma 42 For any t gt 0 there is a constant c gt 0 such that

E(∣∣Jt(X)

∣∣q)le c(int t

0E(1 +

∣∣Xs∣∣q)

1(tminus s)α +

1(tminus s)α2

ds)

0 lt α lt 1 (44)

Proof

∣∣∣∣int t

0F(tsXs

)ds∣∣∣∣q

le cint t

0

∣∣F(tsXs)∣∣q(tminus s)α +

ksumi=1

∣∣Gprimei(h tsXs)|qh=Ht

∣∣DisHt

∣∣q(tminus s)α2 ds

le cint t

0

(1 +

∣∣Xs∣∣q)

1(tminus s)α +

1(tminus s)α2

ds

(45)

where we have used (H2) (H3) and (H4)

∣∣It(X Ht)∣∣le sup

hisinB

∣∣It(X h)∣∣ (46)

(c c are positive constants)It is easy to show using in particular (H4) and Lebesguersquos dominated convergence

theorem that the mapping

hminusrarr It(X h) (47)

from Rp into Lq(Ω) is differentiable and that

partIt(X h)parthj

= bint t

0

part

parthj

ksumi=1

(Gi(h tsXs

)(tminus s)α2q

)dWi

s (48)

Since q gt p we can infer from Sobolevrsquos embedding theorem (see [13]) that

E

(suphisinB

∣∣It(X h)∣∣q)le cE

intB

(∣∣It(X h)∣∣q +

psumj=1

∣∣∣∣ partItparthj(X h)

∣∣∣∣q)dh (49)

It then follows from the Burkholder-Gundy inequality that

E

(suphisinB

∣∣It(X h)∣∣q)le cE

intB

int t0

ksumi=1

(∣∣Gi(h tsXs

)∣∣q(tminus s)α2 +

psumj=1

∣∣∣∣ part

parthj

Gi(h tsXs

)(tminus s)α2q

∣∣∣∣q)dsdh

le c(int

Bdh)(Eint t

0

(1 +

∣∣Xs∣∣q)(tminus s)α2 ds

)

le cint t

0

E(1 +

∣∣Xs∣∣q)(tminus s)α2 ds

(410)

where we have used (H4) and B is bounded From (45) and (410) the proof is complete

A similar argument using (H5) instead of (H4) yields the following lemma

Mahmoud M El-Borai et al 461

Lemma 43 For any 0 lt t le T there exists a constant c gt 0 such that

E(∣∣Jt(X)minus Jt(Y)

∣∣q)le cint t

0E∣∣XsminusYs∣∣q

1

(tminus s)α +1

(tminus s)α2ds (411)

We are now in a position to prove the main result of this section

Theorem 44 Under conditions (H1) (H2) (H3) (H4) and (H5) there exists a uniqueelement X isin ⋂tgt0L

qprog(Ωtimes (0 t)) which solves (31) Moreover if τ is a stopping time

uniqueness holds on the random interval [0τ]

Proof Equation (31) can be rewritten as

Xt = X0 + Jt(X) t ge 0 (412)

Uniqueness Let X Y isin⋂tgt0Lqprog(Ωtimes (0 t)) and let τ be a stopping time such that

Xt = X0 + Jt(X) Yt = X0 + Jt(Y) 0le t le τ (413)

From Lemma 43

E(∣∣Xt minusYt∣∣q)= E(∣∣Jt(X)minus Jt(Y)

∣∣q)le c

int t0E∣∣XsminusYs∣∣q

1

(tminus s)α +1

(tminus s)α2ds

(414)

Set (E|Xt minusYt|q)= γt It is easy to see that

γt le 2cη(t)int t

0

γs(tminus s)α ds (415)

where η(t)=Max(1 tα) and c is a positive constant Thus (see [5 21])

γt le M2(Γ(tminusα)

)2

Γ(2(1minusα)

) int t0

γs(tminus s)2αminus1

ds (416)

and by a classical argument we get

γt le Mn(Γ(tminusα)

)nΓ(n(1minusα)

) int t0

γs(tminus s)nαminus(nminus1)

ds (417)

where M = 2cη(T) Thus for sufficiently large n n gt 1(1minusα) we get from (417)

γt le Kn

Γ(n(1minusα)

) int t0γs ds (418)

where K is a positive constantTaking the limit as nrarrinfin we find that (418) leads to γt = 0To prove the existence we define a sequence Xn

t 0le t le T n= 012 as follows

Xot = Xo Xn+1

t = Xo + Jt(Xn) T le t le 0 (419)

462 Volterra equations with fractional stochastic integrals

Using Lemma 42 we can see that

Xn isin⋂tgt0

Lqprog

(Ωtimes [0T]

) (420)

It then follows from Lemma 43 that

gn+1(t)le 2cη(t)int t

0

gn(s)(tminus s)α ds (421)

where

gn+1(t)= E(∣∣Xn+1t minusXn

t

∣∣q) (422)

By using a similar argument we can write

gn le Kn

Γ(n(1minusα)

) int t0go(s)ds (423)

The last estimation implies that Xn is a Cauchy sequence in Lqprog(Ωtimes [0T]) Then there

exists X such that Xn rarr X in⋂tgt0L

qprog(Ωtimes [0T]) and again using Lemma 43 we can

pass to the limit in (420) yielding that X solves (412)

5 An existence and uniqueness result under weaker assumptions

We formulate a new set of weaker hypotheses

(H1prime) X0 is η0 measurable(H2prime) H isin (L12

loc)pHt is a progressively measurable process which can be localized in(L12)p by a progressively measurable sequence

We assume that there exists an increasing progressively measurable process Ut t ge 0with values in R+ such that

(H3prime) |Ht|+sumk

i=1 |DisHt| leUt as 0le s lt t

Finally we suppose that for any N gt 0 there exists an increasing progressively measur-able process VN

t t ge 0 with values in R+ such that

(H4prime) |F(tsx)|+sumk

i=1 |Gi(h tsx)|+sumk

i=1 |Gprimei (h tsx)| le VNt (1+|x|) for all |h|leN

0le s lt t and x isinRd(H5prime) |F(tsx) minus F(ts y)| +

sumki=1 |Gi(h tsx) minus Gi(h ts y)| +

sumki=1 |Gprimei (h tsx)

minusGprimei (h ts y)| leVNt |xminus y| for all |h| leN 0le s lt t x y isinRd

Let again q be a fixed real number with q gt p and set 1minusβ = αq 0 lt α lt 1 We havethe following theorem

Theorem 51 Equation (31) has a unique solution in the class of progressively measurableprocesses which satisfy

X isin⋂tgt0

Lq(0 t) as (51)

Mahmoud M El-Borai et al 463

Proof (a) We first see how (31) makes sense if X isin⋂tgt0Lq(0 t) as

That is we have to show that for fixed t gt 0

int t0

Gi(h tsXs

)(tminus s)α2q dWi

s hisinRP

(52)

is a well-defined random field which possesses an as continuous versionFor that sake we define

τn = inftint t

0

∣∣Xs∣∣q(tminus s)α dsge n or VN

t ge n (53)

The argument of Lemma 41 can be used to show that

hminusrarrint tandτn

0

Gi(h tsXs

)(tminus s)α2q dWi

s (54)

possesses an as continuous modification on |h| le N Since this is true for any n andN and cupnτn ge t =Ω as the result follows

(b) Existence we want to show existence on an arbitrary interval [0T] (T will be fixedbelow)

Let Hn nisinN denote a progressively measurable localizing sequence forH in (L12)p

on [0T] Since from (H3prime) suptleT |Ht| is as finite we can and do assume without lossof generality that

∣∣Hnt (ω)

∣∣le n forall(tω)isin [0T]timesΩ (55)

Note that Ht(ω) =Hnt (ω) as on ΩT

n for all t isin [0T] where ΩTn uarrΩ as as nrarrinfin We

moreover define Xn0 = X01|X0|len Sn = inft supsltt |DsH

nt |orVn

t ge n We consider theequation

Xnt = Xn

0 +int t

0Fn(tsXn

s

)ds+ b

ksumi=1

int t0

Gni

(h tsXn

s

)(tminus s)α2q dWi

s|h=Hnt (56)

where

Fn(tsx)= 1[0Sn](s)

[aF(tsx)(tminus s)αq minus b

ksumi=1

Gprimei(Hnt tsx

)DisH

nt

(tminus s)α2q]

Gni (h tsx)= 1[0Sn](s)Gi(h tsx)

(57)

It is easy to see that theorem (415) applies to (56)Define

Sn(ω)=Sn(ω)and inf

t le T

int t0

∣∣Hs(ω)minusHns (ω)

∣∣ds gt 0

if∣∣X0(ω)

∣∣ lt n

0 otherwise(58)

464 Volterra equations with fractional stochastic integrals

Sn is a stopping time and it follows from the uniqueness part of Theorem 44 that ifmgt n

Xmt = Xn

t on[0 Sn

]as (59)

Since moreover Sn = T uarrΩ as we can define the process Xt on [0T] by Xt = Xnt

on [0 Sn] for all nisinNClearly X isin Lq(0T) as and solves (31) on [0T] Since T is arbitrary the existence

is proved(c) Uniqueness it suffices to prove uniqueness on an arbitrary interval [0T] Let

Xt t isin [0T] be a progressively measurable process such that X isin Lq(0T) as andX solves (31) It suffices to show that X coincides with the solution we have just con-structed

Let

Sn(ω)= S(ω)and inft le T

int t0

∣∣Xs(ω)∣∣q

(tminus s)α ds gt n

Xnt = XtandSn

(510)

Sn isin Lq(Ωtimes [0T]) and it solves (56) with Sn replaced by SnThen

Xnt (ω)= Xt(ω)timesP ae on

[0 Sn

] (511)

where is the Lebesgue measure on the real lineThe result follows from the fact that Sn ge T uarrΩ asNote that the above solution satisfies in fact X isin⋂qgt1

⋂tgt0L

q(0 t) as

6 Continuity of the solution

We want to give additional conditions under which the solution of (31) is an as contin-uous process

(H6) For all (sx)isinR+timesRd trarr F(tsx)(tminus s)αq is as continuous on (s+infin)(H7) Ht t ge 0 is as continuous(H8) For all iisin 1 k sisinR+ trarrDi

sHt is as continuous on (s+infin)(H9) For all (sx) isin R+ timesRd i isin 1 k (th)rarr Gprimei (h tsx)(tminus s)α2q is as con-

tinuous on (s+infin)timesRP

We also suppose that there exist δ gt 0 gt 0 such that for all N gt 0 |h| le N 0 le s lttand r x isinRd |tminus r| le 1

(H10) There exists an increasing process VNt t ge 0 such that

∣∣Gi(h tsx)minusGi(hrsx)∣∣leVN

t |tminus r|δ(1 + |x|) (61)

Theorem 61 Under conditions (H1prime) (H2prime) (H3prime) (H4prime) (H5prime) and (H6) (H7) (H8)(H9) (H10) the unique solution of (31) (which is progressively measurable and belongs asto⋂qgt1⋂tgt0L

q(0 t)) has as continuous modification

Mahmoud M El-Borai et al 465

Proof We need to show only that whenever X isin⋂qgt1⋂tgt0L

q(0 t) as Jt(X) t gt 0 hasan as continuous modification

(a) We first show that trarr int t0 F(tsXs)ds is as continuous

Note that (H6) (H7) (H8) and (H9) imply that for all (sx)isinR+timesRd trarr F(tsx)is as continuous on (s+infin)

Moreover from (H2prime) (H3prime) (H4prime) and the fact that X isin⋂qgt1⋂tgt0L

q(0 t) as forany T gt 0 there exists a process ZTs sisin [0T] such that

∣∣F(tsXs)∣∣le ZTs ∣∣Gprimei(Ht tsXs

)DisHt

∣∣le ZTs 0le s lt t le T asint t

0ZTs

1

(tminus s)α2q +1

(tminus s)αqds ltinfin as

(62)

Let first tn nisinN be a sequence such that tn lt t for any n and tnrarr t as nrarrinfin then

int t0F(tsXs

)dsminus

int tn0F(tnsXs

)ds=

int ttnF(tsXs

)ds+

int tn0

[F(tsXs

)minus F(tnsXs)]ds

∣∣∣∣int ttnF(tsXs

)ds∣∣∣∣le

int ttnZTs

1

(tminus s)α2q +1

(tminus s)αqds

(63)

and the latter tends as to 0 as nrarrinfin

∣∣∣∣int tn

0

[F(tsXs

)minus F(tnsXs)]ds∣∣∣∣le

int t0

∣∣F(tsXs)minus F(tnsXs)∣∣ds (64)

which tends to 0 as nrarrinfin A similar argument gives the same result when tn gt t tnrarr t(b) We next show that t rarr It(X Ht) possesses an as continuous modification This

follows from (H7) and

(th)minusrarr It(X h) (65)

has an as continuous modificationBy localization it suffices to prove (65) under assumptions (H2) (H3) (H4) (H5)

and (H6) (H7) (H8) (H9) (H10) with VNt (ω) in (H10) replaced by a constant K

and in case X0 isin⋂qgt1L

q(ΩRd)It then suffices to show that under the above hypotheses there exists Cq gt 0 such that

for any hk isinRP the numbers tr will be positive Suppose to fix the ideas that 0le r lt tthen

It(X h)minus Ir(X k)=int tr

Gi(h tsXs

)(tminus s)α2q dWi

s +int r

0

[Gi(h tsXs

)(tminus s)α2q minus Gi

(hrsXs

)(rminus s)α2q

]dWi

s

+int r

0

[Gi(h tsXs

)(rminus s)α2q minus Gi

(krsXs

)(rminus s)α2q

]dWi

s

(66)

466 Volterra equations with fractional stochastic integrals

It follows from the Burkholder-Gundy inequality that

E(∣∣∣∣int tr

Gi(h tsXs

)(tminus s)α2q dWi

s

∣∣∣∣q)le cq

ksumi=1

E

[(int tr

∣∣Gi(h tsXs

)∣∣2

(tminus s)αq ds)q2]

le cq(tminus r)(qminus2)2ksumi=1

Eint tr

∣∣Gi(h tsXs

)∣∣q(tminus s)α2 ds

le cq(tminus r)(qminus2)2int tr

E(1 +

∣∣Xs∣∣q)(tminus s)α2 ds

le Cq(tminus r)(qminus2)2(tminus r)1minusα2

(67)

From (H4) for Gi we deduce as in Lemma 41 that

E

(∣∣∣∣int r

0

[Gi(hrsXs

)minusGi(krsXs

)(rminus s)α2q

]dWi

s

∣∣∣∣q)

le cq(hminus k)qint r

0

E(1 +

∣∣Xs∣∣q)(rminus s)α2 dsle Cq(hminus k)qr1minusα2

(68)

From (H10) and the fact that

X isin⋂qgt1

⋂tgt0

Lq(Ωtimes (0 t)

)

E

(∣∣∣∣int r

0

[Gi(h tsXs

)(tminus s)α2q minus Gi

(hrsXs

)(rminus s)α2q

]dWi

s

∣∣∣∣q)

= E(∣∣∣∣int r

0

(1

(rminus s)α2q[Gi(h tsXs

)minusGi(hrsXs

)]

+Gi(h tsXs

)[ 1(tminus s)α2q minus

1(rminus s)α2q

])dWi

s

∣∣∣∣q)

le E(cq

int r0

(∣∣Gi(h tsXs

)minusGi(hrsXs

)∣∣q(rminus s)α

+∣∣Gi

(h tsXs

)∣∣q((rminus s)minusα2qminus (tminus s)minusα2q)q)ds

)

le cqKq(int r

0

( |tminus r|δq(rminus s)α E

(1 +

∣∣Xs∣∣q))ds

+int r

0

(E(1 +

∣∣Xs∣∣q)((rminus s)minusα2minus (tminus s)minusα2))ds)

le cq[|tminus r|δq

int r0

(rminus s)minusα2ds+int r

0

((rminus s)minusα2minus (tminus s)minusα2)ds]

le Cq(|tminus r|δqr1minusα2 +

(r1minusα2 + (tminus r)1minusα2minus t1minusα2))

(69)

Mahmoud M El-Borai et al 467

which from the above estimate yields

E(∣∣It(X h)minus Ir(X k)

∣∣q)le Cq

[|tminus r|1minusα2 + |tminus r|(qminusα)2

+ r1minusα2(|tminus r|δq + |hminus k|q)+(r1minusα2minus t1minusα2)]

(610)

this completes the proof

Acknowledgment

We are very grateful to the referees for valuable comments

References

[1] C Bender and R J Elliott Binary market models and discrete wick products preprint 2002[2] M A Berger and V J Mizel Volterra equations with Ito integrals I J Integral Equations 2

(1980) no 3 187ndash245[3] Volterra equations with Ito integrals II J Integral Equations 2 (1980) no 4 319ndash337[4] An extension of the stochastic integral Ann Probab 10 (1982) no 2 435ndash450[5] M M El-Borai Some probability densities and fundamental solutions of fractional evolution

equations Chaos Solitons Fractals 14 (2002) no 3 433ndash440[6] A M Kolodh On the existence solutions of stochastic Volterra equations Theory of Random

Processes 11 (1983) 51ndash57 (Russian)[7] B B Mandelbrot and J W Van Ness Fractional Brownian motions fractional noises and appli-

cations SIAM Rev 10 (1968) 422ndash437[8] P-A Meyer Flot drsquoune equation differentielle stochastique (drsquoapres Malliavin Bismut Kunita)

[Flow of a stochastic differential equation (following Malliavin Bismut Kunita)] Seminar onProbability XV (University of Strasbourg Strasbourg 19791980) Lecture Notes in Mathvol 850 Springer-Verlag Berlin 1981 pp 103ndash117 (French)

[9] D Nualart Noncausal stochastic integrals and calculus Stochastic Analysis and Related Topics(Silivri 1986) Lecture Notes in Math vol 1316 Springer-Verlag Berlin 1988 pp 80ndash129

[10] D Nualart and E Pardoux Stochastic calculus with anticipating integrands Probab TheoryRelated Fields 78 (1988) no 4 535ndash581

[11] D Nualart and M Zakai Generalized stochastic integrals and the Malliavin calculus ProbabTheory Related Fields 73 (1986) no 2 255ndash280

[12] D Ocone and E Pardoux A generalized Ito-Ventzell formula Application to a class of anticipat-ing stochastic differential equations Ann Inst H Poincare Probab Statist 25 (1989) no 139ndash71

[13] Linear stochastic differential equations with boundary conditions Probab Theory Re-lated Fields 82 (1989) no 4 489ndash526

[14] S Ogawa Sur la question drsquoexistence de solutions drsquoune equation differentielle stochastique dutype noncausal [On the existence of solutions of a stochastic differential equation of noncausaltype] J Math Kyoto Univ 24 (1984) no 4 699ndash704 (French)

[15] E Pardoux and P Protter A two-sided stochastic integral and its calculus Probab Theory RelatedFields 76 (1987) no 1 15ndash49

[16] Stochastic Volterra equations with anticipating coefficients Ann Probab 18 (1990)no 4 1635ndash1655

[17] P Protter Volterra equations driven by semimartingales Ann Probab 13 (1985) no 2 519ndash530

468 Volterra equations with fractional stochastic integrals

[18] A N V Rao and C P Tsokos On the existence uniqueness and stability behavior of a randomsolution to a nonlinear perturbed stochastic integro-differential equation Information andControl 27 (1975) 61ndash74

[19] Y Shiota A linear stochastic integral equation containing the extended Ito integral Math RepToyama Univ 9 (1986) 43ndash65

[20] A V Skorohod On a generalization of the stochastic integral Teor Verojatnost i Primenen 20(1975) no 2 223ndash238 (Russian)

[21] A-S Sznitman Martingales dependant drsquoun parametre une formule drsquoIto [Martingales depend-ing on a parameter an Ito formula] Z Wahrsch Verw Gebiete 60 (1982) no 1 41ndash70(French)

[22] W Wyss The fractional diffusion equation J Math Phys 27 (1986) no 11 2782ndash2785

Mahmoud M El-Borai Faculty of Science Alexandria University Alexandria EgyptE-mail address m m elboraiyahoocom

Khairia El-Said El-Nadi Faculty of Science Alexandria University Alexandria EgyptE-mail address khairia el saidhotmailcom

Osama L Mostafa Faculty of Science Alexandria University Alexandria EgyptE-mail address olmoustafayahoocom

Hamdy M Ahmed Faculty of Science Alexandria University Alexandria EgyptE-mail address hamdy 17egyahoocom

456 Volterra equations with fractional stochastic integrals

We identify Di and D with their closed extensions (D12 is the domain of D L2(Ω)rarrL2(ΩtimesR+Rk))

We denote byD12iloc the set of measurable Frsquos which are such that there exists a sequence

(ΩnFn) nisinN sub ηtimesD12i with the two properties

(i) Ωn uarrΩ as nrarrinfin(ii) Fn|Ωn = F|Ωn nisinN

For F isinD12iloc we define without ambiguity Di

tF =DitFn on ΩntimesR+ for all nisinN D12

loc isdefined similarly

For i = 1 k we define δi the Skorohod integral with respect to Wit as the adjoint

of Di that is Domδi (the set of adapted processes for the Skorohod integral) is the set ofuisin L2(ΩtimesR+) which are such that there exists a constant c with

∣∣∣∣Eintinfin

0DitFut dt

∣∣∣∣le cF2 forallF isin A (28)

If uisinDomδi δi(u) is defined as the unique element of L2(Ω) which satisfies

E(δi(u)F

)= Eintinfin

0DitFut dt forallF isinA (29)

Let L12i = L2(R+D12

i ) We have that L12i subDomδi and for uisin L12

i

E[δi(u)2]= E

intinfin0u2t dt+E

intinfin0

intinfin0DisutD

itus dsdt (210)

Note that if uisin L2loc(R+D12

i ) then u1[0T] isin L12i for any T gt 0 and we can write

int T0utdW

it = δi

(u1[0T]

) (211)

The Skorohod integral is a local operation on L2loc(R+D12

i ) in the sense that if uv isinL2

loc(R+D12i ) then

int t0 usdW

is =

int t0 vsdW

is as on ω us(ω)= vs(ω) for almost all s lt t

Let L12iloc denote the set of measurable processes u which are such that for any T gt 0

there exists a sequence

(ΩTn uTn

) nisinN

sub ηtimesL12i (212)

such that

(i) ΩTn uarrΩ as as nrarrinfin

(ii) u= uTn dPtimesdt ae on ΩTn times [0T] nisinN

Mahmoud M El-Borai et al 457

For uisin L12iloc we can define its Skorohod integral with respect to Wi

t by

int t0usdW

is =

int t0uTnsdW

is on ΩT

n times [0T] (213)

Finally L12 =⋂ki=1L

12i and L12

loc is defined similarly as L12iloc

We now introduce the particular class of integrands which we will use belowLet u R+timesΩtimesRP rarrR satisfy the following

(i) For all x isinRp (tω)rarr (tωx) is ηt progressively measurable(ii) For all (tω)isinR+timesΩ u(tωmiddot)isin C1(Rp)

(iii) For some increasing function φ R+rarrR+ |u(tωx)|+|uprime(tωx)|leφ(|x|) forall (tωx)isinR+timesΩtimesRp where uprime(tx) stands for the gradient (partupartx)(tx)

Let θ be a p-dimensional random vector such that

(iv) θ j isinD12i

⋂Linfin(Ω) j = 1 p

We fix T gt 0 and consider

Ii(x)=int T

0

u(tx)(T minus t)(1minusβ)2 dW

it 0 lt β lt 1 (214)

Define moreover vt = u(tθ)Under conditions (i) (ii) (iii) and (iv) the following proposition holdsIt is proved in [10] that

int T0u(tθ)dWi

t =int T

0u(tx)dWi

t |x=θ minusint T

0uprime(tθ)Di

tθdt (215)

The same relation is proved in [16] under slightly different conditions Equation (215) isused to define the Skorohod integral

int T0 u(tθ)dWi

t

Proposition 22 The random field Ii(x) x isinRp defined above possesses an as continu-ous modification so that the random variable Ii(θ) can be defined v isinDomδi (see [10 16])

Condition (iv) can be replaced by (ivprime) θ j isinD12iloc j = 1 p

Under conditions (i) (ii) (iii) and (ivprime) v isin (Domδi)loc in the sense that there existsa sequence (Ωnvn) nisinN sub ηtimesDomδi such that Ωn uarrΩ as and vn|Ωn = v|Ωn

Indeed let (Ωprimenθn) be a localizing sequence for θ in (D12

i )p and let ψn nisinN subCinfinc (RpRp) satisfy ψn(x)= x whenever |x| le n

Define vn(t) = u(tψn(θn)) Ωn =Ωprimen

⋂|θ| le n then (Ωnvn) n isin N satisfies theabove conditions

It is then natural to define for every εgt0 the Skorohod integralint Tminusε

0 (vt(T minus t)(1minusβ)2)

dWit by formula (215) and the latter coincides with

int Tminusε0 (vn(t)(T minus t)(1minusβ)2)dWi

t on Ωn

It is clear that limεrarr0int Tminusε

0 (vt(T minus t)(1minusβ)2)dWit exists in the mean by using the norm

middot 2

458 Volterra equations with fractional stochastic integrals

3 Statement of the problem interpretation of (12)

Our aim is to study the equation

Xt = X0 + aint t

0

F(tsXs

)(tminus s)1minusβ ds+ b

ksumi=1

int t0

Gi(Ht tsXs

)(tminus s)(1minusβ)2 dWi

s

0 lt β lt 1 a= 1Γ(β)

b = 1Γ((β+ 1)2

) (31)

We define D = (ts)isinR2+ 0le s lt t

The coefficients F and G are given as follows F DtimesRd rarrRd is measurable and foreach (sx)isinR+timesRd F(middotsx) is ηt progressively measurable on Ωtimes [s+infin)

For i= 1 k Gi RptimesDtimesRd rarrRd is measurable for each (h tx) Gi(h tmiddotx) is ηsprogressively measurable on Ωtimes [0 t] and for each (ω tsx) Gi(middot tsx) is of class C1Ht is a given progressively measurable p-dimensional process It will follow from thesehypotheses that we will be able to construct a progressively measurable solution Xt

Therefore for each t the process Gi(h tsXs) s isin [0T] is of the form vs = u(sθ)with u(sh)=Gi(h tsXs) and θ =Ht We will impose below conditions on G Ht andthe solution Xt so as to satisfy requirements (i) (ii) (iii) and (ivprime) of Section 2

In particular we will consider only nonanticipating solutions Therefore the stochasticintegrals in (31) will be interpreted according to (215) that is

int t0

Gi(Ht tsXs

)(tminus s)(1minusβ)2 dWi

s =int t

0

Gi(h tsXs

)(tminus s)(1minusβ)2 dW

is|h=Ht minus

int t0

Gprimei(Ht tsXs

)(tminus s)(1minusβ)2 Di

sHt ds (32)

In other words we can rewrite (31) as

Xt = X0 +int t

0F(tsXs

)ds+ b

ksumi=1

int t0

Gi(h tsXs

)(tminus s)(1minusβ)2 dW

is|h=Ht (33)

where

F(tsx)= a F(tsx)(tminus s)1minusβ minus b

ksumi=1

Gprimei(Ht tsx

)DisHt

(tminus s)(1minusβ)2 (34)

and the stochastic integrals are now the usual Ito integralsWe will show below that (33) makes sense for any progressively measurable process X

which satisfies X isin⋂tgt0Lq(0 t) as for some q gt p We will find such a solution to (33)

it will then follow from (32) that it is a solution to (31) Similarly uniqueness for (31)in the above class will follow from uniqueness for (33) in that class

4 Existence and uniqueness under strong hypotheses

We formulate a set of further hypotheses (those stated in Section 3 are assumed to holdthroughout the paper) under which we will establish a first result of the existence anduniqueness of a solution of (12)

Mahmoud M El-Borai et al 459

Let B be an open bounded subset of Rp K gt 0 and q gt p such that

(H1) X0 isin Lq(Ωη0PRd)(H2) P(Ht isin Bforallt ge 0)= 1(H3) H isin (L12)p |DsHt| le K as 0le s lt t(H4) |F(tsx)|+

sumki=1 |Gi(h tsx)|+

sumki=1 |Gprimei (h tsx)| le K(1 + |x|) for any 0le s lt t

hisin B x isinRd(H5) |F(tsx) minus F(ts y)| +

sumki=1 |Gi(h tsx) minus Gi(h ts y)| +

sumki=1 |Gprimei (h tsx)

minusGprimei (h ts y)| le K|xminus y| for any 0le s lt t hisin B and x y isinRd as

Note that q will be a fixed real number such that q gt p and Lqprog(Ωtimes (0 t)) will

stand for the space Lq(Ωtimes (0 t)ξtP times λ) where ξt denotes the σ-algebra of progres-sively measurable subsets of Ωtimes (0 t) and λ denotes the Lebesgue measure on (0 t) andset 1minusβ = αq 0 lt α lt 1

Lemma 41 Let X isin⋂tgt0Lqprog(Ωtimes (0 t)) where q gt p and suppose that (H4) is in force

Then for any T ge t gt 0 and iisin 1 k the random field

int t0

Gi(h tsXs

)(tminus s)α2q dWi

s hisin B

(41)

possesses an as continuous modification

Proof Using Burkholder-Gundy and Holderrsquos inequalities together with (H4) we obtain

E

(∣∣∣∣int t

0

Gi(h tsXs

)(tminus s)α2q dWi

s minusint t

0

Gi(k tsXs

)(tminus s)α2q dWi

s

∣∣∣∣q)

le c1Eint t

0

∣∣Gi(h tsXs

)minusGi(k tsXs

)∣∣q(tminus s)α2 ds

le c1Kq|hminus k|q

int t0

E(1 +

∣∣Xs∣∣q)(tminus s)α2 ds

le c2Kq|hminus k|q

int t0(tminus s)minusα2ds

le c3|hminus k|qt1minusα2

(42)

where c1 c2 and c3 are positive constants The result now follows from the multidimen-sional generalization of Kolmogrovrsquos lemma (see [1 8])

We can now assume that for fixed t the random field (41) is as continuous in hprovided X isin Lqprog(Ωtimes (0 t))

From X isin⋂tgt0Lqprog(Ωtimes (0 t)) define

It(X h)= bksumi=1

int t0

Gi(h tsXs

)(tminus s)α2q dWi

s hisinRp t gt 0

Jt(X)=int t

0F(tsXs

)ds+ It

(X Ht

)

(43)

460 Volterra equations with fractional stochastic integrals

Lemma 42 For any t gt 0 there is a constant c gt 0 such that

E(∣∣Jt(X)

∣∣q)le c(int t

0E(1 +

∣∣Xs∣∣q)

1(tminus s)α +

1(tminus s)α2

ds)

0 lt α lt 1 (44)

Proof

∣∣∣∣int t

0F(tsXs

)ds∣∣∣∣q

le cint t

0

∣∣F(tsXs)∣∣q(tminus s)α +

ksumi=1

∣∣Gprimei(h tsXs)|qh=Ht

∣∣DisHt

∣∣q(tminus s)α2 ds

le cint t

0

(1 +

∣∣Xs∣∣q)

1(tminus s)α +

1(tminus s)α2

ds

(45)

where we have used (H2) (H3) and (H4)

∣∣It(X Ht)∣∣le sup

hisinB

∣∣It(X h)∣∣ (46)

(c c are positive constants)It is easy to show using in particular (H4) and Lebesguersquos dominated convergence

theorem that the mapping

hminusrarr It(X h) (47)

from Rp into Lq(Ω) is differentiable and that

partIt(X h)parthj

= bint t

0

part

parthj

ksumi=1

(Gi(h tsXs

)(tminus s)α2q

)dWi

s (48)

Since q gt p we can infer from Sobolevrsquos embedding theorem (see [13]) that

E

(suphisinB

∣∣It(X h)∣∣q)le cE

intB

(∣∣It(X h)∣∣q +

psumj=1

∣∣∣∣ partItparthj(X h)

∣∣∣∣q)dh (49)

It then follows from the Burkholder-Gundy inequality that

E

(suphisinB

∣∣It(X h)∣∣q)le cE

intB

int t0

ksumi=1

(∣∣Gi(h tsXs

)∣∣q(tminus s)α2 +

psumj=1

∣∣∣∣ part

parthj

Gi(h tsXs

)(tminus s)α2q

∣∣∣∣q)dsdh

le c(int

Bdh)(Eint t

0

(1 +

∣∣Xs∣∣q)(tminus s)α2 ds

)

le cint t

0

E(1 +

∣∣Xs∣∣q)(tminus s)α2 ds

(410)

where we have used (H4) and B is bounded From (45) and (410) the proof is complete

A similar argument using (H5) instead of (H4) yields the following lemma

Mahmoud M El-Borai et al 461

Lemma 43 For any 0 lt t le T there exists a constant c gt 0 such that

E(∣∣Jt(X)minus Jt(Y)

∣∣q)le cint t

0E∣∣XsminusYs∣∣q

1

(tminus s)α +1

(tminus s)α2ds (411)

We are now in a position to prove the main result of this section

Theorem 44 Under conditions (H1) (H2) (H3) (H4) and (H5) there exists a uniqueelement X isin ⋂tgt0L

qprog(Ωtimes (0 t)) which solves (31) Moreover if τ is a stopping time

uniqueness holds on the random interval [0τ]

Proof Equation (31) can be rewritten as

Xt = X0 + Jt(X) t ge 0 (412)

Uniqueness Let X Y isin⋂tgt0Lqprog(Ωtimes (0 t)) and let τ be a stopping time such that

Xt = X0 + Jt(X) Yt = X0 + Jt(Y) 0le t le τ (413)

From Lemma 43

E(∣∣Xt minusYt∣∣q)= E(∣∣Jt(X)minus Jt(Y)

∣∣q)le c

int t0E∣∣XsminusYs∣∣q

1

(tminus s)α +1

(tminus s)α2ds

(414)

Set (E|Xt minusYt|q)= γt It is easy to see that

γt le 2cη(t)int t

0

γs(tminus s)α ds (415)

where η(t)=Max(1 tα) and c is a positive constant Thus (see [5 21])

γt le M2(Γ(tminusα)

)2

Γ(2(1minusα)

) int t0

γs(tminus s)2αminus1

ds (416)

and by a classical argument we get

γt le Mn(Γ(tminusα)

)nΓ(n(1minusα)

) int t0

γs(tminus s)nαminus(nminus1)

ds (417)

where M = 2cη(T) Thus for sufficiently large n n gt 1(1minusα) we get from (417)

γt le Kn

Γ(n(1minusα)

) int t0γs ds (418)

where K is a positive constantTaking the limit as nrarrinfin we find that (418) leads to γt = 0To prove the existence we define a sequence Xn

t 0le t le T n= 012 as follows

Xot = Xo Xn+1

t = Xo + Jt(Xn) T le t le 0 (419)

462 Volterra equations with fractional stochastic integrals

Using Lemma 42 we can see that

Xn isin⋂tgt0

Lqprog

(Ωtimes [0T]

) (420)

It then follows from Lemma 43 that

gn+1(t)le 2cη(t)int t

0

gn(s)(tminus s)α ds (421)

where

gn+1(t)= E(∣∣Xn+1t minusXn

t

∣∣q) (422)

By using a similar argument we can write

gn le Kn

Γ(n(1minusα)

) int t0go(s)ds (423)

The last estimation implies that Xn is a Cauchy sequence in Lqprog(Ωtimes [0T]) Then there

exists X such that Xn rarr X in⋂tgt0L

qprog(Ωtimes [0T]) and again using Lemma 43 we can

pass to the limit in (420) yielding that X solves (412)

5 An existence and uniqueness result under weaker assumptions

We formulate a new set of weaker hypotheses

(H1prime) X0 is η0 measurable(H2prime) H isin (L12

loc)pHt is a progressively measurable process which can be localized in(L12)p by a progressively measurable sequence

We assume that there exists an increasing progressively measurable process Ut t ge 0with values in R+ such that

(H3prime) |Ht|+sumk

i=1 |DisHt| leUt as 0le s lt t

Finally we suppose that for any N gt 0 there exists an increasing progressively measur-able process VN

t t ge 0 with values in R+ such that

(H4prime) |F(tsx)|+sumk

i=1 |Gi(h tsx)|+sumk

i=1 |Gprimei (h tsx)| le VNt (1+|x|) for all |h|leN

0le s lt t and x isinRd(H5prime) |F(tsx) minus F(ts y)| +

sumki=1 |Gi(h tsx) minus Gi(h ts y)| +

sumki=1 |Gprimei (h tsx)

minusGprimei (h ts y)| leVNt |xminus y| for all |h| leN 0le s lt t x y isinRd

Let again q be a fixed real number with q gt p and set 1minusβ = αq 0 lt α lt 1 We havethe following theorem

Theorem 51 Equation (31) has a unique solution in the class of progressively measurableprocesses which satisfy

X isin⋂tgt0

Lq(0 t) as (51)

Mahmoud M El-Borai et al 463

Proof (a) We first see how (31) makes sense if X isin⋂tgt0Lq(0 t) as

That is we have to show that for fixed t gt 0

int t0

Gi(h tsXs

)(tminus s)α2q dWi

s hisinRP

(52)

is a well-defined random field which possesses an as continuous versionFor that sake we define

τn = inftint t

0

∣∣Xs∣∣q(tminus s)α dsge n or VN

t ge n (53)

The argument of Lemma 41 can be used to show that

hminusrarrint tandτn

0

Gi(h tsXs

)(tminus s)α2q dWi

s (54)

possesses an as continuous modification on |h| le N Since this is true for any n andN and cupnτn ge t =Ω as the result follows

(b) Existence we want to show existence on an arbitrary interval [0T] (T will be fixedbelow)

Let Hn nisinN denote a progressively measurable localizing sequence forH in (L12)p

on [0T] Since from (H3prime) suptleT |Ht| is as finite we can and do assume without lossof generality that

∣∣Hnt (ω)

∣∣le n forall(tω)isin [0T]timesΩ (55)

Note that Ht(ω) =Hnt (ω) as on ΩT

n for all t isin [0T] where ΩTn uarrΩ as as nrarrinfin We

moreover define Xn0 = X01|X0|len Sn = inft supsltt |DsH

nt |orVn

t ge n We consider theequation

Xnt = Xn

0 +int t

0Fn(tsXn

s

)ds+ b

ksumi=1

int t0

Gni

(h tsXn

s

)(tminus s)α2q dWi

s|h=Hnt (56)

where

Fn(tsx)= 1[0Sn](s)

[aF(tsx)(tminus s)αq minus b

ksumi=1

Gprimei(Hnt tsx

)DisH

nt

(tminus s)α2q]

Gni (h tsx)= 1[0Sn](s)Gi(h tsx)

(57)

It is easy to see that theorem (415) applies to (56)Define

Sn(ω)=Sn(ω)and inf

t le T

int t0

∣∣Hs(ω)minusHns (ω)

∣∣ds gt 0

if∣∣X0(ω)

∣∣ lt n

0 otherwise(58)

464 Volterra equations with fractional stochastic integrals

Sn is a stopping time and it follows from the uniqueness part of Theorem 44 that ifmgt n

Xmt = Xn

t on[0 Sn

]as (59)

Since moreover Sn = T uarrΩ as we can define the process Xt on [0T] by Xt = Xnt

on [0 Sn] for all nisinNClearly X isin Lq(0T) as and solves (31) on [0T] Since T is arbitrary the existence

is proved(c) Uniqueness it suffices to prove uniqueness on an arbitrary interval [0T] Let

Xt t isin [0T] be a progressively measurable process such that X isin Lq(0T) as andX solves (31) It suffices to show that X coincides with the solution we have just con-structed

Let

Sn(ω)= S(ω)and inft le T

int t0

∣∣Xs(ω)∣∣q

(tminus s)α ds gt n

Xnt = XtandSn

(510)

Sn isin Lq(Ωtimes [0T]) and it solves (56) with Sn replaced by SnThen

Xnt (ω)= Xt(ω)timesP ae on

[0 Sn

] (511)

where is the Lebesgue measure on the real lineThe result follows from the fact that Sn ge T uarrΩ asNote that the above solution satisfies in fact X isin⋂qgt1

⋂tgt0L

q(0 t) as

6 Continuity of the solution

We want to give additional conditions under which the solution of (31) is an as contin-uous process

(H6) For all (sx)isinR+timesRd trarr F(tsx)(tminus s)αq is as continuous on (s+infin)(H7) Ht t ge 0 is as continuous(H8) For all iisin 1 k sisinR+ trarrDi

sHt is as continuous on (s+infin)(H9) For all (sx) isin R+ timesRd i isin 1 k (th)rarr Gprimei (h tsx)(tminus s)α2q is as con-

tinuous on (s+infin)timesRP

We also suppose that there exist δ gt 0 gt 0 such that for all N gt 0 |h| le N 0 le s lttand r x isinRd |tminus r| le 1

(H10) There exists an increasing process VNt t ge 0 such that

∣∣Gi(h tsx)minusGi(hrsx)∣∣leVN

t |tminus r|δ(1 + |x|) (61)

Theorem 61 Under conditions (H1prime) (H2prime) (H3prime) (H4prime) (H5prime) and (H6) (H7) (H8)(H9) (H10) the unique solution of (31) (which is progressively measurable and belongs asto⋂qgt1⋂tgt0L

q(0 t)) has as continuous modification

Mahmoud M El-Borai et al 465

Proof We need to show only that whenever X isin⋂qgt1⋂tgt0L

q(0 t) as Jt(X) t gt 0 hasan as continuous modification

(a) We first show that trarr int t0 F(tsXs)ds is as continuous

Note that (H6) (H7) (H8) and (H9) imply that for all (sx)isinR+timesRd trarr F(tsx)is as continuous on (s+infin)

Moreover from (H2prime) (H3prime) (H4prime) and the fact that X isin⋂qgt1⋂tgt0L

q(0 t) as forany T gt 0 there exists a process ZTs sisin [0T] such that

∣∣F(tsXs)∣∣le ZTs ∣∣Gprimei(Ht tsXs

)DisHt

∣∣le ZTs 0le s lt t le T asint t

0ZTs

1

(tminus s)α2q +1

(tminus s)αqds ltinfin as

(62)

Let first tn nisinN be a sequence such that tn lt t for any n and tnrarr t as nrarrinfin then

int t0F(tsXs

)dsminus

int tn0F(tnsXs

)ds=

int ttnF(tsXs

)ds+

int tn0

[F(tsXs

)minus F(tnsXs)]ds

∣∣∣∣int ttnF(tsXs

)ds∣∣∣∣le

int ttnZTs

1

(tminus s)α2q +1

(tminus s)αqds

(63)

and the latter tends as to 0 as nrarrinfin

∣∣∣∣int tn

0

[F(tsXs

)minus F(tnsXs)]ds∣∣∣∣le

int t0

∣∣F(tsXs)minus F(tnsXs)∣∣ds (64)

which tends to 0 as nrarrinfin A similar argument gives the same result when tn gt t tnrarr t(b) We next show that t rarr It(X Ht) possesses an as continuous modification This

follows from (H7) and

(th)minusrarr It(X h) (65)

has an as continuous modificationBy localization it suffices to prove (65) under assumptions (H2) (H3) (H4) (H5)

and (H6) (H7) (H8) (H9) (H10) with VNt (ω) in (H10) replaced by a constant K

and in case X0 isin⋂qgt1L

q(ΩRd)It then suffices to show that under the above hypotheses there exists Cq gt 0 such that

for any hk isinRP the numbers tr will be positive Suppose to fix the ideas that 0le r lt tthen

It(X h)minus Ir(X k)=int tr

Gi(h tsXs

)(tminus s)α2q dWi

s +int r

0

[Gi(h tsXs

)(tminus s)α2q minus Gi

(hrsXs

)(rminus s)α2q

]dWi

s

+int r

0

[Gi(h tsXs

)(rminus s)α2q minus Gi

(krsXs

)(rminus s)α2q

]dWi

s

(66)

466 Volterra equations with fractional stochastic integrals

It follows from the Burkholder-Gundy inequality that

E(∣∣∣∣int tr

Gi(h tsXs

)(tminus s)α2q dWi

s

∣∣∣∣q)le cq

ksumi=1

E

[(int tr

∣∣Gi(h tsXs

)∣∣2

(tminus s)αq ds)q2]

le cq(tminus r)(qminus2)2ksumi=1

Eint tr

∣∣Gi(h tsXs

)∣∣q(tminus s)α2 ds

le cq(tminus r)(qminus2)2int tr

E(1 +

∣∣Xs∣∣q)(tminus s)α2 ds

le Cq(tminus r)(qminus2)2(tminus r)1minusα2

(67)

From (H4) for Gi we deduce as in Lemma 41 that

E

(∣∣∣∣int r

0

[Gi(hrsXs

)minusGi(krsXs

)(rminus s)α2q

]dWi

s

∣∣∣∣q)

le cq(hminus k)qint r

0

E(1 +

∣∣Xs∣∣q)(rminus s)α2 dsle Cq(hminus k)qr1minusα2

(68)

From (H10) and the fact that

X isin⋂qgt1

⋂tgt0

Lq(Ωtimes (0 t)

)

E

(∣∣∣∣int r

0

[Gi(h tsXs

)(tminus s)α2q minus Gi

(hrsXs

)(rminus s)α2q

]dWi

s

∣∣∣∣q)

= E(∣∣∣∣int r

0

(1

(rminus s)α2q[Gi(h tsXs

)minusGi(hrsXs

)]

+Gi(h tsXs

)[ 1(tminus s)α2q minus

1(rminus s)α2q

])dWi

s

∣∣∣∣q)

le E(cq

int r0

(∣∣Gi(h tsXs

)minusGi(hrsXs

)∣∣q(rminus s)α

+∣∣Gi

(h tsXs

)∣∣q((rminus s)minusα2qminus (tminus s)minusα2q)q)ds

)

le cqKq(int r

0

( |tminus r|δq(rminus s)α E

(1 +

∣∣Xs∣∣q))ds

+int r

0

(E(1 +

∣∣Xs∣∣q)((rminus s)minusα2minus (tminus s)minusα2))ds)

le cq[|tminus r|δq

int r0

(rminus s)minusα2ds+int r

0

((rminus s)minusα2minus (tminus s)minusα2)ds]

le Cq(|tminus r|δqr1minusα2 +

(r1minusα2 + (tminus r)1minusα2minus t1minusα2))

(69)

Mahmoud M El-Borai et al 467

which from the above estimate yields

E(∣∣It(X h)minus Ir(X k)

∣∣q)le Cq

[|tminus r|1minusα2 + |tminus r|(qminusα)2

+ r1minusα2(|tminus r|δq + |hminus k|q)+(r1minusα2minus t1minusα2)]

(610)

this completes the proof

Acknowledgment

We are very grateful to the referees for valuable comments

References

[1] C Bender and R J Elliott Binary market models and discrete wick products preprint 2002[2] M A Berger and V J Mizel Volterra equations with Ito integrals I J Integral Equations 2

(1980) no 3 187ndash245[3] Volterra equations with Ito integrals II J Integral Equations 2 (1980) no 4 319ndash337[4] An extension of the stochastic integral Ann Probab 10 (1982) no 2 435ndash450[5] M M El-Borai Some probability densities and fundamental solutions of fractional evolution

equations Chaos Solitons Fractals 14 (2002) no 3 433ndash440[6] A M Kolodh On the existence solutions of stochastic Volterra equations Theory of Random

Processes 11 (1983) 51ndash57 (Russian)[7] B B Mandelbrot and J W Van Ness Fractional Brownian motions fractional noises and appli-

cations SIAM Rev 10 (1968) 422ndash437[8] P-A Meyer Flot drsquoune equation differentielle stochastique (drsquoapres Malliavin Bismut Kunita)

[Flow of a stochastic differential equation (following Malliavin Bismut Kunita)] Seminar onProbability XV (University of Strasbourg Strasbourg 19791980) Lecture Notes in Mathvol 850 Springer-Verlag Berlin 1981 pp 103ndash117 (French)

[9] D Nualart Noncausal stochastic integrals and calculus Stochastic Analysis and Related Topics(Silivri 1986) Lecture Notes in Math vol 1316 Springer-Verlag Berlin 1988 pp 80ndash129

[10] D Nualart and E Pardoux Stochastic calculus with anticipating integrands Probab TheoryRelated Fields 78 (1988) no 4 535ndash581

[11] D Nualart and M Zakai Generalized stochastic integrals and the Malliavin calculus ProbabTheory Related Fields 73 (1986) no 2 255ndash280

[12] D Ocone and E Pardoux A generalized Ito-Ventzell formula Application to a class of anticipat-ing stochastic differential equations Ann Inst H Poincare Probab Statist 25 (1989) no 139ndash71

[13] Linear stochastic differential equations with boundary conditions Probab Theory Re-lated Fields 82 (1989) no 4 489ndash526

[14] S Ogawa Sur la question drsquoexistence de solutions drsquoune equation differentielle stochastique dutype noncausal [On the existence of solutions of a stochastic differential equation of noncausaltype] J Math Kyoto Univ 24 (1984) no 4 699ndash704 (French)

[15] E Pardoux and P Protter A two-sided stochastic integral and its calculus Probab Theory RelatedFields 76 (1987) no 1 15ndash49

[16] Stochastic Volterra equations with anticipating coefficients Ann Probab 18 (1990)no 4 1635ndash1655

[17] P Protter Volterra equations driven by semimartingales Ann Probab 13 (1985) no 2 519ndash530

468 Volterra equations with fractional stochastic integrals

[18] A N V Rao and C P Tsokos On the existence uniqueness and stability behavior of a randomsolution to a nonlinear perturbed stochastic integro-differential equation Information andControl 27 (1975) 61ndash74

[19] Y Shiota A linear stochastic integral equation containing the extended Ito integral Math RepToyama Univ 9 (1986) 43ndash65

[20] A V Skorohod On a generalization of the stochastic integral Teor Verojatnost i Primenen 20(1975) no 2 223ndash238 (Russian)

[21] A-S Sznitman Martingales dependant drsquoun parametre une formule drsquoIto [Martingales depend-ing on a parameter an Ito formula] Z Wahrsch Verw Gebiete 60 (1982) no 1 41ndash70(French)

[22] W Wyss The fractional diffusion equation J Math Phys 27 (1986) no 11 2782ndash2785

Mahmoud M El-Borai Faculty of Science Alexandria University Alexandria EgyptE-mail address m m elboraiyahoocom

Khairia El-Said El-Nadi Faculty of Science Alexandria University Alexandria EgyptE-mail address khairia el saidhotmailcom

Osama L Mostafa Faculty of Science Alexandria University Alexandria EgyptE-mail address olmoustafayahoocom

Hamdy M Ahmed Faculty of Science Alexandria University Alexandria EgyptE-mail address hamdy 17egyahoocom

Mahmoud M El-Borai et al 457

For uisin L12iloc we can define its Skorohod integral with respect to Wi

t by

int t0usdW

is =

int t0uTnsdW

is on ΩT

n times [0T] (213)

Finally L12 =⋂ki=1L

12i and L12

loc is defined similarly as L12iloc

We now introduce the particular class of integrands which we will use belowLet u R+timesΩtimesRP rarrR satisfy the following

(i) For all x isinRp (tω)rarr (tωx) is ηt progressively measurable(ii) For all (tω)isinR+timesΩ u(tωmiddot)isin C1(Rp)

(iii) For some increasing function φ R+rarrR+ |u(tωx)|+|uprime(tωx)|leφ(|x|) forall (tωx)isinR+timesΩtimesRp where uprime(tx) stands for the gradient (partupartx)(tx)

Let θ be a p-dimensional random vector such that

(iv) θ j isinD12i

⋂Linfin(Ω) j = 1 p

We fix T gt 0 and consider

Ii(x)=int T

0

u(tx)(T minus t)(1minusβ)2 dW

it 0 lt β lt 1 (214)

Define moreover vt = u(tθ)Under conditions (i) (ii) (iii) and (iv) the following proposition holdsIt is proved in [10] that

int T0u(tθ)dWi

t =int T

0u(tx)dWi

t |x=θ minusint T

0uprime(tθ)Di

tθdt (215)

The same relation is proved in [16] under slightly different conditions Equation (215) isused to define the Skorohod integral

int T0 u(tθ)dWi

t

Proposition 22 The random field Ii(x) x isinRp defined above possesses an as continu-ous modification so that the random variable Ii(θ) can be defined v isinDomδi (see [10 16])

Condition (iv) can be replaced by (ivprime) θ j isinD12iloc j = 1 p

Under conditions (i) (ii) (iii) and (ivprime) v isin (Domδi)loc in the sense that there existsa sequence (Ωnvn) nisinN sub ηtimesDomδi such that Ωn uarrΩ as and vn|Ωn = v|Ωn

Indeed let (Ωprimenθn) be a localizing sequence for θ in (D12

i )p and let ψn nisinN subCinfinc (RpRp) satisfy ψn(x)= x whenever |x| le n

Define vn(t) = u(tψn(θn)) Ωn =Ωprimen

⋂|θ| le n then (Ωnvn) n isin N satisfies theabove conditions

It is then natural to define for every εgt0 the Skorohod integralint Tminusε

0 (vt(T minus t)(1minusβ)2)

dWit by formula (215) and the latter coincides with

int Tminusε0 (vn(t)(T minus t)(1minusβ)2)dWi

t on Ωn

It is clear that limεrarr0int Tminusε

0 (vt(T minus t)(1minusβ)2)dWit exists in the mean by using the norm

middot 2

458 Volterra equations with fractional stochastic integrals

3 Statement of the problem interpretation of (12)

Our aim is to study the equation

Xt = X0 + aint t

0

F(tsXs

)(tminus s)1minusβ ds+ b

ksumi=1

int t0

Gi(Ht tsXs

)(tminus s)(1minusβ)2 dWi

s

0 lt β lt 1 a= 1Γ(β)

b = 1Γ((β+ 1)2

) (31)

We define D = (ts)isinR2+ 0le s lt t

The coefficients F and G are given as follows F DtimesRd rarrRd is measurable and foreach (sx)isinR+timesRd F(middotsx) is ηt progressively measurable on Ωtimes [s+infin)

For i= 1 k Gi RptimesDtimesRd rarrRd is measurable for each (h tx) Gi(h tmiddotx) is ηsprogressively measurable on Ωtimes [0 t] and for each (ω tsx) Gi(middot tsx) is of class C1Ht is a given progressively measurable p-dimensional process It will follow from thesehypotheses that we will be able to construct a progressively measurable solution Xt

Therefore for each t the process Gi(h tsXs) s isin [0T] is of the form vs = u(sθ)with u(sh)=Gi(h tsXs) and θ =Ht We will impose below conditions on G Ht andthe solution Xt so as to satisfy requirements (i) (ii) (iii) and (ivprime) of Section 2

In particular we will consider only nonanticipating solutions Therefore the stochasticintegrals in (31) will be interpreted according to (215) that is

int t0

Gi(Ht tsXs

)(tminus s)(1minusβ)2 dWi

s =int t

0

Gi(h tsXs

)(tminus s)(1minusβ)2 dW

is|h=Ht minus

int t0

Gprimei(Ht tsXs

)(tminus s)(1minusβ)2 Di

sHt ds (32)

In other words we can rewrite (31) as

Xt = X0 +int t

0F(tsXs

)ds+ b

ksumi=1

int t0

Gi(h tsXs

)(tminus s)(1minusβ)2 dW

is|h=Ht (33)

where

F(tsx)= a F(tsx)(tminus s)1minusβ minus b

ksumi=1

Gprimei(Ht tsx

)DisHt

(tminus s)(1minusβ)2 (34)

and the stochastic integrals are now the usual Ito integralsWe will show below that (33) makes sense for any progressively measurable process X

which satisfies X isin⋂tgt0Lq(0 t) as for some q gt p We will find such a solution to (33)

it will then follow from (32) that it is a solution to (31) Similarly uniqueness for (31)in the above class will follow from uniqueness for (33) in that class

4 Existence and uniqueness under strong hypotheses

We formulate a set of further hypotheses (those stated in Section 3 are assumed to holdthroughout the paper) under which we will establish a first result of the existence anduniqueness of a solution of (12)

Mahmoud M El-Borai et al 459

Let B be an open bounded subset of Rp K gt 0 and q gt p such that

(H1) X0 isin Lq(Ωη0PRd)(H2) P(Ht isin Bforallt ge 0)= 1(H3) H isin (L12)p |DsHt| le K as 0le s lt t(H4) |F(tsx)|+

sumki=1 |Gi(h tsx)|+

sumki=1 |Gprimei (h tsx)| le K(1 + |x|) for any 0le s lt t

hisin B x isinRd(H5) |F(tsx) minus F(ts y)| +

sumki=1 |Gi(h tsx) minus Gi(h ts y)| +

sumki=1 |Gprimei (h tsx)

minusGprimei (h ts y)| le K|xminus y| for any 0le s lt t hisin B and x y isinRd as

Note that q will be a fixed real number such that q gt p and Lqprog(Ωtimes (0 t)) will

stand for the space Lq(Ωtimes (0 t)ξtP times λ) where ξt denotes the σ-algebra of progres-sively measurable subsets of Ωtimes (0 t) and λ denotes the Lebesgue measure on (0 t) andset 1minusβ = αq 0 lt α lt 1

Lemma 41 Let X isin⋂tgt0Lqprog(Ωtimes (0 t)) where q gt p and suppose that (H4) is in force

Then for any T ge t gt 0 and iisin 1 k the random field

int t0

Gi(h tsXs

)(tminus s)α2q dWi

s hisin B

(41)

possesses an as continuous modification

Proof Using Burkholder-Gundy and Holderrsquos inequalities together with (H4) we obtain

E

(∣∣∣∣int t

0

Gi(h tsXs

)(tminus s)α2q dWi

s minusint t

0

Gi(k tsXs

)(tminus s)α2q dWi

s

∣∣∣∣q)

le c1Eint t

0

∣∣Gi(h tsXs

)minusGi(k tsXs

)∣∣q(tminus s)α2 ds

le c1Kq|hminus k|q

int t0

E(1 +

∣∣Xs∣∣q)(tminus s)α2 ds

le c2Kq|hminus k|q

int t0(tminus s)minusα2ds

le c3|hminus k|qt1minusα2

(42)

where c1 c2 and c3 are positive constants The result now follows from the multidimen-sional generalization of Kolmogrovrsquos lemma (see [1 8])

We can now assume that for fixed t the random field (41) is as continuous in hprovided X isin Lqprog(Ωtimes (0 t))

From X isin⋂tgt0Lqprog(Ωtimes (0 t)) define

It(X h)= bksumi=1

int t0

Gi(h tsXs

)(tminus s)α2q dWi

s hisinRp t gt 0

Jt(X)=int t

0F(tsXs

)ds+ It

(X Ht

)

(43)

460 Volterra equations with fractional stochastic integrals

Lemma 42 For any t gt 0 there is a constant c gt 0 such that

E(∣∣Jt(X)

∣∣q)le c(int t

0E(1 +

∣∣Xs∣∣q)

1(tminus s)α +

1(tminus s)α2

ds)

0 lt α lt 1 (44)

Proof

∣∣∣∣int t

0F(tsXs

)ds∣∣∣∣q

le cint t

0

∣∣F(tsXs)∣∣q(tminus s)α +

ksumi=1

∣∣Gprimei(h tsXs)|qh=Ht

∣∣DisHt

∣∣q(tminus s)α2 ds

le cint t

0

(1 +

∣∣Xs∣∣q)

1(tminus s)α +

1(tminus s)α2

ds

(45)

where we have used (H2) (H3) and (H4)

∣∣It(X Ht)∣∣le sup

hisinB

∣∣It(X h)∣∣ (46)

(c c are positive constants)It is easy to show using in particular (H4) and Lebesguersquos dominated convergence

theorem that the mapping

hminusrarr It(X h) (47)

from Rp into Lq(Ω) is differentiable and that

partIt(X h)parthj

= bint t

0

part

parthj

ksumi=1

(Gi(h tsXs

)(tminus s)α2q

)dWi

s (48)

Since q gt p we can infer from Sobolevrsquos embedding theorem (see [13]) that

E

(suphisinB

∣∣It(X h)∣∣q)le cE

intB

(∣∣It(X h)∣∣q +

psumj=1

∣∣∣∣ partItparthj(X h)

∣∣∣∣q)dh (49)

It then follows from the Burkholder-Gundy inequality that

E

(suphisinB

∣∣It(X h)∣∣q)le cE

intB

int t0

ksumi=1

(∣∣Gi(h tsXs

)∣∣q(tminus s)α2 +

psumj=1

∣∣∣∣ part

parthj

Gi(h tsXs

)(tminus s)α2q

∣∣∣∣q)dsdh

le c(int

Bdh)(Eint t

0

(1 +

∣∣Xs∣∣q)(tminus s)α2 ds

)

le cint t

0

E(1 +

∣∣Xs∣∣q)(tminus s)α2 ds

(410)

where we have used (H4) and B is bounded From (45) and (410) the proof is complete

A similar argument using (H5) instead of (H4) yields the following lemma

Mahmoud M El-Borai et al 461

Lemma 43 For any 0 lt t le T there exists a constant c gt 0 such that

E(∣∣Jt(X)minus Jt(Y)

∣∣q)le cint t

0E∣∣XsminusYs∣∣q

1

(tminus s)α +1

(tminus s)α2ds (411)

We are now in a position to prove the main result of this section

Theorem 44 Under conditions (H1) (H2) (H3) (H4) and (H5) there exists a uniqueelement X isin ⋂tgt0L

qprog(Ωtimes (0 t)) which solves (31) Moreover if τ is a stopping time

uniqueness holds on the random interval [0τ]

Proof Equation (31) can be rewritten as

Xt = X0 + Jt(X) t ge 0 (412)

Uniqueness Let X Y isin⋂tgt0Lqprog(Ωtimes (0 t)) and let τ be a stopping time such that

Xt = X0 + Jt(X) Yt = X0 + Jt(Y) 0le t le τ (413)

From Lemma 43

E(∣∣Xt minusYt∣∣q)= E(∣∣Jt(X)minus Jt(Y)

∣∣q)le c

int t0E∣∣XsminusYs∣∣q

1

(tminus s)α +1

(tminus s)α2ds

(414)

Set (E|Xt minusYt|q)= γt It is easy to see that

γt le 2cη(t)int t

0

γs(tminus s)α ds (415)

where η(t)=Max(1 tα) and c is a positive constant Thus (see [5 21])

γt le M2(Γ(tminusα)

)2

Γ(2(1minusα)

) int t0

γs(tminus s)2αminus1

ds (416)

and by a classical argument we get

γt le Mn(Γ(tminusα)

)nΓ(n(1minusα)

) int t0

γs(tminus s)nαminus(nminus1)

ds (417)

where M = 2cη(T) Thus for sufficiently large n n gt 1(1minusα) we get from (417)

γt le Kn

Γ(n(1minusα)

) int t0γs ds (418)

where K is a positive constantTaking the limit as nrarrinfin we find that (418) leads to γt = 0To prove the existence we define a sequence Xn

t 0le t le T n= 012 as follows

Xot = Xo Xn+1

t = Xo + Jt(Xn) T le t le 0 (419)

462 Volterra equations with fractional stochastic integrals

Using Lemma 42 we can see that

Xn isin⋂tgt0

Lqprog

(Ωtimes [0T]

) (420)

It then follows from Lemma 43 that

gn+1(t)le 2cη(t)int t

0

gn(s)(tminus s)α ds (421)

where

gn+1(t)= E(∣∣Xn+1t minusXn

t

∣∣q) (422)

By using a similar argument we can write

gn le Kn

Γ(n(1minusα)

) int t0go(s)ds (423)

The last estimation implies that Xn is a Cauchy sequence in Lqprog(Ωtimes [0T]) Then there

exists X such that Xn rarr X in⋂tgt0L

qprog(Ωtimes [0T]) and again using Lemma 43 we can

pass to the limit in (420) yielding that X solves (412)

5 An existence and uniqueness result under weaker assumptions

We formulate a new set of weaker hypotheses

(H1prime) X0 is η0 measurable(H2prime) H isin (L12

loc)pHt is a progressively measurable process which can be localized in(L12)p by a progressively measurable sequence

We assume that there exists an increasing progressively measurable process Ut t ge 0with values in R+ such that

(H3prime) |Ht|+sumk

i=1 |DisHt| leUt as 0le s lt t

Finally we suppose that for any N gt 0 there exists an increasing progressively measur-able process VN

t t ge 0 with values in R+ such that

(H4prime) |F(tsx)|+sumk

i=1 |Gi(h tsx)|+sumk

i=1 |Gprimei (h tsx)| le VNt (1+|x|) for all |h|leN

0le s lt t and x isinRd(H5prime) |F(tsx) minus F(ts y)| +

sumki=1 |Gi(h tsx) minus Gi(h ts y)| +

sumki=1 |Gprimei (h tsx)

minusGprimei (h ts y)| leVNt |xminus y| for all |h| leN 0le s lt t x y isinRd

Let again q be a fixed real number with q gt p and set 1minusβ = αq 0 lt α lt 1 We havethe following theorem

Theorem 51 Equation (31) has a unique solution in the class of progressively measurableprocesses which satisfy

X isin⋂tgt0

Lq(0 t) as (51)

Mahmoud M El-Borai et al 463

Proof (a) We first see how (31) makes sense if X isin⋂tgt0Lq(0 t) as

That is we have to show that for fixed t gt 0

int t0

Gi(h tsXs

)(tminus s)α2q dWi

s hisinRP

(52)

is a well-defined random field which possesses an as continuous versionFor that sake we define

τn = inftint t

0

∣∣Xs∣∣q(tminus s)α dsge n or VN

t ge n (53)

The argument of Lemma 41 can be used to show that

hminusrarrint tandτn

0

Gi(h tsXs

)(tminus s)α2q dWi

s (54)

possesses an as continuous modification on |h| le N Since this is true for any n andN and cupnτn ge t =Ω as the result follows

(b) Existence we want to show existence on an arbitrary interval [0T] (T will be fixedbelow)

Let Hn nisinN denote a progressively measurable localizing sequence forH in (L12)p

on [0T] Since from (H3prime) suptleT |Ht| is as finite we can and do assume without lossof generality that

∣∣Hnt (ω)

∣∣le n forall(tω)isin [0T]timesΩ (55)

Note that Ht(ω) =Hnt (ω) as on ΩT

n for all t isin [0T] where ΩTn uarrΩ as as nrarrinfin We

moreover define Xn0 = X01|X0|len Sn = inft supsltt |DsH

nt |orVn

t ge n We consider theequation

Xnt = Xn

0 +int t

0Fn(tsXn

s

)ds+ b

ksumi=1

int t0

Gni

(h tsXn

s

)(tminus s)α2q dWi

s|h=Hnt (56)

where

Fn(tsx)= 1[0Sn](s)

[aF(tsx)(tminus s)αq minus b

ksumi=1

Gprimei(Hnt tsx

)DisH

nt

(tminus s)α2q]

Gni (h tsx)= 1[0Sn](s)Gi(h tsx)

(57)

It is easy to see that theorem (415) applies to (56)Define

Sn(ω)=Sn(ω)and inf

t le T

int t0

∣∣Hs(ω)minusHns (ω)

∣∣ds gt 0

if∣∣X0(ω)

∣∣ lt n

0 otherwise(58)

464 Volterra equations with fractional stochastic integrals

Sn is a stopping time and it follows from the uniqueness part of Theorem 44 that ifmgt n

Xmt = Xn

t on[0 Sn

]as (59)

Since moreover Sn = T uarrΩ as we can define the process Xt on [0T] by Xt = Xnt

on [0 Sn] for all nisinNClearly X isin Lq(0T) as and solves (31) on [0T] Since T is arbitrary the existence

is proved(c) Uniqueness it suffices to prove uniqueness on an arbitrary interval [0T] Let

Xt t isin [0T] be a progressively measurable process such that X isin Lq(0T) as andX solves (31) It suffices to show that X coincides with the solution we have just con-structed

Let

Sn(ω)= S(ω)and inft le T

int t0

∣∣Xs(ω)∣∣q

(tminus s)α ds gt n

Xnt = XtandSn

(510)

Sn isin Lq(Ωtimes [0T]) and it solves (56) with Sn replaced by SnThen

Xnt (ω)= Xt(ω)timesP ae on

[0 Sn

] (511)

where is the Lebesgue measure on the real lineThe result follows from the fact that Sn ge T uarrΩ asNote that the above solution satisfies in fact X isin⋂qgt1

⋂tgt0L

q(0 t) as

6 Continuity of the solution

We want to give additional conditions under which the solution of (31) is an as contin-uous process

(H6) For all (sx)isinR+timesRd trarr F(tsx)(tminus s)αq is as continuous on (s+infin)(H7) Ht t ge 0 is as continuous(H8) For all iisin 1 k sisinR+ trarrDi

sHt is as continuous on (s+infin)(H9) For all (sx) isin R+ timesRd i isin 1 k (th)rarr Gprimei (h tsx)(tminus s)α2q is as con-

tinuous on (s+infin)timesRP

We also suppose that there exist δ gt 0 gt 0 such that for all N gt 0 |h| le N 0 le s lttand r x isinRd |tminus r| le 1

(H10) There exists an increasing process VNt t ge 0 such that

∣∣Gi(h tsx)minusGi(hrsx)∣∣leVN

t |tminus r|δ(1 + |x|) (61)

Theorem 61 Under conditions (H1prime) (H2prime) (H3prime) (H4prime) (H5prime) and (H6) (H7) (H8)(H9) (H10) the unique solution of (31) (which is progressively measurable and belongs asto⋂qgt1⋂tgt0L

q(0 t)) has as continuous modification

Mahmoud M El-Borai et al 465

Proof We need to show only that whenever X isin⋂qgt1⋂tgt0L

q(0 t) as Jt(X) t gt 0 hasan as continuous modification

(a) We first show that trarr int t0 F(tsXs)ds is as continuous

Note that (H6) (H7) (H8) and (H9) imply that for all (sx)isinR+timesRd trarr F(tsx)is as continuous on (s+infin)

Moreover from (H2prime) (H3prime) (H4prime) and the fact that X isin⋂qgt1⋂tgt0L

q(0 t) as forany T gt 0 there exists a process ZTs sisin [0T] such that

∣∣F(tsXs)∣∣le ZTs ∣∣Gprimei(Ht tsXs

)DisHt

∣∣le ZTs 0le s lt t le T asint t

0ZTs

1

(tminus s)α2q +1

(tminus s)αqds ltinfin as

(62)

Let first tn nisinN be a sequence such that tn lt t for any n and tnrarr t as nrarrinfin then

int t0F(tsXs

)dsminus

int tn0F(tnsXs

)ds=

int ttnF(tsXs

)ds+

int tn0

[F(tsXs

)minus F(tnsXs)]ds

∣∣∣∣int ttnF(tsXs

)ds∣∣∣∣le

int ttnZTs

1

(tminus s)α2q +1

(tminus s)αqds

(63)

and the latter tends as to 0 as nrarrinfin

∣∣∣∣int tn

0

[F(tsXs

)minus F(tnsXs)]ds∣∣∣∣le

int t0

∣∣F(tsXs)minus F(tnsXs)∣∣ds (64)

which tends to 0 as nrarrinfin A similar argument gives the same result when tn gt t tnrarr t(b) We next show that t rarr It(X Ht) possesses an as continuous modification This

follows from (H7) and

(th)minusrarr It(X h) (65)

has an as continuous modificationBy localization it suffices to prove (65) under assumptions (H2) (H3) (H4) (H5)

and (H6) (H7) (H8) (H9) (H10) with VNt (ω) in (H10) replaced by a constant K

and in case X0 isin⋂qgt1L

q(ΩRd)It then suffices to show that under the above hypotheses there exists Cq gt 0 such that

for any hk isinRP the numbers tr will be positive Suppose to fix the ideas that 0le r lt tthen

It(X h)minus Ir(X k)=int tr

Gi(h tsXs

)(tminus s)α2q dWi

s +int r

0

[Gi(h tsXs

)(tminus s)α2q minus Gi

(hrsXs

)(rminus s)α2q

]dWi

s

+int r

0

[Gi(h tsXs

)(rminus s)α2q minus Gi

(krsXs

)(rminus s)α2q

]dWi

s

(66)

466 Volterra equations with fractional stochastic integrals

It follows from the Burkholder-Gundy inequality that

E(∣∣∣∣int tr

Gi(h tsXs

)(tminus s)α2q dWi

s

∣∣∣∣q)le cq

ksumi=1

E

[(int tr

∣∣Gi(h tsXs

)∣∣2

(tminus s)αq ds)q2]

le cq(tminus r)(qminus2)2ksumi=1

Eint tr

∣∣Gi(h tsXs

)∣∣q(tminus s)α2 ds

le cq(tminus r)(qminus2)2int tr

E(1 +

∣∣Xs∣∣q)(tminus s)α2 ds

le Cq(tminus r)(qminus2)2(tminus r)1minusα2

(67)

From (H4) for Gi we deduce as in Lemma 41 that

E

(∣∣∣∣int r

0

[Gi(hrsXs

)minusGi(krsXs

)(rminus s)α2q

]dWi

s

∣∣∣∣q)

le cq(hminus k)qint r

0

E(1 +

∣∣Xs∣∣q)(rminus s)α2 dsle Cq(hminus k)qr1minusα2

(68)

From (H10) and the fact that

X isin⋂qgt1

⋂tgt0

Lq(Ωtimes (0 t)

)

E

(∣∣∣∣int r

0

[Gi(h tsXs

)(tminus s)α2q minus Gi

(hrsXs

)(rminus s)α2q

]dWi

s

∣∣∣∣q)

= E(∣∣∣∣int r

0

(1

(rminus s)α2q[Gi(h tsXs

)minusGi(hrsXs

)]

+Gi(h tsXs

)[ 1(tminus s)α2q minus

1(rminus s)α2q

])dWi

s

∣∣∣∣q)

le E(cq

int r0

(∣∣Gi(h tsXs

)minusGi(hrsXs

)∣∣q(rminus s)α

+∣∣Gi

(h tsXs

)∣∣q((rminus s)minusα2qminus (tminus s)minusα2q)q)ds

)

le cqKq(int r

0

( |tminus r|δq(rminus s)α E

(1 +

∣∣Xs∣∣q))ds

+int r

0

(E(1 +

∣∣Xs∣∣q)((rminus s)minusα2minus (tminus s)minusα2))ds)

le cq[|tminus r|δq

int r0

(rminus s)minusα2ds+int r

0

((rminus s)minusα2minus (tminus s)minusα2)ds]

le Cq(|tminus r|δqr1minusα2 +

(r1minusα2 + (tminus r)1minusα2minus t1minusα2))

(69)

Mahmoud M El-Borai et al 467

which from the above estimate yields

E(∣∣It(X h)minus Ir(X k)

∣∣q)le Cq

[|tminus r|1minusα2 + |tminus r|(qminusα)2

+ r1minusα2(|tminus r|δq + |hminus k|q)+(r1minusα2minus t1minusα2)]

(610)

this completes the proof

Acknowledgment

We are very grateful to the referees for valuable comments

References

[1] C Bender and R J Elliott Binary market models and discrete wick products preprint 2002[2] M A Berger and V J Mizel Volterra equations with Ito integrals I J Integral Equations 2

(1980) no 3 187ndash245[3] Volterra equations with Ito integrals II J Integral Equations 2 (1980) no 4 319ndash337[4] An extension of the stochastic integral Ann Probab 10 (1982) no 2 435ndash450[5] M M El-Borai Some probability densities and fundamental solutions of fractional evolution

equations Chaos Solitons Fractals 14 (2002) no 3 433ndash440[6] A M Kolodh On the existence solutions of stochastic Volterra equations Theory of Random

Processes 11 (1983) 51ndash57 (Russian)[7] B B Mandelbrot and J W Van Ness Fractional Brownian motions fractional noises and appli-

cations SIAM Rev 10 (1968) 422ndash437[8] P-A Meyer Flot drsquoune equation differentielle stochastique (drsquoapres Malliavin Bismut Kunita)

[Flow of a stochastic differential equation (following Malliavin Bismut Kunita)] Seminar onProbability XV (University of Strasbourg Strasbourg 19791980) Lecture Notes in Mathvol 850 Springer-Verlag Berlin 1981 pp 103ndash117 (French)

[9] D Nualart Noncausal stochastic integrals and calculus Stochastic Analysis and Related Topics(Silivri 1986) Lecture Notes in Math vol 1316 Springer-Verlag Berlin 1988 pp 80ndash129

[10] D Nualart and E Pardoux Stochastic calculus with anticipating integrands Probab TheoryRelated Fields 78 (1988) no 4 535ndash581

[11] D Nualart and M Zakai Generalized stochastic integrals and the Malliavin calculus ProbabTheory Related Fields 73 (1986) no 2 255ndash280

[12] D Ocone and E Pardoux A generalized Ito-Ventzell formula Application to a class of anticipat-ing stochastic differential equations Ann Inst H Poincare Probab Statist 25 (1989) no 139ndash71

[13] Linear stochastic differential equations with boundary conditions Probab Theory Re-lated Fields 82 (1989) no 4 489ndash526

[14] S Ogawa Sur la question drsquoexistence de solutions drsquoune equation differentielle stochastique dutype noncausal [On the existence of solutions of a stochastic differential equation of noncausaltype] J Math Kyoto Univ 24 (1984) no 4 699ndash704 (French)

[15] E Pardoux and P Protter A two-sided stochastic integral and its calculus Probab Theory RelatedFields 76 (1987) no 1 15ndash49

[16] Stochastic Volterra equations with anticipating coefficients Ann Probab 18 (1990)no 4 1635ndash1655

[17] P Protter Volterra equations driven by semimartingales Ann Probab 13 (1985) no 2 519ndash530

468 Volterra equations with fractional stochastic integrals

[18] A N V Rao and C P Tsokos On the existence uniqueness and stability behavior of a randomsolution to a nonlinear perturbed stochastic integro-differential equation Information andControl 27 (1975) 61ndash74

[19] Y Shiota A linear stochastic integral equation containing the extended Ito integral Math RepToyama Univ 9 (1986) 43ndash65

[20] A V Skorohod On a generalization of the stochastic integral Teor Verojatnost i Primenen 20(1975) no 2 223ndash238 (Russian)

[21] A-S Sznitman Martingales dependant drsquoun parametre une formule drsquoIto [Martingales depend-ing on a parameter an Ito formula] Z Wahrsch Verw Gebiete 60 (1982) no 1 41ndash70(French)

[22] W Wyss The fractional diffusion equation J Math Phys 27 (1986) no 11 2782ndash2785

Mahmoud M El-Borai Faculty of Science Alexandria University Alexandria EgyptE-mail address m m elboraiyahoocom

Khairia El-Said El-Nadi Faculty of Science Alexandria University Alexandria EgyptE-mail address khairia el saidhotmailcom

Osama L Mostafa Faculty of Science Alexandria University Alexandria EgyptE-mail address olmoustafayahoocom

Hamdy M Ahmed Faculty of Science Alexandria University Alexandria EgyptE-mail address hamdy 17egyahoocom

458 Volterra equations with fractional stochastic integrals

3 Statement of the problem interpretation of (12)

Our aim is to study the equation

Xt = X0 + aint t

0

F(tsXs

)(tminus s)1minusβ ds+ b

ksumi=1

int t0

Gi(Ht tsXs

)(tminus s)(1minusβ)2 dWi

s

0 lt β lt 1 a= 1Γ(β)

b = 1Γ((β+ 1)2

) (31)

We define D = (ts)isinR2+ 0le s lt t

The coefficients F and G are given as follows F DtimesRd rarrRd is measurable and foreach (sx)isinR+timesRd F(middotsx) is ηt progressively measurable on Ωtimes [s+infin)

For i= 1 k Gi RptimesDtimesRd rarrRd is measurable for each (h tx) Gi(h tmiddotx) is ηsprogressively measurable on Ωtimes [0 t] and for each (ω tsx) Gi(middot tsx) is of class C1Ht is a given progressively measurable p-dimensional process It will follow from thesehypotheses that we will be able to construct a progressively measurable solution Xt

Therefore for each t the process Gi(h tsXs) s isin [0T] is of the form vs = u(sθ)with u(sh)=Gi(h tsXs) and θ =Ht We will impose below conditions on G Ht andthe solution Xt so as to satisfy requirements (i) (ii) (iii) and (ivprime) of Section 2

In particular we will consider only nonanticipating solutions Therefore the stochasticintegrals in (31) will be interpreted according to (215) that is

int t0

Gi(Ht tsXs

)(tminus s)(1minusβ)2 dWi

s =int t

0

Gi(h tsXs

)(tminus s)(1minusβ)2 dW

is|h=Ht minus

int t0

Gprimei(Ht tsXs

)(tminus s)(1minusβ)2 Di

sHt ds (32)

In other words we can rewrite (31) as

Xt = X0 +int t

0F(tsXs

)ds+ b

ksumi=1

int t0

Gi(h tsXs

)(tminus s)(1minusβ)2 dW

is|h=Ht (33)

where

F(tsx)= a F(tsx)(tminus s)1minusβ minus b

ksumi=1

Gprimei(Ht tsx

)DisHt

(tminus s)(1minusβ)2 (34)

and the stochastic integrals are now the usual Ito integralsWe will show below that (33) makes sense for any progressively measurable process X

which satisfies X isin⋂tgt0Lq(0 t) as for some q gt p We will find such a solution to (33)

it will then follow from (32) that it is a solution to (31) Similarly uniqueness for (31)in the above class will follow from uniqueness for (33) in that class

4 Existence and uniqueness under strong hypotheses

We formulate a set of further hypotheses (those stated in Section 3 are assumed to holdthroughout the paper) under which we will establish a first result of the existence anduniqueness of a solution of (12)

Mahmoud M El-Borai et al 459

Let B be an open bounded subset of Rp K gt 0 and q gt p such that

(H1) X0 isin Lq(Ωη0PRd)(H2) P(Ht isin Bforallt ge 0)= 1(H3) H isin (L12)p |DsHt| le K as 0le s lt t(H4) |F(tsx)|+

sumki=1 |Gi(h tsx)|+

sumki=1 |Gprimei (h tsx)| le K(1 + |x|) for any 0le s lt t

hisin B x isinRd(H5) |F(tsx) minus F(ts y)| +

sumki=1 |Gi(h tsx) minus Gi(h ts y)| +

sumki=1 |Gprimei (h tsx)

minusGprimei (h ts y)| le K|xminus y| for any 0le s lt t hisin B and x y isinRd as

Note that q will be a fixed real number such that q gt p and Lqprog(Ωtimes (0 t)) will

stand for the space Lq(Ωtimes (0 t)ξtP times λ) where ξt denotes the σ-algebra of progres-sively measurable subsets of Ωtimes (0 t) and λ denotes the Lebesgue measure on (0 t) andset 1minusβ = αq 0 lt α lt 1

Lemma 41 Let X isin⋂tgt0Lqprog(Ωtimes (0 t)) where q gt p and suppose that (H4) is in force

Then for any T ge t gt 0 and iisin 1 k the random field

int t0

Gi(h tsXs

)(tminus s)α2q dWi

s hisin B

(41)

possesses an as continuous modification

Proof Using Burkholder-Gundy and Holderrsquos inequalities together with (H4) we obtain

E

(∣∣∣∣int t

0

Gi(h tsXs

)(tminus s)α2q dWi

s minusint t

0

Gi(k tsXs

)(tminus s)α2q dWi

s

∣∣∣∣q)

le c1Eint t

0

∣∣Gi(h tsXs

)minusGi(k tsXs

)∣∣q(tminus s)α2 ds

le c1Kq|hminus k|q

int t0

E(1 +

∣∣Xs∣∣q)(tminus s)α2 ds

le c2Kq|hminus k|q

int t0(tminus s)minusα2ds

le c3|hminus k|qt1minusα2

(42)

where c1 c2 and c3 are positive constants The result now follows from the multidimen-sional generalization of Kolmogrovrsquos lemma (see [1 8])

We can now assume that for fixed t the random field (41) is as continuous in hprovided X isin Lqprog(Ωtimes (0 t))

From X isin⋂tgt0Lqprog(Ωtimes (0 t)) define

It(X h)= bksumi=1

int t0

Gi(h tsXs

)(tminus s)α2q dWi

s hisinRp t gt 0

Jt(X)=int t

0F(tsXs

)ds+ It

(X Ht

)

(43)

460 Volterra equations with fractional stochastic integrals

Lemma 42 For any t gt 0 there is a constant c gt 0 such that

E(∣∣Jt(X)

∣∣q)le c(int t

0E(1 +

∣∣Xs∣∣q)

1(tminus s)α +

1(tminus s)α2

ds)

0 lt α lt 1 (44)

Proof

∣∣∣∣int t

0F(tsXs

)ds∣∣∣∣q

le cint t

0

∣∣F(tsXs)∣∣q(tminus s)α +

ksumi=1

∣∣Gprimei(h tsXs)|qh=Ht

∣∣DisHt

∣∣q(tminus s)α2 ds

le cint t

0

(1 +

∣∣Xs∣∣q)

1(tminus s)α +

1(tminus s)α2

ds

(45)

where we have used (H2) (H3) and (H4)

∣∣It(X Ht)∣∣le sup

hisinB

∣∣It(X h)∣∣ (46)

(c c are positive constants)It is easy to show using in particular (H4) and Lebesguersquos dominated convergence

theorem that the mapping

hminusrarr It(X h) (47)

from Rp into Lq(Ω) is differentiable and that

partIt(X h)parthj

= bint t

0

part

parthj

ksumi=1

(Gi(h tsXs

)(tminus s)α2q

)dWi

s (48)

Since q gt p we can infer from Sobolevrsquos embedding theorem (see [13]) that

E

(suphisinB

∣∣It(X h)∣∣q)le cE

intB

(∣∣It(X h)∣∣q +

psumj=1

∣∣∣∣ partItparthj(X h)

∣∣∣∣q)dh (49)

It then follows from the Burkholder-Gundy inequality that

E

(suphisinB

∣∣It(X h)∣∣q)le cE

intB

int t0

ksumi=1

(∣∣Gi(h tsXs

)∣∣q(tminus s)α2 +

psumj=1

∣∣∣∣ part

parthj

Gi(h tsXs

)(tminus s)α2q

∣∣∣∣q)dsdh

le c(int

Bdh)(Eint t

0

(1 +

∣∣Xs∣∣q)(tminus s)α2 ds

)

le cint t

0

E(1 +

∣∣Xs∣∣q)(tminus s)α2 ds

(410)

where we have used (H4) and B is bounded From (45) and (410) the proof is complete

A similar argument using (H5) instead of (H4) yields the following lemma

Mahmoud M El-Borai et al 461

Lemma 43 For any 0 lt t le T there exists a constant c gt 0 such that

E(∣∣Jt(X)minus Jt(Y)

∣∣q)le cint t

0E∣∣XsminusYs∣∣q

1

(tminus s)α +1

(tminus s)α2ds (411)

We are now in a position to prove the main result of this section

Theorem 44 Under conditions (H1) (H2) (H3) (H4) and (H5) there exists a uniqueelement X isin ⋂tgt0L

qprog(Ωtimes (0 t)) which solves (31) Moreover if τ is a stopping time

uniqueness holds on the random interval [0τ]

Proof Equation (31) can be rewritten as

Xt = X0 + Jt(X) t ge 0 (412)

Uniqueness Let X Y isin⋂tgt0Lqprog(Ωtimes (0 t)) and let τ be a stopping time such that

Xt = X0 + Jt(X) Yt = X0 + Jt(Y) 0le t le τ (413)

From Lemma 43

E(∣∣Xt minusYt∣∣q)= E(∣∣Jt(X)minus Jt(Y)

∣∣q)le c

int t0E∣∣XsminusYs∣∣q

1

(tminus s)α +1

(tminus s)α2ds

(414)

Set (E|Xt minusYt|q)= γt It is easy to see that

γt le 2cη(t)int t

0

γs(tminus s)α ds (415)

where η(t)=Max(1 tα) and c is a positive constant Thus (see [5 21])

γt le M2(Γ(tminusα)

)2

Γ(2(1minusα)

) int t0

γs(tminus s)2αminus1

ds (416)

and by a classical argument we get

γt le Mn(Γ(tminusα)

)nΓ(n(1minusα)

) int t0

γs(tminus s)nαminus(nminus1)

ds (417)

where M = 2cη(T) Thus for sufficiently large n n gt 1(1minusα) we get from (417)

γt le Kn

Γ(n(1minusα)

) int t0γs ds (418)

where K is a positive constantTaking the limit as nrarrinfin we find that (418) leads to γt = 0To prove the existence we define a sequence Xn

t 0le t le T n= 012 as follows

Xot = Xo Xn+1

t = Xo + Jt(Xn) T le t le 0 (419)

462 Volterra equations with fractional stochastic integrals

Using Lemma 42 we can see that

Xn isin⋂tgt0

Lqprog

(Ωtimes [0T]

) (420)

It then follows from Lemma 43 that

gn+1(t)le 2cη(t)int t

0

gn(s)(tminus s)α ds (421)

where

gn+1(t)= E(∣∣Xn+1t minusXn

t

∣∣q) (422)

By using a similar argument we can write

gn le Kn

Γ(n(1minusα)

) int t0go(s)ds (423)

The last estimation implies that Xn is a Cauchy sequence in Lqprog(Ωtimes [0T]) Then there

exists X such that Xn rarr X in⋂tgt0L

qprog(Ωtimes [0T]) and again using Lemma 43 we can

pass to the limit in (420) yielding that X solves (412)

5 An existence and uniqueness result under weaker assumptions

We formulate a new set of weaker hypotheses

(H1prime) X0 is η0 measurable(H2prime) H isin (L12

loc)pHt is a progressively measurable process which can be localized in(L12)p by a progressively measurable sequence

We assume that there exists an increasing progressively measurable process Ut t ge 0with values in R+ such that

(H3prime) |Ht|+sumk

i=1 |DisHt| leUt as 0le s lt t

Finally we suppose that for any N gt 0 there exists an increasing progressively measur-able process VN

t t ge 0 with values in R+ such that

(H4prime) |F(tsx)|+sumk

i=1 |Gi(h tsx)|+sumk

i=1 |Gprimei (h tsx)| le VNt (1+|x|) for all |h|leN

0le s lt t and x isinRd(H5prime) |F(tsx) minus F(ts y)| +

sumki=1 |Gi(h tsx) minus Gi(h ts y)| +

sumki=1 |Gprimei (h tsx)

minusGprimei (h ts y)| leVNt |xminus y| for all |h| leN 0le s lt t x y isinRd

Let again q be a fixed real number with q gt p and set 1minusβ = αq 0 lt α lt 1 We havethe following theorem

Theorem 51 Equation (31) has a unique solution in the class of progressively measurableprocesses which satisfy

X isin⋂tgt0

Lq(0 t) as (51)

Mahmoud M El-Borai et al 463

Proof (a) We first see how (31) makes sense if X isin⋂tgt0Lq(0 t) as

That is we have to show that for fixed t gt 0

int t0

Gi(h tsXs

)(tminus s)α2q dWi

s hisinRP

(52)

is a well-defined random field which possesses an as continuous versionFor that sake we define

τn = inftint t

0

∣∣Xs∣∣q(tminus s)α dsge n or VN

t ge n (53)

The argument of Lemma 41 can be used to show that

hminusrarrint tandτn

0

Gi(h tsXs

)(tminus s)α2q dWi

s (54)

possesses an as continuous modification on |h| le N Since this is true for any n andN and cupnτn ge t =Ω as the result follows

(b) Existence we want to show existence on an arbitrary interval [0T] (T will be fixedbelow)

Let Hn nisinN denote a progressively measurable localizing sequence forH in (L12)p

on [0T] Since from (H3prime) suptleT |Ht| is as finite we can and do assume without lossof generality that

∣∣Hnt (ω)

∣∣le n forall(tω)isin [0T]timesΩ (55)

Note that Ht(ω) =Hnt (ω) as on ΩT

n for all t isin [0T] where ΩTn uarrΩ as as nrarrinfin We

moreover define Xn0 = X01|X0|len Sn = inft supsltt |DsH

nt |orVn

t ge n We consider theequation

Xnt = Xn

0 +int t

0Fn(tsXn

s

)ds+ b

ksumi=1

int t0

Gni

(h tsXn

s

)(tminus s)α2q dWi

s|h=Hnt (56)

where

Fn(tsx)= 1[0Sn](s)

[aF(tsx)(tminus s)αq minus b

ksumi=1

Gprimei(Hnt tsx

)DisH

nt

(tminus s)α2q]

Gni (h tsx)= 1[0Sn](s)Gi(h tsx)

(57)

It is easy to see that theorem (415) applies to (56)Define

Sn(ω)=Sn(ω)and inf

t le T

int t0

∣∣Hs(ω)minusHns (ω)

∣∣ds gt 0

if∣∣X0(ω)

∣∣ lt n

0 otherwise(58)

464 Volterra equations with fractional stochastic integrals

Sn is a stopping time and it follows from the uniqueness part of Theorem 44 that ifmgt n

Xmt = Xn

t on[0 Sn

]as (59)

Since moreover Sn = T uarrΩ as we can define the process Xt on [0T] by Xt = Xnt

on [0 Sn] for all nisinNClearly X isin Lq(0T) as and solves (31) on [0T] Since T is arbitrary the existence

is proved(c) Uniqueness it suffices to prove uniqueness on an arbitrary interval [0T] Let

Xt t isin [0T] be a progressively measurable process such that X isin Lq(0T) as andX solves (31) It suffices to show that X coincides with the solution we have just con-structed

Let

Sn(ω)= S(ω)and inft le T

int t0

∣∣Xs(ω)∣∣q

(tminus s)α ds gt n

Xnt = XtandSn

(510)

Sn isin Lq(Ωtimes [0T]) and it solves (56) with Sn replaced by SnThen

Xnt (ω)= Xt(ω)timesP ae on

[0 Sn

] (511)

where is the Lebesgue measure on the real lineThe result follows from the fact that Sn ge T uarrΩ asNote that the above solution satisfies in fact X isin⋂qgt1

⋂tgt0L

q(0 t) as

6 Continuity of the solution

We want to give additional conditions under which the solution of (31) is an as contin-uous process

(H6) For all (sx)isinR+timesRd trarr F(tsx)(tminus s)αq is as continuous on (s+infin)(H7) Ht t ge 0 is as continuous(H8) For all iisin 1 k sisinR+ trarrDi

sHt is as continuous on (s+infin)(H9) For all (sx) isin R+ timesRd i isin 1 k (th)rarr Gprimei (h tsx)(tminus s)α2q is as con-

tinuous on (s+infin)timesRP

We also suppose that there exist δ gt 0 gt 0 such that for all N gt 0 |h| le N 0 le s lttand r x isinRd |tminus r| le 1

(H10) There exists an increasing process VNt t ge 0 such that

∣∣Gi(h tsx)minusGi(hrsx)∣∣leVN

t |tminus r|δ(1 + |x|) (61)

Theorem 61 Under conditions (H1prime) (H2prime) (H3prime) (H4prime) (H5prime) and (H6) (H7) (H8)(H9) (H10) the unique solution of (31) (which is progressively measurable and belongs asto⋂qgt1⋂tgt0L

q(0 t)) has as continuous modification

Mahmoud M El-Borai et al 465

Proof We need to show only that whenever X isin⋂qgt1⋂tgt0L

q(0 t) as Jt(X) t gt 0 hasan as continuous modification

(a) We first show that trarr int t0 F(tsXs)ds is as continuous

Note that (H6) (H7) (H8) and (H9) imply that for all (sx)isinR+timesRd trarr F(tsx)is as continuous on (s+infin)

Moreover from (H2prime) (H3prime) (H4prime) and the fact that X isin⋂qgt1⋂tgt0L

q(0 t) as forany T gt 0 there exists a process ZTs sisin [0T] such that

∣∣F(tsXs)∣∣le ZTs ∣∣Gprimei(Ht tsXs

)DisHt

∣∣le ZTs 0le s lt t le T asint t

0ZTs

1

(tminus s)α2q +1

(tminus s)αqds ltinfin as

(62)

Let first tn nisinN be a sequence such that tn lt t for any n and tnrarr t as nrarrinfin then

int t0F(tsXs

)dsminus

int tn0F(tnsXs

)ds=

int ttnF(tsXs

)ds+

int tn0

[F(tsXs

)minus F(tnsXs)]ds

∣∣∣∣int ttnF(tsXs

)ds∣∣∣∣le

int ttnZTs

1

(tminus s)α2q +1

(tminus s)αqds

(63)

and the latter tends as to 0 as nrarrinfin

∣∣∣∣int tn

0

[F(tsXs

)minus F(tnsXs)]ds∣∣∣∣le

int t0

∣∣F(tsXs)minus F(tnsXs)∣∣ds (64)

which tends to 0 as nrarrinfin A similar argument gives the same result when tn gt t tnrarr t(b) We next show that t rarr It(X Ht) possesses an as continuous modification This

follows from (H7) and

(th)minusrarr It(X h) (65)

has an as continuous modificationBy localization it suffices to prove (65) under assumptions (H2) (H3) (H4) (H5)

and (H6) (H7) (H8) (H9) (H10) with VNt (ω) in (H10) replaced by a constant K

and in case X0 isin⋂qgt1L

q(ΩRd)It then suffices to show that under the above hypotheses there exists Cq gt 0 such that

for any hk isinRP the numbers tr will be positive Suppose to fix the ideas that 0le r lt tthen

It(X h)minus Ir(X k)=int tr

Gi(h tsXs

)(tminus s)α2q dWi

s +int r

0

[Gi(h tsXs

)(tminus s)α2q minus Gi

(hrsXs

)(rminus s)α2q

]dWi

s

+int r

0

[Gi(h tsXs

)(rminus s)α2q minus Gi

(krsXs

)(rminus s)α2q

]dWi

s

(66)

466 Volterra equations with fractional stochastic integrals

It follows from the Burkholder-Gundy inequality that

E(∣∣∣∣int tr

Gi(h tsXs

)(tminus s)α2q dWi

s

∣∣∣∣q)le cq

ksumi=1

E

[(int tr

∣∣Gi(h tsXs

)∣∣2

(tminus s)αq ds)q2]

le cq(tminus r)(qminus2)2ksumi=1

Eint tr

∣∣Gi(h tsXs

)∣∣q(tminus s)α2 ds

le cq(tminus r)(qminus2)2int tr

E(1 +

∣∣Xs∣∣q)(tminus s)α2 ds

le Cq(tminus r)(qminus2)2(tminus r)1minusα2

(67)

From (H4) for Gi we deduce as in Lemma 41 that

E

(∣∣∣∣int r

0

[Gi(hrsXs

)minusGi(krsXs

)(rminus s)α2q

]dWi

s

∣∣∣∣q)

le cq(hminus k)qint r

0

E(1 +

∣∣Xs∣∣q)(rminus s)α2 dsle Cq(hminus k)qr1minusα2

(68)

From (H10) and the fact that

X isin⋂qgt1

⋂tgt0

Lq(Ωtimes (0 t)

)

E

(∣∣∣∣int r

0

[Gi(h tsXs

)(tminus s)α2q minus Gi

(hrsXs

)(rminus s)α2q

]dWi

s

∣∣∣∣q)

= E(∣∣∣∣int r

0

(1

(rminus s)α2q[Gi(h tsXs

)minusGi(hrsXs

)]

+Gi(h tsXs

)[ 1(tminus s)α2q minus

1(rminus s)α2q

])dWi

s

∣∣∣∣q)

le E(cq

int r0

(∣∣Gi(h tsXs

)minusGi(hrsXs

)∣∣q(rminus s)α

+∣∣Gi

(h tsXs

)∣∣q((rminus s)minusα2qminus (tminus s)minusα2q)q)ds

)

le cqKq(int r

0

( |tminus r|δq(rminus s)α E

(1 +

∣∣Xs∣∣q))ds

+int r

0

(E(1 +

∣∣Xs∣∣q)((rminus s)minusα2minus (tminus s)minusα2))ds)

le cq[|tminus r|δq

int r0

(rminus s)minusα2ds+int r

0

((rminus s)minusα2minus (tminus s)minusα2)ds]

le Cq(|tminus r|δqr1minusα2 +

(r1minusα2 + (tminus r)1minusα2minus t1minusα2))

(69)

Mahmoud M El-Borai et al 467

which from the above estimate yields

E(∣∣It(X h)minus Ir(X k)

∣∣q)le Cq

[|tminus r|1minusα2 + |tminus r|(qminusα)2

+ r1minusα2(|tminus r|δq + |hminus k|q)+(r1minusα2minus t1minusα2)]

(610)

this completes the proof

Acknowledgment

We are very grateful to the referees for valuable comments

References

[1] C Bender and R J Elliott Binary market models and discrete wick products preprint 2002[2] M A Berger and V J Mizel Volterra equations with Ito integrals I J Integral Equations 2

(1980) no 3 187ndash245[3] Volterra equations with Ito integrals II J Integral Equations 2 (1980) no 4 319ndash337[4] An extension of the stochastic integral Ann Probab 10 (1982) no 2 435ndash450[5] M M El-Borai Some probability densities and fundamental solutions of fractional evolution

equations Chaos Solitons Fractals 14 (2002) no 3 433ndash440[6] A M Kolodh On the existence solutions of stochastic Volterra equations Theory of Random

Processes 11 (1983) 51ndash57 (Russian)[7] B B Mandelbrot and J W Van Ness Fractional Brownian motions fractional noises and appli-

cations SIAM Rev 10 (1968) 422ndash437[8] P-A Meyer Flot drsquoune equation differentielle stochastique (drsquoapres Malliavin Bismut Kunita)

[Flow of a stochastic differential equation (following Malliavin Bismut Kunita)] Seminar onProbability XV (University of Strasbourg Strasbourg 19791980) Lecture Notes in Mathvol 850 Springer-Verlag Berlin 1981 pp 103ndash117 (French)

[9] D Nualart Noncausal stochastic integrals and calculus Stochastic Analysis and Related Topics(Silivri 1986) Lecture Notes in Math vol 1316 Springer-Verlag Berlin 1988 pp 80ndash129

[10] D Nualart and E Pardoux Stochastic calculus with anticipating integrands Probab TheoryRelated Fields 78 (1988) no 4 535ndash581

[11] D Nualart and M Zakai Generalized stochastic integrals and the Malliavin calculus ProbabTheory Related Fields 73 (1986) no 2 255ndash280

[12] D Ocone and E Pardoux A generalized Ito-Ventzell formula Application to a class of anticipat-ing stochastic differential equations Ann Inst H Poincare Probab Statist 25 (1989) no 139ndash71

[13] Linear stochastic differential equations with boundary conditions Probab Theory Re-lated Fields 82 (1989) no 4 489ndash526

[14] S Ogawa Sur la question drsquoexistence de solutions drsquoune equation differentielle stochastique dutype noncausal [On the existence of solutions of a stochastic differential equation of noncausaltype] J Math Kyoto Univ 24 (1984) no 4 699ndash704 (French)

[15] E Pardoux and P Protter A two-sided stochastic integral and its calculus Probab Theory RelatedFields 76 (1987) no 1 15ndash49

[16] Stochastic Volterra equations with anticipating coefficients Ann Probab 18 (1990)no 4 1635ndash1655

[17] P Protter Volterra equations driven by semimartingales Ann Probab 13 (1985) no 2 519ndash530

468 Volterra equations with fractional stochastic integrals

[18] A N V Rao and C P Tsokos On the existence uniqueness and stability behavior of a randomsolution to a nonlinear perturbed stochastic integro-differential equation Information andControl 27 (1975) 61ndash74

[19] Y Shiota A linear stochastic integral equation containing the extended Ito integral Math RepToyama Univ 9 (1986) 43ndash65

[20] A V Skorohod On a generalization of the stochastic integral Teor Verojatnost i Primenen 20(1975) no 2 223ndash238 (Russian)

[21] A-S Sznitman Martingales dependant drsquoun parametre une formule drsquoIto [Martingales depend-ing on a parameter an Ito formula] Z Wahrsch Verw Gebiete 60 (1982) no 1 41ndash70(French)

[22] W Wyss The fractional diffusion equation J Math Phys 27 (1986) no 11 2782ndash2785

Mahmoud M El-Borai Faculty of Science Alexandria University Alexandria EgyptE-mail address m m elboraiyahoocom

Khairia El-Said El-Nadi Faculty of Science Alexandria University Alexandria EgyptE-mail address khairia el saidhotmailcom

Osama L Mostafa Faculty of Science Alexandria University Alexandria EgyptE-mail address olmoustafayahoocom

Hamdy M Ahmed Faculty of Science Alexandria University Alexandria EgyptE-mail address hamdy 17egyahoocom

Mahmoud M El-Borai et al 459

Let B be an open bounded subset of Rp K gt 0 and q gt p such that

(H1) X0 isin Lq(Ωη0PRd)(H2) P(Ht isin Bforallt ge 0)= 1(H3) H isin (L12)p |DsHt| le K as 0le s lt t(H4) |F(tsx)|+

sumki=1 |Gi(h tsx)|+

sumki=1 |Gprimei (h tsx)| le K(1 + |x|) for any 0le s lt t

hisin B x isinRd(H5) |F(tsx) minus F(ts y)| +

sumki=1 |Gi(h tsx) minus Gi(h ts y)| +

sumki=1 |Gprimei (h tsx)

minusGprimei (h ts y)| le K|xminus y| for any 0le s lt t hisin B and x y isinRd as

Note that q will be a fixed real number such that q gt p and Lqprog(Ωtimes (0 t)) will

stand for the space Lq(Ωtimes (0 t)ξtP times λ) where ξt denotes the σ-algebra of progres-sively measurable subsets of Ωtimes (0 t) and λ denotes the Lebesgue measure on (0 t) andset 1minusβ = αq 0 lt α lt 1

Lemma 41 Let X isin⋂tgt0Lqprog(Ωtimes (0 t)) where q gt p and suppose that (H4) is in force

Then for any T ge t gt 0 and iisin 1 k the random field

int t0

Gi(h tsXs

)(tminus s)α2q dWi

s hisin B

(41)

possesses an as continuous modification

Proof Using Burkholder-Gundy and Holderrsquos inequalities together with (H4) we obtain

E

(∣∣∣∣int t

0

Gi(h tsXs

)(tminus s)α2q dWi

s minusint t

0

Gi(k tsXs

)(tminus s)α2q dWi

s

∣∣∣∣q)

le c1Eint t

0

∣∣Gi(h tsXs

)minusGi(k tsXs

)∣∣q(tminus s)α2 ds

le c1Kq|hminus k|q

int t0

E(1 +

∣∣Xs∣∣q)(tminus s)α2 ds

le c2Kq|hminus k|q

int t0(tminus s)minusα2ds

le c3|hminus k|qt1minusα2

(42)

where c1 c2 and c3 are positive constants The result now follows from the multidimen-sional generalization of Kolmogrovrsquos lemma (see [1 8])

We can now assume that for fixed t the random field (41) is as continuous in hprovided X isin Lqprog(Ωtimes (0 t))

From X isin⋂tgt0Lqprog(Ωtimes (0 t)) define

It(X h)= bksumi=1

int t0

Gi(h tsXs

)(tminus s)α2q dWi

s hisinRp t gt 0

Jt(X)=int t

0F(tsXs

)ds+ It

(X Ht

)

(43)

460 Volterra equations with fractional stochastic integrals

Lemma 42 For any t gt 0 there is a constant c gt 0 such that

E(∣∣Jt(X)

∣∣q)le c(int t

0E(1 +

∣∣Xs∣∣q)

1(tminus s)α +

1(tminus s)α2

ds)

0 lt α lt 1 (44)

Proof

∣∣∣∣int t

0F(tsXs

)ds∣∣∣∣q

le cint t

0

∣∣F(tsXs)∣∣q(tminus s)α +

ksumi=1

∣∣Gprimei(h tsXs)|qh=Ht

∣∣DisHt

∣∣q(tminus s)α2 ds

le cint t

0

(1 +

∣∣Xs∣∣q)

1(tminus s)α +

1(tminus s)α2

ds

(45)

where we have used (H2) (H3) and (H4)

∣∣It(X Ht)∣∣le sup

hisinB

∣∣It(X h)∣∣ (46)

(c c are positive constants)It is easy to show using in particular (H4) and Lebesguersquos dominated convergence

theorem that the mapping

hminusrarr It(X h) (47)

from Rp into Lq(Ω) is differentiable and that

partIt(X h)parthj

= bint t

0

part

parthj

ksumi=1

(Gi(h tsXs

)(tminus s)α2q

)dWi

s (48)

Since q gt p we can infer from Sobolevrsquos embedding theorem (see [13]) that

E

(suphisinB

∣∣It(X h)∣∣q)le cE

intB

(∣∣It(X h)∣∣q +

psumj=1

∣∣∣∣ partItparthj(X h)

∣∣∣∣q)dh (49)

It then follows from the Burkholder-Gundy inequality that

E

(suphisinB

∣∣It(X h)∣∣q)le cE

intB

int t0

ksumi=1

(∣∣Gi(h tsXs

)∣∣q(tminus s)α2 +

psumj=1

∣∣∣∣ part

parthj

Gi(h tsXs

)(tminus s)α2q

∣∣∣∣q)dsdh

le c(int

Bdh)(Eint t

0

(1 +

∣∣Xs∣∣q)(tminus s)α2 ds

)

le cint t

0

E(1 +

∣∣Xs∣∣q)(tminus s)α2 ds

(410)

where we have used (H4) and B is bounded From (45) and (410) the proof is complete

A similar argument using (H5) instead of (H4) yields the following lemma

Mahmoud M El-Borai et al 461

Lemma 43 For any 0 lt t le T there exists a constant c gt 0 such that

E(∣∣Jt(X)minus Jt(Y)

∣∣q)le cint t

0E∣∣XsminusYs∣∣q

1

(tminus s)α +1

(tminus s)α2ds (411)

We are now in a position to prove the main result of this section

Theorem 44 Under conditions (H1) (H2) (H3) (H4) and (H5) there exists a uniqueelement X isin ⋂tgt0L

qprog(Ωtimes (0 t)) which solves (31) Moreover if τ is a stopping time

uniqueness holds on the random interval [0τ]

Proof Equation (31) can be rewritten as

Xt = X0 + Jt(X) t ge 0 (412)

Uniqueness Let X Y isin⋂tgt0Lqprog(Ωtimes (0 t)) and let τ be a stopping time such that

Xt = X0 + Jt(X) Yt = X0 + Jt(Y) 0le t le τ (413)

From Lemma 43

E(∣∣Xt minusYt∣∣q)= E(∣∣Jt(X)minus Jt(Y)

∣∣q)le c

int t0E∣∣XsminusYs∣∣q

1

(tminus s)α +1

(tminus s)α2ds

(414)

Set (E|Xt minusYt|q)= γt It is easy to see that

γt le 2cη(t)int t

0

γs(tminus s)α ds (415)

where η(t)=Max(1 tα) and c is a positive constant Thus (see [5 21])

γt le M2(Γ(tminusα)

)2

Γ(2(1minusα)

) int t0

γs(tminus s)2αminus1

ds (416)

and by a classical argument we get

γt le Mn(Γ(tminusα)

)nΓ(n(1minusα)

) int t0

γs(tminus s)nαminus(nminus1)

ds (417)

where M = 2cη(T) Thus for sufficiently large n n gt 1(1minusα) we get from (417)

γt le Kn

Γ(n(1minusα)

) int t0γs ds (418)

where K is a positive constantTaking the limit as nrarrinfin we find that (418) leads to γt = 0To prove the existence we define a sequence Xn

t 0le t le T n= 012 as follows

Xot = Xo Xn+1

t = Xo + Jt(Xn) T le t le 0 (419)

462 Volterra equations with fractional stochastic integrals

Using Lemma 42 we can see that

Xn isin⋂tgt0

Lqprog

(Ωtimes [0T]

) (420)

It then follows from Lemma 43 that

gn+1(t)le 2cη(t)int t

0

gn(s)(tminus s)α ds (421)

where

gn+1(t)= E(∣∣Xn+1t minusXn

t

∣∣q) (422)

By using a similar argument we can write

gn le Kn

Γ(n(1minusα)

) int t0go(s)ds (423)

The last estimation implies that Xn is a Cauchy sequence in Lqprog(Ωtimes [0T]) Then there

exists X such that Xn rarr X in⋂tgt0L

qprog(Ωtimes [0T]) and again using Lemma 43 we can

pass to the limit in (420) yielding that X solves (412)

5 An existence and uniqueness result under weaker assumptions

We formulate a new set of weaker hypotheses

(H1prime) X0 is η0 measurable(H2prime) H isin (L12

loc)pHt is a progressively measurable process which can be localized in(L12)p by a progressively measurable sequence

We assume that there exists an increasing progressively measurable process Ut t ge 0with values in R+ such that

(H3prime) |Ht|+sumk

i=1 |DisHt| leUt as 0le s lt t

Finally we suppose that for any N gt 0 there exists an increasing progressively measur-able process VN

t t ge 0 with values in R+ such that

(H4prime) |F(tsx)|+sumk

i=1 |Gi(h tsx)|+sumk

i=1 |Gprimei (h tsx)| le VNt (1+|x|) for all |h|leN

0le s lt t and x isinRd(H5prime) |F(tsx) minus F(ts y)| +

sumki=1 |Gi(h tsx) minus Gi(h ts y)| +

sumki=1 |Gprimei (h tsx)

minusGprimei (h ts y)| leVNt |xminus y| for all |h| leN 0le s lt t x y isinRd

Let again q be a fixed real number with q gt p and set 1minusβ = αq 0 lt α lt 1 We havethe following theorem

Theorem 51 Equation (31) has a unique solution in the class of progressively measurableprocesses which satisfy

X isin⋂tgt0

Lq(0 t) as (51)

Mahmoud M El-Borai et al 463

Proof (a) We first see how (31) makes sense if X isin⋂tgt0Lq(0 t) as

That is we have to show that for fixed t gt 0

int t0

Gi(h tsXs

)(tminus s)α2q dWi

s hisinRP

(52)

is a well-defined random field which possesses an as continuous versionFor that sake we define

τn = inftint t

0

∣∣Xs∣∣q(tminus s)α dsge n or VN

t ge n (53)

The argument of Lemma 41 can be used to show that

hminusrarrint tandτn

0

Gi(h tsXs

)(tminus s)α2q dWi

s (54)

possesses an as continuous modification on |h| le N Since this is true for any n andN and cupnτn ge t =Ω as the result follows

(b) Existence we want to show existence on an arbitrary interval [0T] (T will be fixedbelow)

Let Hn nisinN denote a progressively measurable localizing sequence forH in (L12)p

on [0T] Since from (H3prime) suptleT |Ht| is as finite we can and do assume without lossof generality that

∣∣Hnt (ω)

∣∣le n forall(tω)isin [0T]timesΩ (55)

Note that Ht(ω) =Hnt (ω) as on ΩT

n for all t isin [0T] where ΩTn uarrΩ as as nrarrinfin We

moreover define Xn0 = X01|X0|len Sn = inft supsltt |DsH

nt |orVn

t ge n We consider theequation

Xnt = Xn

0 +int t

0Fn(tsXn

s

)ds+ b

ksumi=1

int t0

Gni

(h tsXn

s

)(tminus s)α2q dWi

s|h=Hnt (56)

where

Fn(tsx)= 1[0Sn](s)

[aF(tsx)(tminus s)αq minus b

ksumi=1

Gprimei(Hnt tsx

)DisH

nt

(tminus s)α2q]

Gni (h tsx)= 1[0Sn](s)Gi(h tsx)

(57)

It is easy to see that theorem (415) applies to (56)Define

Sn(ω)=Sn(ω)and inf

t le T

int t0

∣∣Hs(ω)minusHns (ω)

∣∣ds gt 0

if∣∣X0(ω)

∣∣ lt n

0 otherwise(58)

464 Volterra equations with fractional stochastic integrals

Sn is a stopping time and it follows from the uniqueness part of Theorem 44 that ifmgt n

Xmt = Xn

t on[0 Sn

]as (59)

Since moreover Sn = T uarrΩ as we can define the process Xt on [0T] by Xt = Xnt

on [0 Sn] for all nisinNClearly X isin Lq(0T) as and solves (31) on [0T] Since T is arbitrary the existence

is proved(c) Uniqueness it suffices to prove uniqueness on an arbitrary interval [0T] Let

Xt t isin [0T] be a progressively measurable process such that X isin Lq(0T) as andX solves (31) It suffices to show that X coincides with the solution we have just con-structed

Let

Sn(ω)= S(ω)and inft le T

int t0

∣∣Xs(ω)∣∣q

(tminus s)α ds gt n

Xnt = XtandSn

(510)

Sn isin Lq(Ωtimes [0T]) and it solves (56) with Sn replaced by SnThen

Xnt (ω)= Xt(ω)timesP ae on

[0 Sn

] (511)

where is the Lebesgue measure on the real lineThe result follows from the fact that Sn ge T uarrΩ asNote that the above solution satisfies in fact X isin⋂qgt1

⋂tgt0L

q(0 t) as

6 Continuity of the solution

We want to give additional conditions under which the solution of (31) is an as contin-uous process

(H6) For all (sx)isinR+timesRd trarr F(tsx)(tminus s)αq is as continuous on (s+infin)(H7) Ht t ge 0 is as continuous(H8) For all iisin 1 k sisinR+ trarrDi

sHt is as continuous on (s+infin)(H9) For all (sx) isin R+ timesRd i isin 1 k (th)rarr Gprimei (h tsx)(tminus s)α2q is as con-

tinuous on (s+infin)timesRP

We also suppose that there exist δ gt 0 gt 0 such that for all N gt 0 |h| le N 0 le s lttand r x isinRd |tminus r| le 1

(H10) There exists an increasing process VNt t ge 0 such that

∣∣Gi(h tsx)minusGi(hrsx)∣∣leVN

t |tminus r|δ(1 + |x|) (61)

Theorem 61 Under conditions (H1prime) (H2prime) (H3prime) (H4prime) (H5prime) and (H6) (H7) (H8)(H9) (H10) the unique solution of (31) (which is progressively measurable and belongs asto⋂qgt1⋂tgt0L

q(0 t)) has as continuous modification

Mahmoud M El-Borai et al 465

Proof We need to show only that whenever X isin⋂qgt1⋂tgt0L

q(0 t) as Jt(X) t gt 0 hasan as continuous modification

(a) We first show that trarr int t0 F(tsXs)ds is as continuous

Note that (H6) (H7) (H8) and (H9) imply that for all (sx)isinR+timesRd trarr F(tsx)is as continuous on (s+infin)

Moreover from (H2prime) (H3prime) (H4prime) and the fact that X isin⋂qgt1⋂tgt0L

q(0 t) as forany T gt 0 there exists a process ZTs sisin [0T] such that

∣∣F(tsXs)∣∣le ZTs ∣∣Gprimei(Ht tsXs

)DisHt

∣∣le ZTs 0le s lt t le T asint t

0ZTs

1

(tminus s)α2q +1

(tminus s)αqds ltinfin as

(62)

Let first tn nisinN be a sequence such that tn lt t for any n and tnrarr t as nrarrinfin then

int t0F(tsXs

)dsminus

int tn0F(tnsXs

)ds=

int ttnF(tsXs

)ds+

int tn0

[F(tsXs

)minus F(tnsXs)]ds

∣∣∣∣int ttnF(tsXs

)ds∣∣∣∣le

int ttnZTs

1

(tminus s)α2q +1

(tminus s)αqds

(63)

and the latter tends as to 0 as nrarrinfin

∣∣∣∣int tn

0

[F(tsXs

)minus F(tnsXs)]ds∣∣∣∣le

int t0

∣∣F(tsXs)minus F(tnsXs)∣∣ds (64)

which tends to 0 as nrarrinfin A similar argument gives the same result when tn gt t tnrarr t(b) We next show that t rarr It(X Ht) possesses an as continuous modification This

follows from (H7) and

(th)minusrarr It(X h) (65)

has an as continuous modificationBy localization it suffices to prove (65) under assumptions (H2) (H3) (H4) (H5)

and (H6) (H7) (H8) (H9) (H10) with VNt (ω) in (H10) replaced by a constant K

and in case X0 isin⋂qgt1L

q(ΩRd)It then suffices to show that under the above hypotheses there exists Cq gt 0 such that

for any hk isinRP the numbers tr will be positive Suppose to fix the ideas that 0le r lt tthen

It(X h)minus Ir(X k)=int tr

Gi(h tsXs

)(tminus s)α2q dWi

s +int r

0

[Gi(h tsXs

)(tminus s)α2q minus Gi

(hrsXs

)(rminus s)α2q

]dWi

s

+int r

0

[Gi(h tsXs

)(rminus s)α2q minus Gi

(krsXs

)(rminus s)α2q

]dWi

s

(66)

466 Volterra equations with fractional stochastic integrals

It follows from the Burkholder-Gundy inequality that

E(∣∣∣∣int tr

Gi(h tsXs

)(tminus s)α2q dWi

s

∣∣∣∣q)le cq

ksumi=1

E

[(int tr

∣∣Gi(h tsXs

)∣∣2

(tminus s)αq ds)q2]

le cq(tminus r)(qminus2)2ksumi=1

Eint tr

∣∣Gi(h tsXs

)∣∣q(tminus s)α2 ds

le cq(tminus r)(qminus2)2int tr

E(1 +

∣∣Xs∣∣q)(tminus s)α2 ds

le Cq(tminus r)(qminus2)2(tminus r)1minusα2

(67)

From (H4) for Gi we deduce as in Lemma 41 that

E

(∣∣∣∣int r

0

[Gi(hrsXs

)minusGi(krsXs

)(rminus s)α2q

]dWi

s

∣∣∣∣q)

le cq(hminus k)qint r

0

E(1 +

∣∣Xs∣∣q)(rminus s)α2 dsle Cq(hminus k)qr1minusα2

(68)

From (H10) and the fact that

X isin⋂qgt1

⋂tgt0

Lq(Ωtimes (0 t)

)

E

(∣∣∣∣int r

0

[Gi(h tsXs

)(tminus s)α2q minus Gi

(hrsXs

)(rminus s)α2q

]dWi

s

∣∣∣∣q)

= E(∣∣∣∣int r

0

(1

(rminus s)α2q[Gi(h tsXs

)minusGi(hrsXs

)]

+Gi(h tsXs

)[ 1(tminus s)α2q minus

1(rminus s)α2q

])dWi

s

∣∣∣∣q)

le E(cq

int r0

(∣∣Gi(h tsXs

)minusGi(hrsXs

)∣∣q(rminus s)α

+∣∣Gi

(h tsXs

)∣∣q((rminus s)minusα2qminus (tminus s)minusα2q)q)ds

)

le cqKq(int r

0

( |tminus r|δq(rminus s)α E

(1 +

∣∣Xs∣∣q))ds

+int r

0

(E(1 +

∣∣Xs∣∣q)((rminus s)minusα2minus (tminus s)minusα2))ds)

le cq[|tminus r|δq

int r0

(rminus s)minusα2ds+int r

0

((rminus s)minusα2minus (tminus s)minusα2)ds]

le Cq(|tminus r|δqr1minusα2 +

(r1minusα2 + (tminus r)1minusα2minus t1minusα2))

(69)

Mahmoud M El-Borai et al 467

which from the above estimate yields

E(∣∣It(X h)minus Ir(X k)

∣∣q)le Cq

[|tminus r|1minusα2 + |tminus r|(qminusα)2

+ r1minusα2(|tminus r|δq + |hminus k|q)+(r1minusα2minus t1minusα2)]

(610)

this completes the proof

Acknowledgment

We are very grateful to the referees for valuable comments

References

[1] C Bender and R J Elliott Binary market models and discrete wick products preprint 2002[2] M A Berger and V J Mizel Volterra equations with Ito integrals I J Integral Equations 2

(1980) no 3 187ndash245[3] Volterra equations with Ito integrals II J Integral Equations 2 (1980) no 4 319ndash337[4] An extension of the stochastic integral Ann Probab 10 (1982) no 2 435ndash450[5] M M El-Borai Some probability densities and fundamental solutions of fractional evolution

equations Chaos Solitons Fractals 14 (2002) no 3 433ndash440[6] A M Kolodh On the existence solutions of stochastic Volterra equations Theory of Random

Processes 11 (1983) 51ndash57 (Russian)[7] B B Mandelbrot and J W Van Ness Fractional Brownian motions fractional noises and appli-

cations SIAM Rev 10 (1968) 422ndash437[8] P-A Meyer Flot drsquoune equation differentielle stochastique (drsquoapres Malliavin Bismut Kunita)

[Flow of a stochastic differential equation (following Malliavin Bismut Kunita)] Seminar onProbability XV (University of Strasbourg Strasbourg 19791980) Lecture Notes in Mathvol 850 Springer-Verlag Berlin 1981 pp 103ndash117 (French)

[9] D Nualart Noncausal stochastic integrals and calculus Stochastic Analysis and Related Topics(Silivri 1986) Lecture Notes in Math vol 1316 Springer-Verlag Berlin 1988 pp 80ndash129

[10] D Nualart and E Pardoux Stochastic calculus with anticipating integrands Probab TheoryRelated Fields 78 (1988) no 4 535ndash581

[11] D Nualart and M Zakai Generalized stochastic integrals and the Malliavin calculus ProbabTheory Related Fields 73 (1986) no 2 255ndash280

[12] D Ocone and E Pardoux A generalized Ito-Ventzell formula Application to a class of anticipat-ing stochastic differential equations Ann Inst H Poincare Probab Statist 25 (1989) no 139ndash71

[13] Linear stochastic differential equations with boundary conditions Probab Theory Re-lated Fields 82 (1989) no 4 489ndash526

[14] S Ogawa Sur la question drsquoexistence de solutions drsquoune equation differentielle stochastique dutype noncausal [On the existence of solutions of a stochastic differential equation of noncausaltype] J Math Kyoto Univ 24 (1984) no 4 699ndash704 (French)

[15] E Pardoux and P Protter A two-sided stochastic integral and its calculus Probab Theory RelatedFields 76 (1987) no 1 15ndash49

[16] Stochastic Volterra equations with anticipating coefficients Ann Probab 18 (1990)no 4 1635ndash1655

[17] P Protter Volterra equations driven by semimartingales Ann Probab 13 (1985) no 2 519ndash530

468 Volterra equations with fractional stochastic integrals

[18] A N V Rao and C P Tsokos On the existence uniqueness and stability behavior of a randomsolution to a nonlinear perturbed stochastic integro-differential equation Information andControl 27 (1975) 61ndash74

[19] Y Shiota A linear stochastic integral equation containing the extended Ito integral Math RepToyama Univ 9 (1986) 43ndash65

[20] A V Skorohod On a generalization of the stochastic integral Teor Verojatnost i Primenen 20(1975) no 2 223ndash238 (Russian)

[21] A-S Sznitman Martingales dependant drsquoun parametre une formule drsquoIto [Martingales depend-ing on a parameter an Ito formula] Z Wahrsch Verw Gebiete 60 (1982) no 1 41ndash70(French)

[22] W Wyss The fractional diffusion equation J Math Phys 27 (1986) no 11 2782ndash2785

Mahmoud M El-Borai Faculty of Science Alexandria University Alexandria EgyptE-mail address m m elboraiyahoocom

Khairia El-Said El-Nadi Faculty of Science Alexandria University Alexandria EgyptE-mail address khairia el saidhotmailcom

Osama L Mostafa Faculty of Science Alexandria University Alexandria EgyptE-mail address olmoustafayahoocom

Hamdy M Ahmed Faculty of Science Alexandria University Alexandria EgyptE-mail address hamdy 17egyahoocom

460 Volterra equations with fractional stochastic integrals

Lemma 42 For any t gt 0 there is a constant c gt 0 such that

E(∣∣Jt(X)

∣∣q)le c(int t

0E(1 +

∣∣Xs∣∣q)

1(tminus s)α +

1(tminus s)α2

ds)

0 lt α lt 1 (44)

Proof

∣∣∣∣int t

0F(tsXs

)ds∣∣∣∣q

le cint t

0

∣∣F(tsXs)∣∣q(tminus s)α +

ksumi=1

∣∣Gprimei(h tsXs)|qh=Ht

∣∣DisHt

∣∣q(tminus s)α2 ds

le cint t

0

(1 +

∣∣Xs∣∣q)

1(tminus s)α +

1(tminus s)α2

ds

(45)

where we have used (H2) (H3) and (H4)

∣∣It(X Ht)∣∣le sup

hisinB

∣∣It(X h)∣∣ (46)

(c c are positive constants)It is easy to show using in particular (H4) and Lebesguersquos dominated convergence

theorem that the mapping

hminusrarr It(X h) (47)

from Rp into Lq(Ω) is differentiable and that

partIt(X h)parthj

= bint t

0

part

parthj

ksumi=1

(Gi(h tsXs

)(tminus s)α2q

)dWi

s (48)

Since q gt p we can infer from Sobolevrsquos embedding theorem (see [13]) that

E

(suphisinB

∣∣It(X h)∣∣q)le cE

intB

(∣∣It(X h)∣∣q +

psumj=1

∣∣∣∣ partItparthj(X h)

∣∣∣∣q)dh (49)

It then follows from the Burkholder-Gundy inequality that

E

(suphisinB

∣∣It(X h)∣∣q)le cE

intB

int t0

ksumi=1

(∣∣Gi(h tsXs

)∣∣q(tminus s)α2 +

psumj=1

∣∣∣∣ part

parthj

Gi(h tsXs

)(tminus s)α2q

∣∣∣∣q)dsdh

le c(int

Bdh)(Eint t

0

(1 +

∣∣Xs∣∣q)(tminus s)α2 ds

)

le cint t

0

E(1 +

∣∣Xs∣∣q)(tminus s)α2 ds

(410)

where we have used (H4) and B is bounded From (45) and (410) the proof is complete

A similar argument using (H5) instead of (H4) yields the following lemma

Mahmoud M El-Borai et al 461

Lemma 43 For any 0 lt t le T there exists a constant c gt 0 such that

E(∣∣Jt(X)minus Jt(Y)

∣∣q)le cint t

0E∣∣XsminusYs∣∣q

1

(tminus s)α +1

(tminus s)α2ds (411)

We are now in a position to prove the main result of this section

Theorem 44 Under conditions (H1) (H2) (H3) (H4) and (H5) there exists a uniqueelement X isin ⋂tgt0L

qprog(Ωtimes (0 t)) which solves (31) Moreover if τ is a stopping time

uniqueness holds on the random interval [0τ]

Proof Equation (31) can be rewritten as

Xt = X0 + Jt(X) t ge 0 (412)

Uniqueness Let X Y isin⋂tgt0Lqprog(Ωtimes (0 t)) and let τ be a stopping time such that

Xt = X0 + Jt(X) Yt = X0 + Jt(Y) 0le t le τ (413)

From Lemma 43

E(∣∣Xt minusYt∣∣q)= E(∣∣Jt(X)minus Jt(Y)

∣∣q)le c

int t0E∣∣XsminusYs∣∣q

1

(tminus s)α +1

(tminus s)α2ds

(414)

Set (E|Xt minusYt|q)= γt It is easy to see that

γt le 2cη(t)int t

0

γs(tminus s)α ds (415)

where η(t)=Max(1 tα) and c is a positive constant Thus (see [5 21])

γt le M2(Γ(tminusα)

)2

Γ(2(1minusα)

) int t0

γs(tminus s)2αminus1

ds (416)

and by a classical argument we get

γt le Mn(Γ(tminusα)

)nΓ(n(1minusα)

) int t0

γs(tminus s)nαminus(nminus1)

ds (417)

where M = 2cη(T) Thus for sufficiently large n n gt 1(1minusα) we get from (417)

γt le Kn

Γ(n(1minusα)

) int t0γs ds (418)

where K is a positive constantTaking the limit as nrarrinfin we find that (418) leads to γt = 0To prove the existence we define a sequence Xn

t 0le t le T n= 012 as follows

Xot = Xo Xn+1

t = Xo + Jt(Xn) T le t le 0 (419)

462 Volterra equations with fractional stochastic integrals

Using Lemma 42 we can see that

Xn isin⋂tgt0

Lqprog

(Ωtimes [0T]

) (420)

It then follows from Lemma 43 that

gn+1(t)le 2cη(t)int t

0

gn(s)(tminus s)α ds (421)

where

gn+1(t)= E(∣∣Xn+1t minusXn

t

∣∣q) (422)

By using a similar argument we can write

gn le Kn

Γ(n(1minusα)

) int t0go(s)ds (423)

The last estimation implies that Xn is a Cauchy sequence in Lqprog(Ωtimes [0T]) Then there

exists X such that Xn rarr X in⋂tgt0L

qprog(Ωtimes [0T]) and again using Lemma 43 we can

pass to the limit in (420) yielding that X solves (412)

5 An existence and uniqueness result under weaker assumptions

We formulate a new set of weaker hypotheses

(H1prime) X0 is η0 measurable(H2prime) H isin (L12

loc)pHt is a progressively measurable process which can be localized in(L12)p by a progressively measurable sequence

We assume that there exists an increasing progressively measurable process Ut t ge 0with values in R+ such that

(H3prime) |Ht|+sumk

i=1 |DisHt| leUt as 0le s lt t

Finally we suppose that for any N gt 0 there exists an increasing progressively measur-able process VN

t t ge 0 with values in R+ such that

(H4prime) |F(tsx)|+sumk

i=1 |Gi(h tsx)|+sumk

i=1 |Gprimei (h tsx)| le VNt (1+|x|) for all |h|leN

0le s lt t and x isinRd(H5prime) |F(tsx) minus F(ts y)| +

sumki=1 |Gi(h tsx) minus Gi(h ts y)| +

sumki=1 |Gprimei (h tsx)

minusGprimei (h ts y)| leVNt |xminus y| for all |h| leN 0le s lt t x y isinRd

Let again q be a fixed real number with q gt p and set 1minusβ = αq 0 lt α lt 1 We havethe following theorem

Theorem 51 Equation (31) has a unique solution in the class of progressively measurableprocesses which satisfy

X isin⋂tgt0

Lq(0 t) as (51)

Mahmoud M El-Borai et al 463

Proof (a) We first see how (31) makes sense if X isin⋂tgt0Lq(0 t) as

That is we have to show that for fixed t gt 0

int t0

Gi(h tsXs

)(tminus s)α2q dWi

s hisinRP

(52)

is a well-defined random field which possesses an as continuous versionFor that sake we define

τn = inftint t

0

∣∣Xs∣∣q(tminus s)α dsge n or VN

t ge n (53)

The argument of Lemma 41 can be used to show that

hminusrarrint tandτn

0

Gi(h tsXs

)(tminus s)α2q dWi

s (54)

possesses an as continuous modification on |h| le N Since this is true for any n andN and cupnτn ge t =Ω as the result follows

(b) Existence we want to show existence on an arbitrary interval [0T] (T will be fixedbelow)

Let Hn nisinN denote a progressively measurable localizing sequence forH in (L12)p

on [0T] Since from (H3prime) suptleT |Ht| is as finite we can and do assume without lossof generality that

∣∣Hnt (ω)

∣∣le n forall(tω)isin [0T]timesΩ (55)

Note that Ht(ω) =Hnt (ω) as on ΩT

n for all t isin [0T] where ΩTn uarrΩ as as nrarrinfin We

moreover define Xn0 = X01|X0|len Sn = inft supsltt |DsH

nt |orVn

t ge n We consider theequation

Xnt = Xn

0 +int t

0Fn(tsXn

s

)ds+ b

ksumi=1

int t0

Gni

(h tsXn

s

)(tminus s)α2q dWi

s|h=Hnt (56)

where

Fn(tsx)= 1[0Sn](s)

[aF(tsx)(tminus s)αq minus b

ksumi=1

Gprimei(Hnt tsx

)DisH

nt

(tminus s)α2q]

Gni (h tsx)= 1[0Sn](s)Gi(h tsx)

(57)

It is easy to see that theorem (415) applies to (56)Define

Sn(ω)=Sn(ω)and inf

t le T

int t0

∣∣Hs(ω)minusHns (ω)

∣∣ds gt 0

if∣∣X0(ω)

∣∣ lt n

0 otherwise(58)

464 Volterra equations with fractional stochastic integrals

Sn is a stopping time and it follows from the uniqueness part of Theorem 44 that ifmgt n

Xmt = Xn

t on[0 Sn

]as (59)

Since moreover Sn = T uarrΩ as we can define the process Xt on [0T] by Xt = Xnt

on [0 Sn] for all nisinNClearly X isin Lq(0T) as and solves (31) on [0T] Since T is arbitrary the existence

is proved(c) Uniqueness it suffices to prove uniqueness on an arbitrary interval [0T] Let

Xt t isin [0T] be a progressively measurable process such that X isin Lq(0T) as andX solves (31) It suffices to show that X coincides with the solution we have just con-structed

Let

Sn(ω)= S(ω)and inft le T

int t0

∣∣Xs(ω)∣∣q

(tminus s)α ds gt n

Xnt = XtandSn

(510)

Sn isin Lq(Ωtimes [0T]) and it solves (56) with Sn replaced by SnThen

Xnt (ω)= Xt(ω)timesP ae on

[0 Sn

] (511)

where is the Lebesgue measure on the real lineThe result follows from the fact that Sn ge T uarrΩ asNote that the above solution satisfies in fact X isin⋂qgt1

⋂tgt0L

q(0 t) as

6 Continuity of the solution

We want to give additional conditions under which the solution of (31) is an as contin-uous process

(H6) For all (sx)isinR+timesRd trarr F(tsx)(tminus s)αq is as continuous on (s+infin)(H7) Ht t ge 0 is as continuous(H8) For all iisin 1 k sisinR+ trarrDi

sHt is as continuous on (s+infin)(H9) For all (sx) isin R+ timesRd i isin 1 k (th)rarr Gprimei (h tsx)(tminus s)α2q is as con-

tinuous on (s+infin)timesRP

We also suppose that there exist δ gt 0 gt 0 such that for all N gt 0 |h| le N 0 le s lttand r x isinRd |tminus r| le 1

(H10) There exists an increasing process VNt t ge 0 such that

∣∣Gi(h tsx)minusGi(hrsx)∣∣leVN

t |tminus r|δ(1 + |x|) (61)

Theorem 61 Under conditions (H1prime) (H2prime) (H3prime) (H4prime) (H5prime) and (H6) (H7) (H8)(H9) (H10) the unique solution of (31) (which is progressively measurable and belongs asto⋂qgt1⋂tgt0L

q(0 t)) has as continuous modification

Mahmoud M El-Borai et al 465

Proof We need to show only that whenever X isin⋂qgt1⋂tgt0L

q(0 t) as Jt(X) t gt 0 hasan as continuous modification

(a) We first show that trarr int t0 F(tsXs)ds is as continuous

Note that (H6) (H7) (H8) and (H9) imply that for all (sx)isinR+timesRd trarr F(tsx)is as continuous on (s+infin)

Moreover from (H2prime) (H3prime) (H4prime) and the fact that X isin⋂qgt1⋂tgt0L

q(0 t) as forany T gt 0 there exists a process ZTs sisin [0T] such that

∣∣F(tsXs)∣∣le ZTs ∣∣Gprimei(Ht tsXs

)DisHt

∣∣le ZTs 0le s lt t le T asint t

0ZTs

1

(tminus s)α2q +1

(tminus s)αqds ltinfin as

(62)

Let first tn nisinN be a sequence such that tn lt t for any n and tnrarr t as nrarrinfin then

int t0F(tsXs

)dsminus

int tn0F(tnsXs

)ds=

int ttnF(tsXs

)ds+

int tn0

[F(tsXs

)minus F(tnsXs)]ds

∣∣∣∣int ttnF(tsXs

)ds∣∣∣∣le

int ttnZTs

1

(tminus s)α2q +1

(tminus s)αqds

(63)

and the latter tends as to 0 as nrarrinfin

∣∣∣∣int tn

0

[F(tsXs

)minus F(tnsXs)]ds∣∣∣∣le

int t0

∣∣F(tsXs)minus F(tnsXs)∣∣ds (64)

which tends to 0 as nrarrinfin A similar argument gives the same result when tn gt t tnrarr t(b) We next show that t rarr It(X Ht) possesses an as continuous modification This

follows from (H7) and

(th)minusrarr It(X h) (65)

has an as continuous modificationBy localization it suffices to prove (65) under assumptions (H2) (H3) (H4) (H5)

and (H6) (H7) (H8) (H9) (H10) with VNt (ω) in (H10) replaced by a constant K

and in case X0 isin⋂qgt1L

q(ΩRd)It then suffices to show that under the above hypotheses there exists Cq gt 0 such that

for any hk isinRP the numbers tr will be positive Suppose to fix the ideas that 0le r lt tthen

It(X h)minus Ir(X k)=int tr

Gi(h tsXs

)(tminus s)α2q dWi

s +int r

0

[Gi(h tsXs

)(tminus s)α2q minus Gi

(hrsXs

)(rminus s)α2q

]dWi

s

+int r

0

[Gi(h tsXs

)(rminus s)α2q minus Gi

(krsXs

)(rminus s)α2q

]dWi

s

(66)

466 Volterra equations with fractional stochastic integrals

It follows from the Burkholder-Gundy inequality that

E(∣∣∣∣int tr

Gi(h tsXs

)(tminus s)α2q dWi

s

∣∣∣∣q)le cq

ksumi=1

E

[(int tr

∣∣Gi(h tsXs

)∣∣2

(tminus s)αq ds)q2]

le cq(tminus r)(qminus2)2ksumi=1

Eint tr

∣∣Gi(h tsXs

)∣∣q(tminus s)α2 ds

le cq(tminus r)(qminus2)2int tr

E(1 +

∣∣Xs∣∣q)(tminus s)α2 ds

le Cq(tminus r)(qminus2)2(tminus r)1minusα2

(67)

From (H4) for Gi we deduce as in Lemma 41 that

E

(∣∣∣∣int r

0

[Gi(hrsXs

)minusGi(krsXs

)(rminus s)α2q

]dWi

s

∣∣∣∣q)

le cq(hminus k)qint r

0

E(1 +

∣∣Xs∣∣q)(rminus s)α2 dsle Cq(hminus k)qr1minusα2

(68)

From (H10) and the fact that

X isin⋂qgt1

⋂tgt0

Lq(Ωtimes (0 t)

)

E

(∣∣∣∣int r

0

[Gi(h tsXs

)(tminus s)α2q minus Gi

(hrsXs

)(rminus s)α2q

]dWi

s

∣∣∣∣q)

= E(∣∣∣∣int r

0

(1

(rminus s)α2q[Gi(h tsXs

)minusGi(hrsXs

)]

+Gi(h tsXs

)[ 1(tminus s)α2q minus

1(rminus s)α2q

])dWi

s

∣∣∣∣q)

le E(cq

int r0

(∣∣Gi(h tsXs

)minusGi(hrsXs

)∣∣q(rminus s)α

+∣∣Gi

(h tsXs

)∣∣q((rminus s)minusα2qminus (tminus s)minusα2q)q)ds

)

le cqKq(int r

0

( |tminus r|δq(rminus s)α E

(1 +

∣∣Xs∣∣q))ds

+int r

0

(E(1 +

∣∣Xs∣∣q)((rminus s)minusα2minus (tminus s)minusα2))ds)

le cq[|tminus r|δq

int r0

(rminus s)minusα2ds+int r

0

((rminus s)minusα2minus (tminus s)minusα2)ds]

le Cq(|tminus r|δqr1minusα2 +

(r1minusα2 + (tminus r)1minusα2minus t1minusα2))

(69)

Mahmoud M El-Borai et al 467

which from the above estimate yields

E(∣∣It(X h)minus Ir(X k)

∣∣q)le Cq

[|tminus r|1minusα2 + |tminus r|(qminusα)2

+ r1minusα2(|tminus r|δq + |hminus k|q)+(r1minusα2minus t1minusα2)]

(610)

this completes the proof

Acknowledgment

We are very grateful to the referees for valuable comments

References

[1] C Bender and R J Elliott Binary market models and discrete wick products preprint 2002[2] M A Berger and V J Mizel Volterra equations with Ito integrals I J Integral Equations 2

(1980) no 3 187ndash245[3] Volterra equations with Ito integrals II J Integral Equations 2 (1980) no 4 319ndash337[4] An extension of the stochastic integral Ann Probab 10 (1982) no 2 435ndash450[5] M M El-Borai Some probability densities and fundamental solutions of fractional evolution

equations Chaos Solitons Fractals 14 (2002) no 3 433ndash440[6] A M Kolodh On the existence solutions of stochastic Volterra equations Theory of Random

Processes 11 (1983) 51ndash57 (Russian)[7] B B Mandelbrot and J W Van Ness Fractional Brownian motions fractional noises and appli-

cations SIAM Rev 10 (1968) 422ndash437[8] P-A Meyer Flot drsquoune equation differentielle stochastique (drsquoapres Malliavin Bismut Kunita)

[Flow of a stochastic differential equation (following Malliavin Bismut Kunita)] Seminar onProbability XV (University of Strasbourg Strasbourg 19791980) Lecture Notes in Mathvol 850 Springer-Verlag Berlin 1981 pp 103ndash117 (French)

[9] D Nualart Noncausal stochastic integrals and calculus Stochastic Analysis and Related Topics(Silivri 1986) Lecture Notes in Math vol 1316 Springer-Verlag Berlin 1988 pp 80ndash129

[10] D Nualart and E Pardoux Stochastic calculus with anticipating integrands Probab TheoryRelated Fields 78 (1988) no 4 535ndash581

[11] D Nualart and M Zakai Generalized stochastic integrals and the Malliavin calculus ProbabTheory Related Fields 73 (1986) no 2 255ndash280

[12] D Ocone and E Pardoux A generalized Ito-Ventzell formula Application to a class of anticipat-ing stochastic differential equations Ann Inst H Poincare Probab Statist 25 (1989) no 139ndash71

[13] Linear stochastic differential equations with boundary conditions Probab Theory Re-lated Fields 82 (1989) no 4 489ndash526

[14] S Ogawa Sur la question drsquoexistence de solutions drsquoune equation differentielle stochastique dutype noncausal [On the existence of solutions of a stochastic differential equation of noncausaltype] J Math Kyoto Univ 24 (1984) no 4 699ndash704 (French)

[15] E Pardoux and P Protter A two-sided stochastic integral and its calculus Probab Theory RelatedFields 76 (1987) no 1 15ndash49

[16] Stochastic Volterra equations with anticipating coefficients Ann Probab 18 (1990)no 4 1635ndash1655

[17] P Protter Volterra equations driven by semimartingales Ann Probab 13 (1985) no 2 519ndash530

468 Volterra equations with fractional stochastic integrals

[18] A N V Rao and C P Tsokos On the existence uniqueness and stability behavior of a randomsolution to a nonlinear perturbed stochastic integro-differential equation Information andControl 27 (1975) 61ndash74

[19] Y Shiota A linear stochastic integral equation containing the extended Ito integral Math RepToyama Univ 9 (1986) 43ndash65

[20] A V Skorohod On a generalization of the stochastic integral Teor Verojatnost i Primenen 20(1975) no 2 223ndash238 (Russian)

[21] A-S Sznitman Martingales dependant drsquoun parametre une formule drsquoIto [Martingales depend-ing on a parameter an Ito formula] Z Wahrsch Verw Gebiete 60 (1982) no 1 41ndash70(French)

[22] W Wyss The fractional diffusion equation J Math Phys 27 (1986) no 11 2782ndash2785

Mahmoud M El-Borai Faculty of Science Alexandria University Alexandria EgyptE-mail address m m elboraiyahoocom

Khairia El-Said El-Nadi Faculty of Science Alexandria University Alexandria EgyptE-mail address khairia el saidhotmailcom

Osama L Mostafa Faculty of Science Alexandria University Alexandria EgyptE-mail address olmoustafayahoocom

Hamdy M Ahmed Faculty of Science Alexandria University Alexandria EgyptE-mail address hamdy 17egyahoocom

Mahmoud M El-Borai et al 461

Lemma 43 For any 0 lt t le T there exists a constant c gt 0 such that

E(∣∣Jt(X)minus Jt(Y)

∣∣q)le cint t

0E∣∣XsminusYs∣∣q

1

(tminus s)α +1

(tminus s)α2ds (411)

We are now in a position to prove the main result of this section

Theorem 44 Under conditions (H1) (H2) (H3) (H4) and (H5) there exists a uniqueelement X isin ⋂tgt0L

qprog(Ωtimes (0 t)) which solves (31) Moreover if τ is a stopping time

uniqueness holds on the random interval [0τ]

Proof Equation (31) can be rewritten as

Xt = X0 + Jt(X) t ge 0 (412)

Uniqueness Let X Y isin⋂tgt0Lqprog(Ωtimes (0 t)) and let τ be a stopping time such that

Xt = X0 + Jt(X) Yt = X0 + Jt(Y) 0le t le τ (413)

From Lemma 43

E(∣∣Xt minusYt∣∣q)= E(∣∣Jt(X)minus Jt(Y)

∣∣q)le c

int t0E∣∣XsminusYs∣∣q

1

(tminus s)α +1

(tminus s)α2ds

(414)

Set (E|Xt minusYt|q)= γt It is easy to see that

γt le 2cη(t)int t

0

γs(tminus s)α ds (415)

where η(t)=Max(1 tα) and c is a positive constant Thus (see [5 21])

γt le M2(Γ(tminusα)

)2

Γ(2(1minusα)

) int t0

γs(tminus s)2αminus1

ds (416)

and by a classical argument we get

γt le Mn(Γ(tminusα)

)nΓ(n(1minusα)

) int t0

γs(tminus s)nαminus(nminus1)

ds (417)

where M = 2cη(T) Thus for sufficiently large n n gt 1(1minusα) we get from (417)

γt le Kn

Γ(n(1minusα)

) int t0γs ds (418)

where K is a positive constantTaking the limit as nrarrinfin we find that (418) leads to γt = 0To prove the existence we define a sequence Xn

t 0le t le T n= 012 as follows

Xot = Xo Xn+1

t = Xo + Jt(Xn) T le t le 0 (419)

462 Volterra equations with fractional stochastic integrals

Using Lemma 42 we can see that

Xn isin⋂tgt0

Lqprog

(Ωtimes [0T]

) (420)

It then follows from Lemma 43 that

gn+1(t)le 2cη(t)int t

0

gn(s)(tminus s)α ds (421)

where

gn+1(t)= E(∣∣Xn+1t minusXn

t

∣∣q) (422)

By using a similar argument we can write

gn le Kn

Γ(n(1minusα)

) int t0go(s)ds (423)

The last estimation implies that Xn is a Cauchy sequence in Lqprog(Ωtimes [0T]) Then there

exists X such that Xn rarr X in⋂tgt0L

qprog(Ωtimes [0T]) and again using Lemma 43 we can

pass to the limit in (420) yielding that X solves (412)

5 An existence and uniqueness result under weaker assumptions

We formulate a new set of weaker hypotheses

(H1prime) X0 is η0 measurable(H2prime) H isin (L12

loc)pHt is a progressively measurable process which can be localized in(L12)p by a progressively measurable sequence

We assume that there exists an increasing progressively measurable process Ut t ge 0with values in R+ such that

(H3prime) |Ht|+sumk

i=1 |DisHt| leUt as 0le s lt t

Finally we suppose that for any N gt 0 there exists an increasing progressively measur-able process VN

t t ge 0 with values in R+ such that

(H4prime) |F(tsx)|+sumk

i=1 |Gi(h tsx)|+sumk

i=1 |Gprimei (h tsx)| le VNt (1+|x|) for all |h|leN

0le s lt t and x isinRd(H5prime) |F(tsx) minus F(ts y)| +

sumki=1 |Gi(h tsx) minus Gi(h ts y)| +

sumki=1 |Gprimei (h tsx)

minusGprimei (h ts y)| leVNt |xminus y| for all |h| leN 0le s lt t x y isinRd

Let again q be a fixed real number with q gt p and set 1minusβ = αq 0 lt α lt 1 We havethe following theorem

Theorem 51 Equation (31) has a unique solution in the class of progressively measurableprocesses which satisfy

X isin⋂tgt0

Lq(0 t) as (51)

Mahmoud M El-Borai et al 463

Proof (a) We first see how (31) makes sense if X isin⋂tgt0Lq(0 t) as

That is we have to show that for fixed t gt 0

int t0

Gi(h tsXs

)(tminus s)α2q dWi

s hisinRP

(52)

is a well-defined random field which possesses an as continuous versionFor that sake we define

τn = inftint t

0

∣∣Xs∣∣q(tminus s)α dsge n or VN

t ge n (53)

The argument of Lemma 41 can be used to show that

hminusrarrint tandτn

0

Gi(h tsXs

)(tminus s)α2q dWi

s (54)

possesses an as continuous modification on |h| le N Since this is true for any n andN and cupnτn ge t =Ω as the result follows

(b) Existence we want to show existence on an arbitrary interval [0T] (T will be fixedbelow)

Let Hn nisinN denote a progressively measurable localizing sequence forH in (L12)p

on [0T] Since from (H3prime) suptleT |Ht| is as finite we can and do assume without lossof generality that

∣∣Hnt (ω)

∣∣le n forall(tω)isin [0T]timesΩ (55)

Note that Ht(ω) =Hnt (ω) as on ΩT

n for all t isin [0T] where ΩTn uarrΩ as as nrarrinfin We

moreover define Xn0 = X01|X0|len Sn = inft supsltt |DsH

nt |orVn

t ge n We consider theequation

Xnt = Xn

0 +int t

0Fn(tsXn

s

)ds+ b

ksumi=1

int t0

Gni

(h tsXn

s

)(tminus s)α2q dWi

s|h=Hnt (56)

where

Fn(tsx)= 1[0Sn](s)

[aF(tsx)(tminus s)αq minus b

ksumi=1

Gprimei(Hnt tsx

)DisH

nt

(tminus s)α2q]

Gni (h tsx)= 1[0Sn](s)Gi(h tsx)

(57)

It is easy to see that theorem (415) applies to (56)Define

Sn(ω)=Sn(ω)and inf

t le T

int t0

∣∣Hs(ω)minusHns (ω)

∣∣ds gt 0

if∣∣X0(ω)

∣∣ lt n

0 otherwise(58)

464 Volterra equations with fractional stochastic integrals

Sn is a stopping time and it follows from the uniqueness part of Theorem 44 that ifmgt n

Xmt = Xn

t on[0 Sn

]as (59)

Since moreover Sn = T uarrΩ as we can define the process Xt on [0T] by Xt = Xnt

on [0 Sn] for all nisinNClearly X isin Lq(0T) as and solves (31) on [0T] Since T is arbitrary the existence

is proved(c) Uniqueness it suffices to prove uniqueness on an arbitrary interval [0T] Let

Xt t isin [0T] be a progressively measurable process such that X isin Lq(0T) as andX solves (31) It suffices to show that X coincides with the solution we have just con-structed

Let

Sn(ω)= S(ω)and inft le T

int t0

∣∣Xs(ω)∣∣q

(tminus s)α ds gt n

Xnt = XtandSn

(510)

Sn isin Lq(Ωtimes [0T]) and it solves (56) with Sn replaced by SnThen

Xnt (ω)= Xt(ω)timesP ae on

[0 Sn

] (511)

where is the Lebesgue measure on the real lineThe result follows from the fact that Sn ge T uarrΩ asNote that the above solution satisfies in fact X isin⋂qgt1

⋂tgt0L

q(0 t) as

6 Continuity of the solution

We want to give additional conditions under which the solution of (31) is an as contin-uous process

(H6) For all (sx)isinR+timesRd trarr F(tsx)(tminus s)αq is as continuous on (s+infin)(H7) Ht t ge 0 is as continuous(H8) For all iisin 1 k sisinR+ trarrDi

sHt is as continuous on (s+infin)(H9) For all (sx) isin R+ timesRd i isin 1 k (th)rarr Gprimei (h tsx)(tminus s)α2q is as con-

tinuous on (s+infin)timesRP

We also suppose that there exist δ gt 0 gt 0 such that for all N gt 0 |h| le N 0 le s lttand r x isinRd |tminus r| le 1

(H10) There exists an increasing process VNt t ge 0 such that

∣∣Gi(h tsx)minusGi(hrsx)∣∣leVN

t |tminus r|δ(1 + |x|) (61)

Theorem 61 Under conditions (H1prime) (H2prime) (H3prime) (H4prime) (H5prime) and (H6) (H7) (H8)(H9) (H10) the unique solution of (31) (which is progressively measurable and belongs asto⋂qgt1⋂tgt0L

q(0 t)) has as continuous modification

Mahmoud M El-Borai et al 465

Proof We need to show only that whenever X isin⋂qgt1⋂tgt0L

q(0 t) as Jt(X) t gt 0 hasan as continuous modification

(a) We first show that trarr int t0 F(tsXs)ds is as continuous

Note that (H6) (H7) (H8) and (H9) imply that for all (sx)isinR+timesRd trarr F(tsx)is as continuous on (s+infin)

Moreover from (H2prime) (H3prime) (H4prime) and the fact that X isin⋂qgt1⋂tgt0L

q(0 t) as forany T gt 0 there exists a process ZTs sisin [0T] such that

∣∣F(tsXs)∣∣le ZTs ∣∣Gprimei(Ht tsXs

)DisHt

∣∣le ZTs 0le s lt t le T asint t

0ZTs

1

(tminus s)α2q +1

(tminus s)αqds ltinfin as

(62)

Let first tn nisinN be a sequence such that tn lt t for any n and tnrarr t as nrarrinfin then

int t0F(tsXs

)dsminus

int tn0F(tnsXs

)ds=

int ttnF(tsXs

)ds+

int tn0

[F(tsXs

)minus F(tnsXs)]ds

∣∣∣∣int ttnF(tsXs

)ds∣∣∣∣le

int ttnZTs

1

(tminus s)α2q +1

(tminus s)αqds

(63)

and the latter tends as to 0 as nrarrinfin

∣∣∣∣int tn

0

[F(tsXs

)minus F(tnsXs)]ds∣∣∣∣le

int t0

∣∣F(tsXs)minus F(tnsXs)∣∣ds (64)

which tends to 0 as nrarrinfin A similar argument gives the same result when tn gt t tnrarr t(b) We next show that t rarr It(X Ht) possesses an as continuous modification This

follows from (H7) and

(th)minusrarr It(X h) (65)

has an as continuous modificationBy localization it suffices to prove (65) under assumptions (H2) (H3) (H4) (H5)

and (H6) (H7) (H8) (H9) (H10) with VNt (ω) in (H10) replaced by a constant K

and in case X0 isin⋂qgt1L

q(ΩRd)It then suffices to show that under the above hypotheses there exists Cq gt 0 such that

for any hk isinRP the numbers tr will be positive Suppose to fix the ideas that 0le r lt tthen

It(X h)minus Ir(X k)=int tr

Gi(h tsXs

)(tminus s)α2q dWi

s +int r

0

[Gi(h tsXs

)(tminus s)α2q minus Gi

(hrsXs

)(rminus s)α2q

]dWi

s

+int r

0

[Gi(h tsXs

)(rminus s)α2q minus Gi

(krsXs

)(rminus s)α2q

]dWi

s

(66)

466 Volterra equations with fractional stochastic integrals

It follows from the Burkholder-Gundy inequality that

E(∣∣∣∣int tr

Gi(h tsXs

)(tminus s)α2q dWi

s

∣∣∣∣q)le cq

ksumi=1

E

[(int tr

∣∣Gi(h tsXs

)∣∣2

(tminus s)αq ds)q2]

le cq(tminus r)(qminus2)2ksumi=1

Eint tr

∣∣Gi(h tsXs

)∣∣q(tminus s)α2 ds

le cq(tminus r)(qminus2)2int tr

E(1 +

∣∣Xs∣∣q)(tminus s)α2 ds

le Cq(tminus r)(qminus2)2(tminus r)1minusα2

(67)

From (H4) for Gi we deduce as in Lemma 41 that

E

(∣∣∣∣int r

0

[Gi(hrsXs

)minusGi(krsXs

)(rminus s)α2q

]dWi

s

∣∣∣∣q)

le cq(hminus k)qint r

0

E(1 +

∣∣Xs∣∣q)(rminus s)α2 dsle Cq(hminus k)qr1minusα2

(68)

From (H10) and the fact that

X isin⋂qgt1

⋂tgt0

Lq(Ωtimes (0 t)

)

E

(∣∣∣∣int r

0

[Gi(h tsXs

)(tminus s)α2q minus Gi

(hrsXs

)(rminus s)α2q

]dWi

s

∣∣∣∣q)

= E(∣∣∣∣int r

0

(1

(rminus s)α2q[Gi(h tsXs

)minusGi(hrsXs

)]

+Gi(h tsXs

)[ 1(tminus s)α2q minus

1(rminus s)α2q

])dWi

s

∣∣∣∣q)

le E(cq

int r0

(∣∣Gi(h tsXs

)minusGi(hrsXs

)∣∣q(rminus s)α

+∣∣Gi

(h tsXs

)∣∣q((rminus s)minusα2qminus (tminus s)minusα2q)q)ds

)

le cqKq(int r

0

( |tminus r|δq(rminus s)α E

(1 +

∣∣Xs∣∣q))ds

+int r

0

(E(1 +

∣∣Xs∣∣q)((rminus s)minusα2minus (tminus s)minusα2))ds)

le cq[|tminus r|δq

int r0

(rminus s)minusα2ds+int r

0

((rminus s)minusα2minus (tminus s)minusα2)ds]

le Cq(|tminus r|δqr1minusα2 +

(r1minusα2 + (tminus r)1minusα2minus t1minusα2))

(69)

Mahmoud M El-Borai et al 467

which from the above estimate yields

E(∣∣It(X h)minus Ir(X k)

∣∣q)le Cq

[|tminus r|1minusα2 + |tminus r|(qminusα)2

+ r1minusα2(|tminus r|δq + |hminus k|q)+(r1minusα2minus t1minusα2)]

(610)

this completes the proof

Acknowledgment

We are very grateful to the referees for valuable comments

References

[1] C Bender and R J Elliott Binary market models and discrete wick products preprint 2002[2] M A Berger and V J Mizel Volterra equations with Ito integrals I J Integral Equations 2

(1980) no 3 187ndash245[3] Volterra equations with Ito integrals II J Integral Equations 2 (1980) no 4 319ndash337[4] An extension of the stochastic integral Ann Probab 10 (1982) no 2 435ndash450[5] M M El-Borai Some probability densities and fundamental solutions of fractional evolution

equations Chaos Solitons Fractals 14 (2002) no 3 433ndash440[6] A M Kolodh On the existence solutions of stochastic Volterra equations Theory of Random

Processes 11 (1983) 51ndash57 (Russian)[7] B B Mandelbrot and J W Van Ness Fractional Brownian motions fractional noises and appli-

cations SIAM Rev 10 (1968) 422ndash437[8] P-A Meyer Flot drsquoune equation differentielle stochastique (drsquoapres Malliavin Bismut Kunita)

[Flow of a stochastic differential equation (following Malliavin Bismut Kunita)] Seminar onProbability XV (University of Strasbourg Strasbourg 19791980) Lecture Notes in Mathvol 850 Springer-Verlag Berlin 1981 pp 103ndash117 (French)

[9] D Nualart Noncausal stochastic integrals and calculus Stochastic Analysis and Related Topics(Silivri 1986) Lecture Notes in Math vol 1316 Springer-Verlag Berlin 1988 pp 80ndash129

[10] D Nualart and E Pardoux Stochastic calculus with anticipating integrands Probab TheoryRelated Fields 78 (1988) no 4 535ndash581

[11] D Nualart and M Zakai Generalized stochastic integrals and the Malliavin calculus ProbabTheory Related Fields 73 (1986) no 2 255ndash280

[12] D Ocone and E Pardoux A generalized Ito-Ventzell formula Application to a class of anticipat-ing stochastic differential equations Ann Inst H Poincare Probab Statist 25 (1989) no 139ndash71

[13] Linear stochastic differential equations with boundary conditions Probab Theory Re-lated Fields 82 (1989) no 4 489ndash526

[14] S Ogawa Sur la question drsquoexistence de solutions drsquoune equation differentielle stochastique dutype noncausal [On the existence of solutions of a stochastic differential equation of noncausaltype] J Math Kyoto Univ 24 (1984) no 4 699ndash704 (French)

[15] E Pardoux and P Protter A two-sided stochastic integral and its calculus Probab Theory RelatedFields 76 (1987) no 1 15ndash49

[16] Stochastic Volterra equations with anticipating coefficients Ann Probab 18 (1990)no 4 1635ndash1655

[17] P Protter Volterra equations driven by semimartingales Ann Probab 13 (1985) no 2 519ndash530

468 Volterra equations with fractional stochastic integrals

[18] A N V Rao and C P Tsokos On the existence uniqueness and stability behavior of a randomsolution to a nonlinear perturbed stochastic integro-differential equation Information andControl 27 (1975) 61ndash74

[19] Y Shiota A linear stochastic integral equation containing the extended Ito integral Math RepToyama Univ 9 (1986) 43ndash65

[20] A V Skorohod On a generalization of the stochastic integral Teor Verojatnost i Primenen 20(1975) no 2 223ndash238 (Russian)

[21] A-S Sznitman Martingales dependant drsquoun parametre une formule drsquoIto [Martingales depend-ing on a parameter an Ito formula] Z Wahrsch Verw Gebiete 60 (1982) no 1 41ndash70(French)

[22] W Wyss The fractional diffusion equation J Math Phys 27 (1986) no 11 2782ndash2785

Mahmoud M El-Borai Faculty of Science Alexandria University Alexandria EgyptE-mail address m m elboraiyahoocom

Khairia El-Said El-Nadi Faculty of Science Alexandria University Alexandria EgyptE-mail address khairia el saidhotmailcom

Osama L Mostafa Faculty of Science Alexandria University Alexandria EgyptE-mail address olmoustafayahoocom

Hamdy M Ahmed Faculty of Science Alexandria University Alexandria EgyptE-mail address hamdy 17egyahoocom

462 Volterra equations with fractional stochastic integrals

Using Lemma 42 we can see that

Xn isin⋂tgt0

Lqprog

(Ωtimes [0T]

) (420)

It then follows from Lemma 43 that

gn+1(t)le 2cη(t)int t

0

gn(s)(tminus s)α ds (421)

where

gn+1(t)= E(∣∣Xn+1t minusXn

t

∣∣q) (422)

By using a similar argument we can write

gn le Kn

Γ(n(1minusα)

) int t0go(s)ds (423)

The last estimation implies that Xn is a Cauchy sequence in Lqprog(Ωtimes [0T]) Then there

exists X such that Xn rarr X in⋂tgt0L

qprog(Ωtimes [0T]) and again using Lemma 43 we can

pass to the limit in (420) yielding that X solves (412)

5 An existence and uniqueness result under weaker assumptions

We formulate a new set of weaker hypotheses

(H1prime) X0 is η0 measurable(H2prime) H isin (L12

loc)pHt is a progressively measurable process which can be localized in(L12)p by a progressively measurable sequence

We assume that there exists an increasing progressively measurable process Ut t ge 0with values in R+ such that

(H3prime) |Ht|+sumk

i=1 |DisHt| leUt as 0le s lt t

Finally we suppose that for any N gt 0 there exists an increasing progressively measur-able process VN

t t ge 0 with values in R+ such that

(H4prime) |F(tsx)|+sumk

i=1 |Gi(h tsx)|+sumk

i=1 |Gprimei (h tsx)| le VNt (1+|x|) for all |h|leN

0le s lt t and x isinRd(H5prime) |F(tsx) minus F(ts y)| +

sumki=1 |Gi(h tsx) minus Gi(h ts y)| +

sumki=1 |Gprimei (h tsx)

minusGprimei (h ts y)| leVNt |xminus y| for all |h| leN 0le s lt t x y isinRd

Let again q be a fixed real number with q gt p and set 1minusβ = αq 0 lt α lt 1 We havethe following theorem

Theorem 51 Equation (31) has a unique solution in the class of progressively measurableprocesses which satisfy

X isin⋂tgt0

Lq(0 t) as (51)

Mahmoud M El-Borai et al 463

Proof (a) We first see how (31) makes sense if X isin⋂tgt0Lq(0 t) as

That is we have to show that for fixed t gt 0

int t0

Gi(h tsXs

)(tminus s)α2q dWi

s hisinRP

(52)

is a well-defined random field which possesses an as continuous versionFor that sake we define

τn = inftint t

0

∣∣Xs∣∣q(tminus s)α dsge n or VN

t ge n (53)

The argument of Lemma 41 can be used to show that

hminusrarrint tandτn

0

Gi(h tsXs

)(tminus s)α2q dWi

s (54)

possesses an as continuous modification on |h| le N Since this is true for any n andN and cupnτn ge t =Ω as the result follows

(b) Existence we want to show existence on an arbitrary interval [0T] (T will be fixedbelow)

Let Hn nisinN denote a progressively measurable localizing sequence forH in (L12)p

on [0T] Since from (H3prime) suptleT |Ht| is as finite we can and do assume without lossof generality that

∣∣Hnt (ω)

∣∣le n forall(tω)isin [0T]timesΩ (55)

Note that Ht(ω) =Hnt (ω) as on ΩT

n for all t isin [0T] where ΩTn uarrΩ as as nrarrinfin We

moreover define Xn0 = X01|X0|len Sn = inft supsltt |DsH

nt |orVn

t ge n We consider theequation

Xnt = Xn

0 +int t

0Fn(tsXn

s

)ds+ b

ksumi=1

int t0

Gni

(h tsXn

s

)(tminus s)α2q dWi

s|h=Hnt (56)

where

Fn(tsx)= 1[0Sn](s)

[aF(tsx)(tminus s)αq minus b

ksumi=1

Gprimei(Hnt tsx

)DisH

nt

(tminus s)α2q]

Gni (h tsx)= 1[0Sn](s)Gi(h tsx)

(57)

It is easy to see that theorem (415) applies to (56)Define

Sn(ω)=Sn(ω)and inf

t le T

int t0

∣∣Hs(ω)minusHns (ω)

∣∣ds gt 0

if∣∣X0(ω)

∣∣ lt n

0 otherwise(58)

464 Volterra equations with fractional stochastic integrals

Sn is a stopping time and it follows from the uniqueness part of Theorem 44 that ifmgt n

Xmt = Xn

t on[0 Sn

]as (59)

Since moreover Sn = T uarrΩ as we can define the process Xt on [0T] by Xt = Xnt

on [0 Sn] for all nisinNClearly X isin Lq(0T) as and solves (31) on [0T] Since T is arbitrary the existence

is proved(c) Uniqueness it suffices to prove uniqueness on an arbitrary interval [0T] Let

Xt t isin [0T] be a progressively measurable process such that X isin Lq(0T) as andX solves (31) It suffices to show that X coincides with the solution we have just con-structed

Let

Sn(ω)= S(ω)and inft le T

int t0

∣∣Xs(ω)∣∣q

(tminus s)α ds gt n

Xnt = XtandSn

(510)

Sn isin Lq(Ωtimes [0T]) and it solves (56) with Sn replaced by SnThen

Xnt (ω)= Xt(ω)timesP ae on

[0 Sn

] (511)

where is the Lebesgue measure on the real lineThe result follows from the fact that Sn ge T uarrΩ asNote that the above solution satisfies in fact X isin⋂qgt1

⋂tgt0L

q(0 t) as

6 Continuity of the solution

We want to give additional conditions under which the solution of (31) is an as contin-uous process

(H6) For all (sx)isinR+timesRd trarr F(tsx)(tminus s)αq is as continuous on (s+infin)(H7) Ht t ge 0 is as continuous(H8) For all iisin 1 k sisinR+ trarrDi

sHt is as continuous on (s+infin)(H9) For all (sx) isin R+ timesRd i isin 1 k (th)rarr Gprimei (h tsx)(tminus s)α2q is as con-

tinuous on (s+infin)timesRP

We also suppose that there exist δ gt 0 gt 0 such that for all N gt 0 |h| le N 0 le s lttand r x isinRd |tminus r| le 1

(H10) There exists an increasing process VNt t ge 0 such that

∣∣Gi(h tsx)minusGi(hrsx)∣∣leVN

t |tminus r|δ(1 + |x|) (61)

Theorem 61 Under conditions (H1prime) (H2prime) (H3prime) (H4prime) (H5prime) and (H6) (H7) (H8)(H9) (H10) the unique solution of (31) (which is progressively measurable and belongs asto⋂qgt1⋂tgt0L

q(0 t)) has as continuous modification

Mahmoud M El-Borai et al 465

Proof We need to show only that whenever X isin⋂qgt1⋂tgt0L

q(0 t) as Jt(X) t gt 0 hasan as continuous modification

(a) We first show that trarr int t0 F(tsXs)ds is as continuous

Note that (H6) (H7) (H8) and (H9) imply that for all (sx)isinR+timesRd trarr F(tsx)is as continuous on (s+infin)

Moreover from (H2prime) (H3prime) (H4prime) and the fact that X isin⋂qgt1⋂tgt0L

q(0 t) as forany T gt 0 there exists a process ZTs sisin [0T] such that

∣∣F(tsXs)∣∣le ZTs ∣∣Gprimei(Ht tsXs

)DisHt

∣∣le ZTs 0le s lt t le T asint t

0ZTs

1

(tminus s)α2q +1

(tminus s)αqds ltinfin as

(62)

Let first tn nisinN be a sequence such that tn lt t for any n and tnrarr t as nrarrinfin then

int t0F(tsXs

)dsminus

int tn0F(tnsXs

)ds=

int ttnF(tsXs

)ds+

int tn0

[F(tsXs

)minus F(tnsXs)]ds

∣∣∣∣int ttnF(tsXs

)ds∣∣∣∣le

int ttnZTs

1

(tminus s)α2q +1

(tminus s)αqds

(63)

and the latter tends as to 0 as nrarrinfin

∣∣∣∣int tn

0

[F(tsXs

)minus F(tnsXs)]ds∣∣∣∣le

int t0

∣∣F(tsXs)minus F(tnsXs)∣∣ds (64)

which tends to 0 as nrarrinfin A similar argument gives the same result when tn gt t tnrarr t(b) We next show that t rarr It(X Ht) possesses an as continuous modification This

follows from (H7) and

(th)minusrarr It(X h) (65)

has an as continuous modificationBy localization it suffices to prove (65) under assumptions (H2) (H3) (H4) (H5)

and (H6) (H7) (H8) (H9) (H10) with VNt (ω) in (H10) replaced by a constant K

and in case X0 isin⋂qgt1L

q(ΩRd)It then suffices to show that under the above hypotheses there exists Cq gt 0 such that

for any hk isinRP the numbers tr will be positive Suppose to fix the ideas that 0le r lt tthen

It(X h)minus Ir(X k)=int tr

Gi(h tsXs

)(tminus s)α2q dWi

s +int r

0

[Gi(h tsXs

)(tminus s)α2q minus Gi

(hrsXs

)(rminus s)α2q

]dWi

s

+int r

0

[Gi(h tsXs

)(rminus s)α2q minus Gi

(krsXs

)(rminus s)α2q

]dWi

s

(66)

466 Volterra equations with fractional stochastic integrals

It follows from the Burkholder-Gundy inequality that

E(∣∣∣∣int tr

Gi(h tsXs

)(tminus s)α2q dWi

s

∣∣∣∣q)le cq

ksumi=1

E

[(int tr

∣∣Gi(h tsXs

)∣∣2

(tminus s)αq ds)q2]

le cq(tminus r)(qminus2)2ksumi=1

Eint tr

∣∣Gi(h tsXs

)∣∣q(tminus s)α2 ds

le cq(tminus r)(qminus2)2int tr

E(1 +

∣∣Xs∣∣q)(tminus s)α2 ds

le Cq(tminus r)(qminus2)2(tminus r)1minusα2

(67)

From (H4) for Gi we deduce as in Lemma 41 that

E

(∣∣∣∣int r

0

[Gi(hrsXs

)minusGi(krsXs

)(rminus s)α2q

]dWi

s

∣∣∣∣q)

le cq(hminus k)qint r

0

E(1 +

∣∣Xs∣∣q)(rminus s)α2 dsle Cq(hminus k)qr1minusα2

(68)

From (H10) and the fact that

X isin⋂qgt1

⋂tgt0

Lq(Ωtimes (0 t)

)

E

(∣∣∣∣int r

0

[Gi(h tsXs

)(tminus s)α2q minus Gi

(hrsXs

)(rminus s)α2q

]dWi

s

∣∣∣∣q)

= E(∣∣∣∣int r

0

(1

(rminus s)α2q[Gi(h tsXs

)minusGi(hrsXs

)]

+Gi(h tsXs

)[ 1(tminus s)α2q minus

1(rminus s)α2q

])dWi

s

∣∣∣∣q)

le E(cq

int r0

(∣∣Gi(h tsXs

)minusGi(hrsXs

)∣∣q(rminus s)α

+∣∣Gi

(h tsXs

)∣∣q((rminus s)minusα2qminus (tminus s)minusα2q)q)ds

)

le cqKq(int r

0

( |tminus r|δq(rminus s)α E

(1 +

∣∣Xs∣∣q))ds

+int r

0

(E(1 +

∣∣Xs∣∣q)((rminus s)minusα2minus (tminus s)minusα2))ds)

le cq[|tminus r|δq

int r0

(rminus s)minusα2ds+int r

0

((rminus s)minusα2minus (tminus s)minusα2)ds]

le Cq(|tminus r|δqr1minusα2 +

(r1minusα2 + (tminus r)1minusα2minus t1minusα2))

(69)

Mahmoud M El-Borai et al 467

which from the above estimate yields

E(∣∣It(X h)minus Ir(X k)

∣∣q)le Cq

[|tminus r|1minusα2 + |tminus r|(qminusα)2

+ r1minusα2(|tminus r|δq + |hminus k|q)+(r1minusα2minus t1minusα2)]

(610)

this completes the proof

Acknowledgment

We are very grateful to the referees for valuable comments

References

[1] C Bender and R J Elliott Binary market models and discrete wick products preprint 2002[2] M A Berger and V J Mizel Volterra equations with Ito integrals I J Integral Equations 2

(1980) no 3 187ndash245[3] Volterra equations with Ito integrals II J Integral Equations 2 (1980) no 4 319ndash337[4] An extension of the stochastic integral Ann Probab 10 (1982) no 2 435ndash450[5] M M El-Borai Some probability densities and fundamental solutions of fractional evolution

equations Chaos Solitons Fractals 14 (2002) no 3 433ndash440[6] A M Kolodh On the existence solutions of stochastic Volterra equations Theory of Random

Processes 11 (1983) 51ndash57 (Russian)[7] B B Mandelbrot and J W Van Ness Fractional Brownian motions fractional noises and appli-

cations SIAM Rev 10 (1968) 422ndash437[8] P-A Meyer Flot drsquoune equation differentielle stochastique (drsquoapres Malliavin Bismut Kunita)

[Flow of a stochastic differential equation (following Malliavin Bismut Kunita)] Seminar onProbability XV (University of Strasbourg Strasbourg 19791980) Lecture Notes in Mathvol 850 Springer-Verlag Berlin 1981 pp 103ndash117 (French)

[9] D Nualart Noncausal stochastic integrals and calculus Stochastic Analysis and Related Topics(Silivri 1986) Lecture Notes in Math vol 1316 Springer-Verlag Berlin 1988 pp 80ndash129

[10] D Nualart and E Pardoux Stochastic calculus with anticipating integrands Probab TheoryRelated Fields 78 (1988) no 4 535ndash581

[11] D Nualart and M Zakai Generalized stochastic integrals and the Malliavin calculus ProbabTheory Related Fields 73 (1986) no 2 255ndash280

[12] D Ocone and E Pardoux A generalized Ito-Ventzell formula Application to a class of anticipat-ing stochastic differential equations Ann Inst H Poincare Probab Statist 25 (1989) no 139ndash71

[13] Linear stochastic differential equations with boundary conditions Probab Theory Re-lated Fields 82 (1989) no 4 489ndash526

[14] S Ogawa Sur la question drsquoexistence de solutions drsquoune equation differentielle stochastique dutype noncausal [On the existence of solutions of a stochastic differential equation of noncausaltype] J Math Kyoto Univ 24 (1984) no 4 699ndash704 (French)

[15] E Pardoux and P Protter A two-sided stochastic integral and its calculus Probab Theory RelatedFields 76 (1987) no 1 15ndash49

[16] Stochastic Volterra equations with anticipating coefficients Ann Probab 18 (1990)no 4 1635ndash1655

[17] P Protter Volterra equations driven by semimartingales Ann Probab 13 (1985) no 2 519ndash530

468 Volterra equations with fractional stochastic integrals

[18] A N V Rao and C P Tsokos On the existence uniqueness and stability behavior of a randomsolution to a nonlinear perturbed stochastic integro-differential equation Information andControl 27 (1975) 61ndash74

[19] Y Shiota A linear stochastic integral equation containing the extended Ito integral Math RepToyama Univ 9 (1986) 43ndash65

[20] A V Skorohod On a generalization of the stochastic integral Teor Verojatnost i Primenen 20(1975) no 2 223ndash238 (Russian)

[21] A-S Sznitman Martingales dependant drsquoun parametre une formule drsquoIto [Martingales depend-ing on a parameter an Ito formula] Z Wahrsch Verw Gebiete 60 (1982) no 1 41ndash70(French)

[22] W Wyss The fractional diffusion equation J Math Phys 27 (1986) no 11 2782ndash2785

Mahmoud M El-Borai Faculty of Science Alexandria University Alexandria EgyptE-mail address m m elboraiyahoocom

Khairia El-Said El-Nadi Faculty of Science Alexandria University Alexandria EgyptE-mail address khairia el saidhotmailcom

Osama L Mostafa Faculty of Science Alexandria University Alexandria EgyptE-mail address olmoustafayahoocom

Hamdy M Ahmed Faculty of Science Alexandria University Alexandria EgyptE-mail address hamdy 17egyahoocom

Mahmoud M El-Borai et al 463

Proof (a) We first see how (31) makes sense if X isin⋂tgt0Lq(0 t) as

That is we have to show that for fixed t gt 0

int t0

Gi(h tsXs

)(tminus s)α2q dWi

s hisinRP

(52)

is a well-defined random field which possesses an as continuous versionFor that sake we define

τn = inftint t

0

∣∣Xs∣∣q(tminus s)α dsge n or VN

t ge n (53)

The argument of Lemma 41 can be used to show that

hminusrarrint tandτn

0

Gi(h tsXs

)(tminus s)α2q dWi

s (54)

possesses an as continuous modification on |h| le N Since this is true for any n andN and cupnτn ge t =Ω as the result follows

(b) Existence we want to show existence on an arbitrary interval [0T] (T will be fixedbelow)

Let Hn nisinN denote a progressively measurable localizing sequence forH in (L12)p

on [0T] Since from (H3prime) suptleT |Ht| is as finite we can and do assume without lossof generality that

∣∣Hnt (ω)

∣∣le n forall(tω)isin [0T]timesΩ (55)

Note that Ht(ω) =Hnt (ω) as on ΩT

n for all t isin [0T] where ΩTn uarrΩ as as nrarrinfin We

moreover define Xn0 = X01|X0|len Sn = inft supsltt |DsH

nt |orVn

t ge n We consider theequation

Xnt = Xn

0 +int t

0Fn(tsXn

s

)ds+ b

ksumi=1

int t0

Gni

(h tsXn

s

)(tminus s)α2q dWi

s|h=Hnt (56)

where

Fn(tsx)= 1[0Sn](s)

[aF(tsx)(tminus s)αq minus b

ksumi=1

Gprimei(Hnt tsx

)DisH

nt

(tminus s)α2q]

Gni (h tsx)= 1[0Sn](s)Gi(h tsx)

(57)

It is easy to see that theorem (415) applies to (56)Define

Sn(ω)=Sn(ω)and inf

t le T

int t0

∣∣Hs(ω)minusHns (ω)

∣∣ds gt 0

if∣∣X0(ω)

∣∣ lt n

0 otherwise(58)

464 Volterra equations with fractional stochastic integrals

Sn is a stopping time and it follows from the uniqueness part of Theorem 44 that ifmgt n

Xmt = Xn

t on[0 Sn

]as (59)

Since moreover Sn = T uarrΩ as we can define the process Xt on [0T] by Xt = Xnt

on [0 Sn] for all nisinNClearly X isin Lq(0T) as and solves (31) on [0T] Since T is arbitrary the existence

is proved(c) Uniqueness it suffices to prove uniqueness on an arbitrary interval [0T] Let

Xt t isin [0T] be a progressively measurable process such that X isin Lq(0T) as andX solves (31) It suffices to show that X coincides with the solution we have just con-structed

Let

Sn(ω)= S(ω)and inft le T

int t0

∣∣Xs(ω)∣∣q

(tminus s)α ds gt n

Xnt = XtandSn

(510)

Sn isin Lq(Ωtimes [0T]) and it solves (56) with Sn replaced by SnThen

Xnt (ω)= Xt(ω)timesP ae on

[0 Sn

] (511)

where is the Lebesgue measure on the real lineThe result follows from the fact that Sn ge T uarrΩ asNote that the above solution satisfies in fact X isin⋂qgt1

⋂tgt0L

q(0 t) as

6 Continuity of the solution

We want to give additional conditions under which the solution of (31) is an as contin-uous process

(H6) For all (sx)isinR+timesRd trarr F(tsx)(tminus s)αq is as continuous on (s+infin)(H7) Ht t ge 0 is as continuous(H8) For all iisin 1 k sisinR+ trarrDi

sHt is as continuous on (s+infin)(H9) For all (sx) isin R+ timesRd i isin 1 k (th)rarr Gprimei (h tsx)(tminus s)α2q is as con-

tinuous on (s+infin)timesRP

We also suppose that there exist δ gt 0 gt 0 such that for all N gt 0 |h| le N 0 le s lttand r x isinRd |tminus r| le 1

(H10) There exists an increasing process VNt t ge 0 such that

∣∣Gi(h tsx)minusGi(hrsx)∣∣leVN

t |tminus r|δ(1 + |x|) (61)

Theorem 61 Under conditions (H1prime) (H2prime) (H3prime) (H4prime) (H5prime) and (H6) (H7) (H8)(H9) (H10) the unique solution of (31) (which is progressively measurable and belongs asto⋂qgt1⋂tgt0L

q(0 t)) has as continuous modification

Mahmoud M El-Borai et al 465

Proof We need to show only that whenever X isin⋂qgt1⋂tgt0L

q(0 t) as Jt(X) t gt 0 hasan as continuous modification

(a) We first show that trarr int t0 F(tsXs)ds is as continuous

Note that (H6) (H7) (H8) and (H9) imply that for all (sx)isinR+timesRd trarr F(tsx)is as continuous on (s+infin)

Moreover from (H2prime) (H3prime) (H4prime) and the fact that X isin⋂qgt1⋂tgt0L

q(0 t) as forany T gt 0 there exists a process ZTs sisin [0T] such that

∣∣F(tsXs)∣∣le ZTs ∣∣Gprimei(Ht tsXs

)DisHt

∣∣le ZTs 0le s lt t le T asint t

0ZTs

1

(tminus s)α2q +1

(tminus s)αqds ltinfin as

(62)

Let first tn nisinN be a sequence such that tn lt t for any n and tnrarr t as nrarrinfin then

int t0F(tsXs

)dsminus

int tn0F(tnsXs

)ds=

int ttnF(tsXs

)ds+

int tn0

[F(tsXs

)minus F(tnsXs)]ds

∣∣∣∣int ttnF(tsXs

)ds∣∣∣∣le

int ttnZTs

1

(tminus s)α2q +1

(tminus s)αqds

(63)

and the latter tends as to 0 as nrarrinfin

∣∣∣∣int tn

0

[F(tsXs

)minus F(tnsXs)]ds∣∣∣∣le

int t0

∣∣F(tsXs)minus F(tnsXs)∣∣ds (64)

which tends to 0 as nrarrinfin A similar argument gives the same result when tn gt t tnrarr t(b) We next show that t rarr It(X Ht) possesses an as continuous modification This

follows from (H7) and

(th)minusrarr It(X h) (65)

has an as continuous modificationBy localization it suffices to prove (65) under assumptions (H2) (H3) (H4) (H5)

and (H6) (H7) (H8) (H9) (H10) with VNt (ω) in (H10) replaced by a constant K

and in case X0 isin⋂qgt1L

q(ΩRd)It then suffices to show that under the above hypotheses there exists Cq gt 0 such that

for any hk isinRP the numbers tr will be positive Suppose to fix the ideas that 0le r lt tthen

It(X h)minus Ir(X k)=int tr

Gi(h tsXs

)(tminus s)α2q dWi

s +int r

0

[Gi(h tsXs

)(tminus s)α2q minus Gi

(hrsXs

)(rminus s)α2q

]dWi

s

+int r

0

[Gi(h tsXs

)(rminus s)α2q minus Gi

(krsXs

)(rminus s)α2q

]dWi

s

(66)

466 Volterra equations with fractional stochastic integrals

It follows from the Burkholder-Gundy inequality that

E(∣∣∣∣int tr

Gi(h tsXs

)(tminus s)α2q dWi

s

∣∣∣∣q)le cq

ksumi=1

E

[(int tr

∣∣Gi(h tsXs

)∣∣2

(tminus s)αq ds)q2]

le cq(tminus r)(qminus2)2ksumi=1

Eint tr

∣∣Gi(h tsXs

)∣∣q(tminus s)α2 ds

le cq(tminus r)(qminus2)2int tr

E(1 +

∣∣Xs∣∣q)(tminus s)α2 ds

le Cq(tminus r)(qminus2)2(tminus r)1minusα2

(67)

From (H4) for Gi we deduce as in Lemma 41 that

E

(∣∣∣∣int r

0

[Gi(hrsXs

)minusGi(krsXs

)(rminus s)α2q

]dWi

s

∣∣∣∣q)

le cq(hminus k)qint r

0

E(1 +

∣∣Xs∣∣q)(rminus s)α2 dsle Cq(hminus k)qr1minusα2

(68)

From (H10) and the fact that

X isin⋂qgt1

⋂tgt0

Lq(Ωtimes (0 t)

)

E

(∣∣∣∣int r

0

[Gi(h tsXs

)(tminus s)α2q minus Gi

(hrsXs

)(rminus s)α2q

]dWi

s

∣∣∣∣q)

= E(∣∣∣∣int r

0

(1

(rminus s)α2q[Gi(h tsXs

)minusGi(hrsXs

)]

+Gi(h tsXs

)[ 1(tminus s)α2q minus

1(rminus s)α2q

])dWi

s

∣∣∣∣q)

le E(cq

int r0

(∣∣Gi(h tsXs

)minusGi(hrsXs

)∣∣q(rminus s)α

+∣∣Gi

(h tsXs

)∣∣q((rminus s)minusα2qminus (tminus s)minusα2q)q)ds

)

le cqKq(int r

0

( |tminus r|δq(rminus s)α E

(1 +

∣∣Xs∣∣q))ds

+int r

0

(E(1 +

∣∣Xs∣∣q)((rminus s)minusα2minus (tminus s)minusα2))ds)

le cq[|tminus r|δq

int r0

(rminus s)minusα2ds+int r

0

((rminus s)minusα2minus (tminus s)minusα2)ds]

le Cq(|tminus r|δqr1minusα2 +

(r1minusα2 + (tminus r)1minusα2minus t1minusα2))

(69)

Mahmoud M El-Borai et al 467

which from the above estimate yields

E(∣∣It(X h)minus Ir(X k)

∣∣q)le Cq

[|tminus r|1minusα2 + |tminus r|(qminusα)2

+ r1minusα2(|tminus r|δq + |hminus k|q)+(r1minusα2minus t1minusα2)]

(610)

this completes the proof

Acknowledgment

We are very grateful to the referees for valuable comments

References

[1] C Bender and R J Elliott Binary market models and discrete wick products preprint 2002[2] M A Berger and V J Mizel Volterra equations with Ito integrals I J Integral Equations 2

(1980) no 3 187ndash245[3] Volterra equations with Ito integrals II J Integral Equations 2 (1980) no 4 319ndash337[4] An extension of the stochastic integral Ann Probab 10 (1982) no 2 435ndash450[5] M M El-Borai Some probability densities and fundamental solutions of fractional evolution

equations Chaos Solitons Fractals 14 (2002) no 3 433ndash440[6] A M Kolodh On the existence solutions of stochastic Volterra equations Theory of Random

Processes 11 (1983) 51ndash57 (Russian)[7] B B Mandelbrot and J W Van Ness Fractional Brownian motions fractional noises and appli-

cations SIAM Rev 10 (1968) 422ndash437[8] P-A Meyer Flot drsquoune equation differentielle stochastique (drsquoapres Malliavin Bismut Kunita)

[Flow of a stochastic differential equation (following Malliavin Bismut Kunita)] Seminar onProbability XV (University of Strasbourg Strasbourg 19791980) Lecture Notes in Mathvol 850 Springer-Verlag Berlin 1981 pp 103ndash117 (French)

[9] D Nualart Noncausal stochastic integrals and calculus Stochastic Analysis and Related Topics(Silivri 1986) Lecture Notes in Math vol 1316 Springer-Verlag Berlin 1988 pp 80ndash129

[10] D Nualart and E Pardoux Stochastic calculus with anticipating integrands Probab TheoryRelated Fields 78 (1988) no 4 535ndash581

[11] D Nualart and M Zakai Generalized stochastic integrals and the Malliavin calculus ProbabTheory Related Fields 73 (1986) no 2 255ndash280

[12] D Ocone and E Pardoux A generalized Ito-Ventzell formula Application to a class of anticipat-ing stochastic differential equations Ann Inst H Poincare Probab Statist 25 (1989) no 139ndash71

[13] Linear stochastic differential equations with boundary conditions Probab Theory Re-lated Fields 82 (1989) no 4 489ndash526

[14] S Ogawa Sur la question drsquoexistence de solutions drsquoune equation differentielle stochastique dutype noncausal [On the existence of solutions of a stochastic differential equation of noncausaltype] J Math Kyoto Univ 24 (1984) no 4 699ndash704 (French)

[15] E Pardoux and P Protter A two-sided stochastic integral and its calculus Probab Theory RelatedFields 76 (1987) no 1 15ndash49

[16] Stochastic Volterra equations with anticipating coefficients Ann Probab 18 (1990)no 4 1635ndash1655

[17] P Protter Volterra equations driven by semimartingales Ann Probab 13 (1985) no 2 519ndash530

468 Volterra equations with fractional stochastic integrals

[18] A N V Rao and C P Tsokos On the existence uniqueness and stability behavior of a randomsolution to a nonlinear perturbed stochastic integro-differential equation Information andControl 27 (1975) 61ndash74

[19] Y Shiota A linear stochastic integral equation containing the extended Ito integral Math RepToyama Univ 9 (1986) 43ndash65

[20] A V Skorohod On a generalization of the stochastic integral Teor Verojatnost i Primenen 20(1975) no 2 223ndash238 (Russian)

[21] A-S Sznitman Martingales dependant drsquoun parametre une formule drsquoIto [Martingales depend-ing on a parameter an Ito formula] Z Wahrsch Verw Gebiete 60 (1982) no 1 41ndash70(French)

[22] W Wyss The fractional diffusion equation J Math Phys 27 (1986) no 11 2782ndash2785

Mahmoud M El-Borai Faculty of Science Alexandria University Alexandria EgyptE-mail address m m elboraiyahoocom

Khairia El-Said El-Nadi Faculty of Science Alexandria University Alexandria EgyptE-mail address khairia el saidhotmailcom

Osama L Mostafa Faculty of Science Alexandria University Alexandria EgyptE-mail address olmoustafayahoocom

Hamdy M Ahmed Faculty of Science Alexandria University Alexandria EgyptE-mail address hamdy 17egyahoocom

464 Volterra equations with fractional stochastic integrals

Sn is a stopping time and it follows from the uniqueness part of Theorem 44 that ifmgt n

Xmt = Xn

t on[0 Sn

]as (59)

Since moreover Sn = T uarrΩ as we can define the process Xt on [0T] by Xt = Xnt

on [0 Sn] for all nisinNClearly X isin Lq(0T) as and solves (31) on [0T] Since T is arbitrary the existence

is proved(c) Uniqueness it suffices to prove uniqueness on an arbitrary interval [0T] Let

Xt t isin [0T] be a progressively measurable process such that X isin Lq(0T) as andX solves (31) It suffices to show that X coincides with the solution we have just con-structed

Let

Sn(ω)= S(ω)and inft le T

int t0

∣∣Xs(ω)∣∣q

(tminus s)α ds gt n

Xnt = XtandSn

(510)

Sn isin Lq(Ωtimes [0T]) and it solves (56) with Sn replaced by SnThen

Xnt (ω)= Xt(ω)timesP ae on

[0 Sn

] (511)

where is the Lebesgue measure on the real lineThe result follows from the fact that Sn ge T uarrΩ asNote that the above solution satisfies in fact X isin⋂qgt1

⋂tgt0L

q(0 t) as

6 Continuity of the solution

We want to give additional conditions under which the solution of (31) is an as contin-uous process

(H6) For all (sx)isinR+timesRd trarr F(tsx)(tminus s)αq is as continuous on (s+infin)(H7) Ht t ge 0 is as continuous(H8) For all iisin 1 k sisinR+ trarrDi

sHt is as continuous on (s+infin)(H9) For all (sx) isin R+ timesRd i isin 1 k (th)rarr Gprimei (h tsx)(tminus s)α2q is as con-

tinuous on (s+infin)timesRP

We also suppose that there exist δ gt 0 gt 0 such that for all N gt 0 |h| le N 0 le s lttand r x isinRd |tminus r| le 1

(H10) There exists an increasing process VNt t ge 0 such that

∣∣Gi(h tsx)minusGi(hrsx)∣∣leVN

t |tminus r|δ(1 + |x|) (61)

Theorem 61 Under conditions (H1prime) (H2prime) (H3prime) (H4prime) (H5prime) and (H6) (H7) (H8)(H9) (H10) the unique solution of (31) (which is progressively measurable and belongs asto⋂qgt1⋂tgt0L

q(0 t)) has as continuous modification

Mahmoud M El-Borai et al 465

Proof We need to show only that whenever X isin⋂qgt1⋂tgt0L

q(0 t) as Jt(X) t gt 0 hasan as continuous modification

(a) We first show that trarr int t0 F(tsXs)ds is as continuous

Note that (H6) (H7) (H8) and (H9) imply that for all (sx)isinR+timesRd trarr F(tsx)is as continuous on (s+infin)

Moreover from (H2prime) (H3prime) (H4prime) and the fact that X isin⋂qgt1⋂tgt0L

q(0 t) as forany T gt 0 there exists a process ZTs sisin [0T] such that

∣∣F(tsXs)∣∣le ZTs ∣∣Gprimei(Ht tsXs

)DisHt

∣∣le ZTs 0le s lt t le T asint t

0ZTs

1

(tminus s)α2q +1

(tminus s)αqds ltinfin as

(62)

Let first tn nisinN be a sequence such that tn lt t for any n and tnrarr t as nrarrinfin then

int t0F(tsXs

)dsminus

int tn0F(tnsXs

)ds=

int ttnF(tsXs

)ds+

int tn0

[F(tsXs

)minus F(tnsXs)]ds

∣∣∣∣int ttnF(tsXs

)ds∣∣∣∣le

int ttnZTs

1

(tminus s)α2q +1

(tminus s)αqds

(63)

and the latter tends as to 0 as nrarrinfin

∣∣∣∣int tn

0

[F(tsXs

)minus F(tnsXs)]ds∣∣∣∣le

int t0

∣∣F(tsXs)minus F(tnsXs)∣∣ds (64)

which tends to 0 as nrarrinfin A similar argument gives the same result when tn gt t tnrarr t(b) We next show that t rarr It(X Ht) possesses an as continuous modification This

follows from (H7) and

(th)minusrarr It(X h) (65)

has an as continuous modificationBy localization it suffices to prove (65) under assumptions (H2) (H3) (H4) (H5)

and (H6) (H7) (H8) (H9) (H10) with VNt (ω) in (H10) replaced by a constant K

and in case X0 isin⋂qgt1L

q(ΩRd)It then suffices to show that under the above hypotheses there exists Cq gt 0 such that

for any hk isinRP the numbers tr will be positive Suppose to fix the ideas that 0le r lt tthen

It(X h)minus Ir(X k)=int tr

Gi(h tsXs

)(tminus s)α2q dWi

s +int r

0

[Gi(h tsXs

)(tminus s)α2q minus Gi

(hrsXs

)(rminus s)α2q

]dWi

s

+int r

0

[Gi(h tsXs

)(rminus s)α2q minus Gi

(krsXs

)(rminus s)α2q

]dWi

s

(66)

466 Volterra equations with fractional stochastic integrals

It follows from the Burkholder-Gundy inequality that

E(∣∣∣∣int tr

Gi(h tsXs

)(tminus s)α2q dWi

s

∣∣∣∣q)le cq

ksumi=1

E

[(int tr

∣∣Gi(h tsXs

)∣∣2

(tminus s)αq ds)q2]

le cq(tminus r)(qminus2)2ksumi=1

Eint tr

∣∣Gi(h tsXs

)∣∣q(tminus s)α2 ds

le cq(tminus r)(qminus2)2int tr

E(1 +

∣∣Xs∣∣q)(tminus s)α2 ds

le Cq(tminus r)(qminus2)2(tminus r)1minusα2

(67)

From (H4) for Gi we deduce as in Lemma 41 that

E

(∣∣∣∣int r

0

[Gi(hrsXs

)minusGi(krsXs

)(rminus s)α2q

]dWi

s

∣∣∣∣q)

le cq(hminus k)qint r

0

E(1 +

∣∣Xs∣∣q)(rminus s)α2 dsle Cq(hminus k)qr1minusα2

(68)

From (H10) and the fact that

X isin⋂qgt1

⋂tgt0

Lq(Ωtimes (0 t)

)

E

(∣∣∣∣int r

0

[Gi(h tsXs

)(tminus s)α2q minus Gi

(hrsXs

)(rminus s)α2q

]dWi

s

∣∣∣∣q)

= E(∣∣∣∣int r

0

(1

(rminus s)α2q[Gi(h tsXs

)minusGi(hrsXs

)]

+Gi(h tsXs

)[ 1(tminus s)α2q minus

1(rminus s)α2q

])dWi

s

∣∣∣∣q)

le E(cq

int r0

(∣∣Gi(h tsXs

)minusGi(hrsXs

)∣∣q(rminus s)α

+∣∣Gi

(h tsXs

)∣∣q((rminus s)minusα2qminus (tminus s)minusα2q)q)ds

)

le cqKq(int r

0

( |tminus r|δq(rminus s)α E

(1 +

∣∣Xs∣∣q))ds

+int r

0

(E(1 +

∣∣Xs∣∣q)((rminus s)minusα2minus (tminus s)minusα2))ds)

le cq[|tminus r|δq

int r0

(rminus s)minusα2ds+int r

0

((rminus s)minusα2minus (tminus s)minusα2)ds]

le Cq(|tminus r|δqr1minusα2 +

(r1minusα2 + (tminus r)1minusα2minus t1minusα2))

(69)

Mahmoud M El-Borai et al 467

which from the above estimate yields

E(∣∣It(X h)minus Ir(X k)

∣∣q)le Cq

[|tminus r|1minusα2 + |tminus r|(qminusα)2

+ r1minusα2(|tminus r|δq + |hminus k|q)+(r1minusα2minus t1minusα2)]

(610)

this completes the proof

Acknowledgment

We are very grateful to the referees for valuable comments

References

[1] C Bender and R J Elliott Binary market models and discrete wick products preprint 2002[2] M A Berger and V J Mizel Volterra equations with Ito integrals I J Integral Equations 2

(1980) no 3 187ndash245[3] Volterra equations with Ito integrals II J Integral Equations 2 (1980) no 4 319ndash337[4] An extension of the stochastic integral Ann Probab 10 (1982) no 2 435ndash450[5] M M El-Borai Some probability densities and fundamental solutions of fractional evolution

equations Chaos Solitons Fractals 14 (2002) no 3 433ndash440[6] A M Kolodh On the existence solutions of stochastic Volterra equations Theory of Random

Processes 11 (1983) 51ndash57 (Russian)[7] B B Mandelbrot and J W Van Ness Fractional Brownian motions fractional noises and appli-

cations SIAM Rev 10 (1968) 422ndash437[8] P-A Meyer Flot drsquoune equation differentielle stochastique (drsquoapres Malliavin Bismut Kunita)

[Flow of a stochastic differential equation (following Malliavin Bismut Kunita)] Seminar onProbability XV (University of Strasbourg Strasbourg 19791980) Lecture Notes in Mathvol 850 Springer-Verlag Berlin 1981 pp 103ndash117 (French)

[9] D Nualart Noncausal stochastic integrals and calculus Stochastic Analysis and Related Topics(Silivri 1986) Lecture Notes in Math vol 1316 Springer-Verlag Berlin 1988 pp 80ndash129

[10] D Nualart and E Pardoux Stochastic calculus with anticipating integrands Probab TheoryRelated Fields 78 (1988) no 4 535ndash581

[11] D Nualart and M Zakai Generalized stochastic integrals and the Malliavin calculus ProbabTheory Related Fields 73 (1986) no 2 255ndash280

[12] D Ocone and E Pardoux A generalized Ito-Ventzell formula Application to a class of anticipat-ing stochastic differential equations Ann Inst H Poincare Probab Statist 25 (1989) no 139ndash71

[13] Linear stochastic differential equations with boundary conditions Probab Theory Re-lated Fields 82 (1989) no 4 489ndash526

[14] S Ogawa Sur la question drsquoexistence de solutions drsquoune equation differentielle stochastique dutype noncausal [On the existence of solutions of a stochastic differential equation of noncausaltype] J Math Kyoto Univ 24 (1984) no 4 699ndash704 (French)

[15] E Pardoux and P Protter A two-sided stochastic integral and its calculus Probab Theory RelatedFields 76 (1987) no 1 15ndash49

[16] Stochastic Volterra equations with anticipating coefficients Ann Probab 18 (1990)no 4 1635ndash1655

[17] P Protter Volterra equations driven by semimartingales Ann Probab 13 (1985) no 2 519ndash530

468 Volterra equations with fractional stochastic integrals

[18] A N V Rao and C P Tsokos On the existence uniqueness and stability behavior of a randomsolution to a nonlinear perturbed stochastic integro-differential equation Information andControl 27 (1975) 61ndash74

[19] Y Shiota A linear stochastic integral equation containing the extended Ito integral Math RepToyama Univ 9 (1986) 43ndash65

[20] A V Skorohod On a generalization of the stochastic integral Teor Verojatnost i Primenen 20(1975) no 2 223ndash238 (Russian)

[21] A-S Sznitman Martingales dependant drsquoun parametre une formule drsquoIto [Martingales depend-ing on a parameter an Ito formula] Z Wahrsch Verw Gebiete 60 (1982) no 1 41ndash70(French)

[22] W Wyss The fractional diffusion equation J Math Phys 27 (1986) no 11 2782ndash2785

Mahmoud M El-Borai Faculty of Science Alexandria University Alexandria EgyptE-mail address m m elboraiyahoocom

Khairia El-Said El-Nadi Faculty of Science Alexandria University Alexandria EgyptE-mail address khairia el saidhotmailcom

Osama L Mostafa Faculty of Science Alexandria University Alexandria EgyptE-mail address olmoustafayahoocom

Hamdy M Ahmed Faculty of Science Alexandria University Alexandria EgyptE-mail address hamdy 17egyahoocom

Mahmoud M El-Borai et al 465

Proof We need to show only that whenever X isin⋂qgt1⋂tgt0L

q(0 t) as Jt(X) t gt 0 hasan as continuous modification

(a) We first show that trarr int t0 F(tsXs)ds is as continuous

Note that (H6) (H7) (H8) and (H9) imply that for all (sx)isinR+timesRd trarr F(tsx)is as continuous on (s+infin)

Moreover from (H2prime) (H3prime) (H4prime) and the fact that X isin⋂qgt1⋂tgt0L

q(0 t) as forany T gt 0 there exists a process ZTs sisin [0T] such that

∣∣F(tsXs)∣∣le ZTs ∣∣Gprimei(Ht tsXs

)DisHt

∣∣le ZTs 0le s lt t le T asint t

0ZTs

1

(tminus s)α2q +1

(tminus s)αqds ltinfin as

(62)

Let first tn nisinN be a sequence such that tn lt t for any n and tnrarr t as nrarrinfin then

int t0F(tsXs

)dsminus

int tn0F(tnsXs

)ds=

int ttnF(tsXs

)ds+

int tn0

[F(tsXs

)minus F(tnsXs)]ds

∣∣∣∣int ttnF(tsXs

)ds∣∣∣∣le

int ttnZTs

1

(tminus s)α2q +1

(tminus s)αqds

(63)

and the latter tends as to 0 as nrarrinfin

∣∣∣∣int tn

0

[F(tsXs

)minus F(tnsXs)]ds∣∣∣∣le

int t0

∣∣F(tsXs)minus F(tnsXs)∣∣ds (64)

which tends to 0 as nrarrinfin A similar argument gives the same result when tn gt t tnrarr t(b) We next show that t rarr It(X Ht) possesses an as continuous modification This

follows from (H7) and

(th)minusrarr It(X h) (65)

has an as continuous modificationBy localization it suffices to prove (65) under assumptions (H2) (H3) (H4) (H5)

and (H6) (H7) (H8) (H9) (H10) with VNt (ω) in (H10) replaced by a constant K

and in case X0 isin⋂qgt1L

q(ΩRd)It then suffices to show that under the above hypotheses there exists Cq gt 0 such that

for any hk isinRP the numbers tr will be positive Suppose to fix the ideas that 0le r lt tthen

It(X h)minus Ir(X k)=int tr

Gi(h tsXs

)(tminus s)α2q dWi

s +int r

0

[Gi(h tsXs

)(tminus s)α2q minus Gi

(hrsXs

)(rminus s)α2q

]dWi

s

+int r

0

[Gi(h tsXs

)(rminus s)α2q minus Gi

(krsXs

)(rminus s)α2q

]dWi

s

(66)

466 Volterra equations with fractional stochastic integrals

It follows from the Burkholder-Gundy inequality that

E(∣∣∣∣int tr

Gi(h tsXs

)(tminus s)α2q dWi

s

∣∣∣∣q)le cq

ksumi=1

E

[(int tr

∣∣Gi(h tsXs

)∣∣2

(tminus s)αq ds)q2]

le cq(tminus r)(qminus2)2ksumi=1

Eint tr

∣∣Gi(h tsXs

)∣∣q(tminus s)α2 ds

le cq(tminus r)(qminus2)2int tr

E(1 +

∣∣Xs∣∣q)(tminus s)α2 ds

le Cq(tminus r)(qminus2)2(tminus r)1minusα2

(67)

From (H4) for Gi we deduce as in Lemma 41 that

E

(∣∣∣∣int r

0

[Gi(hrsXs

)minusGi(krsXs

)(rminus s)α2q

]dWi

s

∣∣∣∣q)

le cq(hminus k)qint r

0

E(1 +

∣∣Xs∣∣q)(rminus s)α2 dsle Cq(hminus k)qr1minusα2

(68)

From (H10) and the fact that

X isin⋂qgt1

⋂tgt0

Lq(Ωtimes (0 t)

)

E

(∣∣∣∣int r

0

[Gi(h tsXs

)(tminus s)α2q minus Gi

(hrsXs

)(rminus s)α2q

]dWi

s

∣∣∣∣q)

= E(∣∣∣∣int r

0

(1

(rminus s)α2q[Gi(h tsXs

)minusGi(hrsXs

)]

+Gi(h tsXs

)[ 1(tminus s)α2q minus

1(rminus s)α2q

])dWi

s

∣∣∣∣q)

le E(cq

int r0

(∣∣Gi(h tsXs

)minusGi(hrsXs

)∣∣q(rminus s)α

+∣∣Gi

(h tsXs

)∣∣q((rminus s)minusα2qminus (tminus s)minusα2q)q)ds

)

le cqKq(int r

0

( |tminus r|δq(rminus s)α E

(1 +

∣∣Xs∣∣q))ds

+int r

0

(E(1 +

∣∣Xs∣∣q)((rminus s)minusα2minus (tminus s)minusα2))ds)

le cq[|tminus r|δq

int r0

(rminus s)minusα2ds+int r

0

((rminus s)minusα2minus (tminus s)minusα2)ds]

le Cq(|tminus r|δqr1minusα2 +

(r1minusα2 + (tminus r)1minusα2minus t1minusα2))

(69)

Mahmoud M El-Borai et al 467

which from the above estimate yields

E(∣∣It(X h)minus Ir(X k)

∣∣q)le Cq

[|tminus r|1minusα2 + |tminus r|(qminusα)2

+ r1minusα2(|tminus r|δq + |hminus k|q)+(r1minusα2minus t1minusα2)]

(610)

this completes the proof

Acknowledgment

We are very grateful to the referees for valuable comments

References

[1] C Bender and R J Elliott Binary market models and discrete wick products preprint 2002[2] M A Berger and V J Mizel Volterra equations with Ito integrals I J Integral Equations 2

(1980) no 3 187ndash245[3] Volterra equations with Ito integrals II J Integral Equations 2 (1980) no 4 319ndash337[4] An extension of the stochastic integral Ann Probab 10 (1982) no 2 435ndash450[5] M M El-Borai Some probability densities and fundamental solutions of fractional evolution

equations Chaos Solitons Fractals 14 (2002) no 3 433ndash440[6] A M Kolodh On the existence solutions of stochastic Volterra equations Theory of Random

Processes 11 (1983) 51ndash57 (Russian)[7] B B Mandelbrot and J W Van Ness Fractional Brownian motions fractional noises and appli-

cations SIAM Rev 10 (1968) 422ndash437[8] P-A Meyer Flot drsquoune equation differentielle stochastique (drsquoapres Malliavin Bismut Kunita)

[Flow of a stochastic differential equation (following Malliavin Bismut Kunita)] Seminar onProbability XV (University of Strasbourg Strasbourg 19791980) Lecture Notes in Mathvol 850 Springer-Verlag Berlin 1981 pp 103ndash117 (French)

[9] D Nualart Noncausal stochastic integrals and calculus Stochastic Analysis and Related Topics(Silivri 1986) Lecture Notes in Math vol 1316 Springer-Verlag Berlin 1988 pp 80ndash129

[10] D Nualart and E Pardoux Stochastic calculus with anticipating integrands Probab TheoryRelated Fields 78 (1988) no 4 535ndash581

[11] D Nualart and M Zakai Generalized stochastic integrals and the Malliavin calculus ProbabTheory Related Fields 73 (1986) no 2 255ndash280

[12] D Ocone and E Pardoux A generalized Ito-Ventzell formula Application to a class of anticipat-ing stochastic differential equations Ann Inst H Poincare Probab Statist 25 (1989) no 139ndash71

[13] Linear stochastic differential equations with boundary conditions Probab Theory Re-lated Fields 82 (1989) no 4 489ndash526

[14] S Ogawa Sur la question drsquoexistence de solutions drsquoune equation differentielle stochastique dutype noncausal [On the existence of solutions of a stochastic differential equation of noncausaltype] J Math Kyoto Univ 24 (1984) no 4 699ndash704 (French)

[15] E Pardoux and P Protter A two-sided stochastic integral and its calculus Probab Theory RelatedFields 76 (1987) no 1 15ndash49

[16] Stochastic Volterra equations with anticipating coefficients Ann Probab 18 (1990)no 4 1635ndash1655

[17] P Protter Volterra equations driven by semimartingales Ann Probab 13 (1985) no 2 519ndash530

468 Volterra equations with fractional stochastic integrals

[18] A N V Rao and C P Tsokos On the existence uniqueness and stability behavior of a randomsolution to a nonlinear perturbed stochastic integro-differential equation Information andControl 27 (1975) 61ndash74

[19] Y Shiota A linear stochastic integral equation containing the extended Ito integral Math RepToyama Univ 9 (1986) 43ndash65

[20] A V Skorohod On a generalization of the stochastic integral Teor Verojatnost i Primenen 20(1975) no 2 223ndash238 (Russian)

[21] A-S Sznitman Martingales dependant drsquoun parametre une formule drsquoIto [Martingales depend-ing on a parameter an Ito formula] Z Wahrsch Verw Gebiete 60 (1982) no 1 41ndash70(French)

[22] W Wyss The fractional diffusion equation J Math Phys 27 (1986) no 11 2782ndash2785

Mahmoud M El-Borai Faculty of Science Alexandria University Alexandria EgyptE-mail address m m elboraiyahoocom

Khairia El-Said El-Nadi Faculty of Science Alexandria University Alexandria EgyptE-mail address khairia el saidhotmailcom

Osama L Mostafa Faculty of Science Alexandria University Alexandria EgyptE-mail address olmoustafayahoocom

Hamdy M Ahmed Faculty of Science Alexandria University Alexandria EgyptE-mail address hamdy 17egyahoocom

466 Volterra equations with fractional stochastic integrals

It follows from the Burkholder-Gundy inequality that

E(∣∣∣∣int tr

Gi(h tsXs

)(tminus s)α2q dWi

s

∣∣∣∣q)le cq

ksumi=1

E

[(int tr

∣∣Gi(h tsXs

)∣∣2

(tminus s)αq ds)q2]

le cq(tminus r)(qminus2)2ksumi=1

Eint tr

∣∣Gi(h tsXs

)∣∣q(tminus s)α2 ds

le cq(tminus r)(qminus2)2int tr

E(1 +

∣∣Xs∣∣q)(tminus s)α2 ds

le Cq(tminus r)(qminus2)2(tminus r)1minusα2

(67)

From (H4) for Gi we deduce as in Lemma 41 that

E

(∣∣∣∣int r

0

[Gi(hrsXs

)minusGi(krsXs

)(rminus s)α2q

]dWi

s

∣∣∣∣q)

le cq(hminus k)qint r

0

E(1 +

∣∣Xs∣∣q)(rminus s)α2 dsle Cq(hminus k)qr1minusα2

(68)

From (H10) and the fact that

X isin⋂qgt1

⋂tgt0

Lq(Ωtimes (0 t)

)

E

(∣∣∣∣int r

0

[Gi(h tsXs

)(tminus s)α2q minus Gi

(hrsXs

)(rminus s)α2q

]dWi

s

∣∣∣∣q)

= E(∣∣∣∣int r

0

(1

(rminus s)α2q[Gi(h tsXs

)minusGi(hrsXs

)]

+Gi(h tsXs

)[ 1(tminus s)α2q minus

1(rminus s)α2q

])dWi

s

∣∣∣∣q)

le E(cq

int r0

(∣∣Gi(h tsXs

)minusGi(hrsXs

)∣∣q(rminus s)α

+∣∣Gi

(h tsXs

)∣∣q((rminus s)minusα2qminus (tminus s)minusα2q)q)ds

)

le cqKq(int r

0

( |tminus r|δq(rminus s)α E

(1 +

∣∣Xs∣∣q))ds

+int r

0

(E(1 +

∣∣Xs∣∣q)((rminus s)minusα2minus (tminus s)minusα2))ds)

le cq[|tminus r|δq

int r0

(rminus s)minusα2ds+int r

0

((rminus s)minusα2minus (tminus s)minusα2)ds]

le Cq(|tminus r|δqr1minusα2 +

(r1minusα2 + (tminus r)1minusα2minus t1minusα2))

(69)

Mahmoud M El-Borai et al 467

which from the above estimate yields

E(∣∣It(X h)minus Ir(X k)

∣∣q)le Cq

[|tminus r|1minusα2 + |tminus r|(qminusα)2

+ r1minusα2(|tminus r|δq + |hminus k|q)+(r1minusα2minus t1minusα2)]

(610)

this completes the proof

Acknowledgment

We are very grateful to the referees for valuable comments

References

[1] C Bender and R J Elliott Binary market models and discrete wick products preprint 2002[2] M A Berger and V J Mizel Volterra equations with Ito integrals I J Integral Equations 2

(1980) no 3 187ndash245[3] Volterra equations with Ito integrals II J Integral Equations 2 (1980) no 4 319ndash337[4] An extension of the stochastic integral Ann Probab 10 (1982) no 2 435ndash450[5] M M El-Borai Some probability densities and fundamental solutions of fractional evolution

equations Chaos Solitons Fractals 14 (2002) no 3 433ndash440[6] A M Kolodh On the existence solutions of stochastic Volterra equations Theory of Random

Processes 11 (1983) 51ndash57 (Russian)[7] B B Mandelbrot and J W Van Ness Fractional Brownian motions fractional noises and appli-

cations SIAM Rev 10 (1968) 422ndash437[8] P-A Meyer Flot drsquoune equation differentielle stochastique (drsquoapres Malliavin Bismut Kunita)

[Flow of a stochastic differential equation (following Malliavin Bismut Kunita)] Seminar onProbability XV (University of Strasbourg Strasbourg 19791980) Lecture Notes in Mathvol 850 Springer-Verlag Berlin 1981 pp 103ndash117 (French)

[9] D Nualart Noncausal stochastic integrals and calculus Stochastic Analysis and Related Topics(Silivri 1986) Lecture Notes in Math vol 1316 Springer-Verlag Berlin 1988 pp 80ndash129

[10] D Nualart and E Pardoux Stochastic calculus with anticipating integrands Probab TheoryRelated Fields 78 (1988) no 4 535ndash581

[11] D Nualart and M Zakai Generalized stochastic integrals and the Malliavin calculus ProbabTheory Related Fields 73 (1986) no 2 255ndash280

[12] D Ocone and E Pardoux A generalized Ito-Ventzell formula Application to a class of anticipat-ing stochastic differential equations Ann Inst H Poincare Probab Statist 25 (1989) no 139ndash71

[13] Linear stochastic differential equations with boundary conditions Probab Theory Re-lated Fields 82 (1989) no 4 489ndash526

[14] S Ogawa Sur la question drsquoexistence de solutions drsquoune equation differentielle stochastique dutype noncausal [On the existence of solutions of a stochastic differential equation of noncausaltype] J Math Kyoto Univ 24 (1984) no 4 699ndash704 (French)

[15] E Pardoux and P Protter A two-sided stochastic integral and its calculus Probab Theory RelatedFields 76 (1987) no 1 15ndash49

[16] Stochastic Volterra equations with anticipating coefficients Ann Probab 18 (1990)no 4 1635ndash1655

[17] P Protter Volterra equations driven by semimartingales Ann Probab 13 (1985) no 2 519ndash530

468 Volterra equations with fractional stochastic integrals

[18] A N V Rao and C P Tsokos On the existence uniqueness and stability behavior of a randomsolution to a nonlinear perturbed stochastic integro-differential equation Information andControl 27 (1975) 61ndash74

[19] Y Shiota A linear stochastic integral equation containing the extended Ito integral Math RepToyama Univ 9 (1986) 43ndash65

[20] A V Skorohod On a generalization of the stochastic integral Teor Verojatnost i Primenen 20(1975) no 2 223ndash238 (Russian)

[21] A-S Sznitman Martingales dependant drsquoun parametre une formule drsquoIto [Martingales depend-ing on a parameter an Ito formula] Z Wahrsch Verw Gebiete 60 (1982) no 1 41ndash70(French)

[22] W Wyss The fractional diffusion equation J Math Phys 27 (1986) no 11 2782ndash2785

Mahmoud M El-Borai Faculty of Science Alexandria University Alexandria EgyptE-mail address m m elboraiyahoocom

Khairia El-Said El-Nadi Faculty of Science Alexandria University Alexandria EgyptE-mail address khairia el saidhotmailcom

Osama L Mostafa Faculty of Science Alexandria University Alexandria EgyptE-mail address olmoustafayahoocom

Hamdy M Ahmed Faculty of Science Alexandria University Alexandria EgyptE-mail address hamdy 17egyahoocom

Mahmoud M El-Borai et al 467

which from the above estimate yields

E(∣∣It(X h)minus Ir(X k)

∣∣q)le Cq

[|tminus r|1minusα2 + |tminus r|(qminusα)2

+ r1minusα2(|tminus r|δq + |hminus k|q)+(r1minusα2minus t1minusα2)]

(610)

this completes the proof

Acknowledgment

We are very grateful to the referees for valuable comments

References

[1] C Bender and R J Elliott Binary market models and discrete wick products preprint 2002[2] M A Berger and V J Mizel Volterra equations with Ito integrals I J Integral Equations 2

(1980) no 3 187ndash245[3] Volterra equations with Ito integrals II J Integral Equations 2 (1980) no 4 319ndash337[4] An extension of the stochastic integral Ann Probab 10 (1982) no 2 435ndash450[5] M M El-Borai Some probability densities and fundamental solutions of fractional evolution

equations Chaos Solitons Fractals 14 (2002) no 3 433ndash440[6] A M Kolodh On the existence solutions of stochastic Volterra equations Theory of Random

Processes 11 (1983) 51ndash57 (Russian)[7] B B Mandelbrot and J W Van Ness Fractional Brownian motions fractional noises and appli-

cations SIAM Rev 10 (1968) 422ndash437[8] P-A Meyer Flot drsquoune equation differentielle stochastique (drsquoapres Malliavin Bismut Kunita)

[Flow of a stochastic differential equation (following Malliavin Bismut Kunita)] Seminar onProbability XV (University of Strasbourg Strasbourg 19791980) Lecture Notes in Mathvol 850 Springer-Verlag Berlin 1981 pp 103ndash117 (French)

[9] D Nualart Noncausal stochastic integrals and calculus Stochastic Analysis and Related Topics(Silivri 1986) Lecture Notes in Math vol 1316 Springer-Verlag Berlin 1988 pp 80ndash129

[10] D Nualart and E Pardoux Stochastic calculus with anticipating integrands Probab TheoryRelated Fields 78 (1988) no 4 535ndash581

[11] D Nualart and M Zakai Generalized stochastic integrals and the Malliavin calculus ProbabTheory Related Fields 73 (1986) no 2 255ndash280

[12] D Ocone and E Pardoux A generalized Ito-Ventzell formula Application to a class of anticipat-ing stochastic differential equations Ann Inst H Poincare Probab Statist 25 (1989) no 139ndash71

[13] Linear stochastic differential equations with boundary conditions Probab Theory Re-lated Fields 82 (1989) no 4 489ndash526

[14] S Ogawa Sur la question drsquoexistence de solutions drsquoune equation differentielle stochastique dutype noncausal [On the existence of solutions of a stochastic differential equation of noncausaltype] J Math Kyoto Univ 24 (1984) no 4 699ndash704 (French)

[15] E Pardoux and P Protter A two-sided stochastic integral and its calculus Probab Theory RelatedFields 76 (1987) no 1 15ndash49

[16] Stochastic Volterra equations with anticipating coefficients Ann Probab 18 (1990)no 4 1635ndash1655

[17] P Protter Volterra equations driven by semimartingales Ann Probab 13 (1985) no 2 519ndash530

468 Volterra equations with fractional stochastic integrals

[18] A N V Rao and C P Tsokos On the existence uniqueness and stability behavior of a randomsolution to a nonlinear perturbed stochastic integro-differential equation Information andControl 27 (1975) 61ndash74

[19] Y Shiota A linear stochastic integral equation containing the extended Ito integral Math RepToyama Univ 9 (1986) 43ndash65

[20] A V Skorohod On a generalization of the stochastic integral Teor Verojatnost i Primenen 20(1975) no 2 223ndash238 (Russian)

[21] A-S Sznitman Martingales dependant drsquoun parametre une formule drsquoIto [Martingales depend-ing on a parameter an Ito formula] Z Wahrsch Verw Gebiete 60 (1982) no 1 41ndash70(French)

[22] W Wyss The fractional diffusion equation J Math Phys 27 (1986) no 11 2782ndash2785

Mahmoud M El-Borai Faculty of Science Alexandria University Alexandria EgyptE-mail address m m elboraiyahoocom

Khairia El-Said El-Nadi Faculty of Science Alexandria University Alexandria EgyptE-mail address khairia el saidhotmailcom

Osama L Mostafa Faculty of Science Alexandria University Alexandria EgyptE-mail address olmoustafayahoocom

Hamdy M Ahmed Faculty of Science Alexandria University Alexandria EgyptE-mail address hamdy 17egyahoocom

468 Volterra equations with fractional stochastic integrals

[18] A N V Rao and C P Tsokos On the existence uniqueness and stability behavior of a randomsolution to a nonlinear perturbed stochastic integro-differential equation Information andControl 27 (1975) 61ndash74

[19] Y Shiota A linear stochastic integral equation containing the extended Ito integral Math RepToyama Univ 9 (1986) 43ndash65

[20] A V Skorohod On a generalization of the stochastic integral Teor Verojatnost i Primenen 20(1975) no 2 223ndash238 (Russian)

[21] A-S Sznitman Martingales dependant drsquoun parametre une formule drsquoIto [Martingales depend-ing on a parameter an Ito formula] Z Wahrsch Verw Gebiete 60 (1982) no 1 41ndash70(French)

[22] W Wyss The fractional diffusion equation J Math Phys 27 (1986) no 11 2782ndash2785

Mahmoud M El-Borai Faculty of Science Alexandria University Alexandria EgyptE-mail address m m elboraiyahoocom

Khairia El-Said El-Nadi Faculty of Science Alexandria University Alexandria EgyptE-mail address khairia el saidhotmailcom

Osama L Mostafa Faculty of Science Alexandria University Alexandria EgyptE-mail address olmoustafayahoocom

Hamdy M Ahmed Faculty of Science Alexandria University Alexandria EgyptE-mail address hamdy 17egyahoocom

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