stability problems in modern actuarial...

Stability problems in modern actuarial sciences

Ekaterina Bulinskaya

(Lomonosov Moscow State University)

ACMPT 2017dedicated to 90th anniversary of professor A.D.Solovyev

Moscow, Russia, October 23-27, 2017

E.Bulinskaya Stability problems in modern actuarial sciences

Professor A.D.Solovyev (1927-2001)


Outline of the talk ”Stability problems in modern actuarial sciences”

1) Motivation

2) Historical background

3) New research directions

4) Stability with respect to small perturbations ofunderlying processes and parameters fluctuations

5) Further results


Motivation

Recall the following well-known facts:

To study a real-life process or a system it is useful toconstruct a mathematical model.

There are a lot of models describing more or lessprecisely a given system.

The same model can describe the processes arisingin different research domains.


Motivation

Applied probability research domains such as

insurance, inventory and dams, finance, queueing,reliability and some otherscan be considered as special cases of decisionmaking under uncertainty (or risk management)aimed at the systems performance optimization, thuseliminating or minimizing risk.

For correct decision making one needs anappropriate mathematical model.

The model stability is a must.


Motivation

A crucial question in all investigations pertaining todecision making is:

How to choosean appropriate mathematical model?

The usual procedure is as follows


Real system investigation


Motivation

Concise OxfordEnglish Dictionary:

Risk is a hazard, a chance of badconsequences, loss or exposure tomischance.


Notation for description of input-output model (T ,Z ,Y ,U,Ψ,L)

T – planning horizon,Z = {Z (t), t ∈ [0,T ]} – input process,Y = {Y (t), t ∈ [0,T ]} – output process,U = {U(t), t ∈ [0,T ]} – control,Ψ – represents the system configuration andoperation mode, X = Ψ(Z ,Y ,U),X = {X (t), t ∈ [0,T ]} – the system state,LT (U) = L(Z ,Y ,U,X ,T ) – objective function (target,valuation criterion, risk measure) evaluating thesystem performance quality.


Models description


Interpretation of model parameters for different research domains

Research Input Output System statefield Z (t) Y (t) X (t)

Insurance Premium Indemnity SurplusFinance Money inflow Money outflow CapitalInventory Supply Demand Inventory

levelStorage Water inflow Water outflow Water levelReliability New & repaired Broken Working

elements elements elementsQueueing Customers Served Queue length

arrival customersPopulation Birth & Death & Populationgrowth immigration emigration size


Models classification

Discrete or continuous time

Deterministic, stochastic, mixed

Configuration, dimension of underlying processes

Known or unknown parameters and distributions

Set of feasible controls (static or dynamic)

Objective function type


Objective functions

Cost approachInventory, finance

No objective functionQueueing, dams

Reliability approachInsurance, reliability


Optimality

Nowadays the risk measures are introduced in all the researchfields to optimize the systems performance

Definition

A control U∗T = {U∗(t), t ∈ [0,T ]} is called optimal if

LT (U∗T ) = infUT∈UT

LT (UT ), (or LT (U∗T ) = supUT∈UT

LT (UT )),

(1)where UT is a class of all feasible controls.

Furthermore, U∗ = {U∗T ,T ≥ 0} is called an optimal policy(or strategy).


Definitions

The choice of inf or sup in (1) is determined by theproblem we want to solve. Namely,

if we are interested in minimization of losses (or ruinprobability) we use the first expression,

whereas for profit (or system life-time) maximizationwe use the second one in (1).


Definitions

Since extremum in (1) may be not attained we introducethe following

DefinitionA control Uε

T is ε-optimal if

LT (UεT ) < inf

UT∈UT

LT (UT ) + ε

(or LT (UεT ) > sup

UT∈UT

LT (UT )− ε).


Definitions

Definition

A policy U = {UT ,T ≥ 0} is stationary if for any T ,S ≥ 0

UT (t) = US(t), t ≤ min(T ,S).

Definition

A policy U = (UT ,T ≥ 0) is asymptotically optimal if

limT→∞

T−1LT (UT ) = limT→∞

T−1LT (U∗T ).

The changes necessary for discrete-time models areobvious.


Two-fold nature of insurance company

Insurance has a long and interesting history.

Methods for transferring or distributing risk were practicedby Chinese and Babylonian traders 3 and 2 thousandsBC.

Code of Hammurabi, c. 1750 BC.

Mutual societies, run by their members with no externalshareholders to pay.

Next step is joint stock companies.


4 periods in actuarial sciences

Why actuarial science emerged significantly later (in the17th century) one can read in the interesting book∗

Deterministic period (E.Halley, D.Bernoulli)

Stochastic period (collective risk)

Financial period (investment side of insurance)

Modern period (enterprise risk management)

—————————————————————————–∗Bernstein, P.L. Against the Gods: The Remarkable Story of Risk.John Wiley and Sons, New York (1996)


Hans Buhlmann in 1987 has given 3 periods classification


Deterministic period: E.Haley’s mortality tables appeared in 1693


Deterministic period: D.Bernoulli’s utility functions appeared in 1738


Stochastic period: F.Lundberg dissertation 1903 (Poisson process)


Stochastic period: Harald Cramer 1930s


Paul Embrechts announced the fourth period in 2005


President of CAS d’Arcy and his colleagues


Enterprise risk management (ERM)

Study of new types of risk: operating and strategicE.Bulinskaya Stability problems in modern actuarial sciences

Two-fold nature of insurance company

The primary task is indemnification ofpolicyholders claims.

The secondary task is dividend payments toshareholders.


Insurance models (mixed type or purely stochastic)

Insurance models are of input-output type.

We have to specify the premiums inflow P(t) and claimpayments to customers (outflow) S(t), as well as theplanning horizon T ≤ ∞.

Thus, the company surplus (capital) X (t) at time t ≤ Thas the form X (t) = x + P(t)− S(t) where x is the initialsurplus.

To accomplish the optimization of the insurance companyperformance one needs to introduce the set of feasiblecontrols and an objective function.


Objective functions

The most popular objective function in non-life insurance(since 1903) was the company ruin probability for theclassical (collective risk) Cramer-Lundberg model. Inother words, the main goal was to achieve the highreliability of the company.

However in practice it turned out that the negative surpluslevel not always leads to bankruptcy, since the companymay use, e.g. a bank loan to avoid insolvency.

So, there were defined and studied ”Absolute ruin”,”Parisian ruin”, as well as, ”Omega models”, in theframework of reliability approach.


Objective functions

Being a corporation insurance company is interested individend payments to its shareholders.

Thus, the so-called cost approach arose in the middle ofthe last century due to the pioneering Bruno de Finettipaper (1957). He proposed to maximize the expecteddiscounted dividends paid out until ruin.

Modern period in actuarial sciences is characterized byinterplay of financial and actuarial methods leading tounification of reliability and cost approaches.—————————————————–De Finetti, B.: Su un’impostazione alternativa della teoria collettivadel rischio. Transactions of the XV-th International Congress ofActuaries. 2, 433–443 (1957)


Optimization tools

For optimization of a company functioning one can usevarious tools such as investment, bank loans andreinsurance.

We have investigated the asymptotic behavior of severalinsurance models and applied the results for establishingthe optimal investment and reinsurance strategies.

The discrete-time models which are more realistic insome situations were considered along withcontinuous-time ones.———————————-Bulinskaya E. New research directions in modern actuarialsciences. // In: Modern problems of stochastic analysis and statistics– Selected contributions in honor of Valentin Konakov (ed. V.Panov).Springer, 2017.


Stability

There are a lot of methods for investigation ofsystems stability:

Lyapunov stability in differential equations theory,

stability of dynamic systems associated withtransport (or inventory) networks,

probability metrics to measure the systems responseto perturbations of underlying stochastic processes,

local (differential importance measure) and global(Sobol decomposition and FAST) Sensitivity Analysistechniques for uncertain scalar parameters.


For the dual models arising in life-insurance one gets thefollowing relation X (t) = x − ct + S(t). There exist otherpossible interpretations for this model. For example, onecan treat the surplus as amount of the capital of abusiness engaged in research and development1. Thecompany pays continuously expenses for research, andoccasional profit of random amounts (such as the awardof a patent or a sudden increase in sales) arisesaccording to a Poisson process. A similar model wasused in 2 to model the functioning of a venture capitalinvestment company.1Avanzi B., Gerber H. U. and Shiu E. S. W.: Optimal Dividends in theDual Model. Insurance: Mathematics and Economics. 41 (1),111–123 (2007)2Bayraktar E. and Egami M.: Optimizing venture capital investment ina jump diffusion model. Mathematical Methods of OperationsResearch. 67 (1), 21–42 (2008)


We mention in passing a so-called Gerber-Shiu functionestimating ruin severity and its generalizations, see, e.g.,Breuer L. and Badescu A. (2014). The use of suchfunctions demonstrates the unification of reliability andcost approaches.Consideration of solvency problems, see, e.g.,Sandstrom A. (2011), gave rise to new ruin notions suchas Parisian ruin, absolute ruin and Omega models. Dueto their practical importance, these problems haveattracted growing attention in risk theory.


Parisian type ruin will occur if the surplus falls below aprescribed critical level (red zone) and stays there for acontinuous time interval of length d. In some respects,this might be a more appropriate measure of risk thanclassical ruin as it gives the office some time to put itsfinances in order, see, e.g., Czarna I. and Palmowski Z.(2011). Another type of Parisian ruin includes a stochasticdelay (clock) in bankruptcy implementation, see, e.g.,Landriault D., Renaud J.-F. and Zhou X. (2014). Thesetwo types of Parisian ruin start a new clock each time thesurplus enters the red zone, either deterministic orstochastic. Proposed in Guerin H., Renaud J.-F.(2015)the third type of Parisian ruin (called cumulative)includes the race between a single deterministic clockand the sum of the excursions below the critical level.


One of the first papers treating the absolute ruin is GerberH.U. (1971). When the surplus is below zero or theinsurer is on deficit, the insurer could borrow money at adebit interest rate to pay claims. Meanwhile, the insurerwill repay the debts from the premium income. Thenegative surplus may return to a positive level. However,when the negative surplus is below a certain critical level,the surplus is no longer able to become positive. Absoluteruin occurs at this moment, see, e.g., Fu D., Guo Y.(2016).


In the Omega model, there is a distinction between ruin(negative surplus) and bankruptcy (going out ofbusiness). It is assumed that even with a negativesurplus, the company can do business as usual andcontinue until bankruptcy occurs. The probability forbankruptcy is quantified by a bankruptcy rate functionω(x), where x is the value of the negative surplus. Thesymbol for this function leads to the name Omega model,see, e.g., Albrecher H., Gerber H.U., Shiu E.S.W. (2011).


The first aim of presentation is to carry out asymptoticanalysis and optimization of some models of thedescribed above type. In particular, we introduce a newindicator of insurance company performance, namely, thefirst time ηX

l when the interval of the surplus staying abovezero (before the Parisian ruin) becomes greater than l .Then for the Cramer-Lundberg case the explicit form ofthe Laplace transform of ηX

l is calculated as a function ofthe model’s parameters.


The second aim is to study the systems stability withrespect to underlying processes perturbations andparameters fluctuations. Under assumption that claimamounts have exponential distribution with parameter αwe perform the sensitivity analysis of the probability ofParisian ruin with a deterministic clock d . For this purposewe use some local and global methods gathered inSaltelli A. et al. Thus, we begin by calculating the partialderivatives with respect to all the parameters α, λ, x , cand d . Then the scatterplots were obtained byMonte-Carlo simulation of ruin probability, as well asfirst-order and total-effect sensitivity indices.


Now we consider the classical Cramer-Lundberg model,its surplus is described by equation where Nt is a Poissonprocess with parameter λ. It is supposed further on thatthe net profit relation c > λµ is satisfied, here c is thepremium rate and µ is the mean claim amount.


To treat the Parisian ruin with a fixed clock d , we define,as in Dassios A., Wu Sh., two random variablesgX

t = sup{s < t | sign(Xs)sign(Xt) ≤ 0},dX

t = inf{s > t | sign(Xs)sign(Xt) ≤ 0}where sup{∅} = 0, inf{∅} =∞ and

sign(x) =

1, x > 0,0, x = 0,−1, x < 0.

The trajectory between gXt and dX

t is the excursion ofprocess X which straddles time t .


Now it is possible to give the following

Definition

The Parisian ruin time τXd is the first time that the length of

the excursion of process X below 0 reaches a given leveld , that is,

τXd = inf{t > 0 | 1{Xt<0}(t − gX

t ) ≥ d}.


Obviously, P(τXd <∞) and P(τX

d ≤ t) are the probabilitiesof ultimate Parisian ruin and finite-time Parisian ruinrespectively.

To simplify the further reasoning we assume that theclaim amounts have exponential distribution withparameter α, namely, the corresponding density isp(x) = αe−αx for x ≥ 0. Therefore the net profit conditioncan be written as c > λ/α. For the most part, it issupposed that initial surplus x = 0.


Introduce an auxiliary process Z Xt with two states by the

following relation

Z Xt =

{1, Xt > 0,2, Xt < 0.

In this definition, we deliberately ignore the situation whenXt = 0. The reason is the equality

∫ t0 1{Xu=0} du = 0.


It is easy to rewrite the defined above random variables interms of the process Z X

gXt = sup{s < t | Z X

t 6= Z Xs }, dX

t = inf{s > t | Z Xt 6= Z X

s }

τXd = inf{t > 0 | 1{Z X

t =2}(t − gXt ) ≥ d}

and introduce the time V Xt = t − gX

t spent by Z X in thecurrent state. It is not difficult to establish that thetwo-dimensional process (Z X

t ,V Xt ) is Markov. Hence Z X

tis semi-Markov with a state space {1,2}, where the state1 corresponds to the process X being above zero andstate 2 to its being below zero.


In order to find transition probabilities of Z X we considertwo sequences of random variables UX

i,k , i = 1,2, k ≥ 1.Here UX

i,k is the time spent in the state i during the k thvisit of this state (the lengths of excursions above andbelow zero). Moreover, for any fixed i and k there existssuch t that UX

i,k = V XdX

t= dX

t − gXt . Since X is compound

Poisson with exponential jumps, each of sequences UX1,k

and UX2,k consists of i.i.d. r.v.’s, moreover, both sequences

are independent.


We therefore define the transition density for Z X by thefollowing relation

pij(t) = lim∆t→0

P(t < UXi,k < t + ∆t)

∆t, j 6= i .

Furthermore, the probability that the process will stay instate i no longer than time t is given by

Pij(t) =

∫ t

0pij(u) du = P(UX

i,k < t) = 1−Pij(t) = 1−P(UXi,k > t).

Note, that according to the law of large numbers for theprocess X , under condition of net profit, X (t)→∞, ast →∞, with probability 1.


Thus, it is possible that the process Z X will stay forever inthe state 1 during its k th visit, for any k ≥ 1, that is,P(UX

1,k =∞) > 0.For calculation of pij(t) we use its Laplace transform

Pij(β) =

∫ ∞0

e−βtpij(t)dt = E(

e−βUXi,k

).

For dealing with UX2,k , k = 1,2,3, . . . the stoping-time

Tx = inf{t > 0,Xt = 0 | X0 = x , x < 0} is useful.


LemmaThe following equality E (exp(−βTx )) = exp

(υ+β x)

holdswith

υ+β =

√(cα + β + λ)2 − 4cαλ− (cα− β − λ)

2c.


DefinitionModified Bessel function of the first kind is introduced bythe relation

Iν(z) =∞∑

k=0

(z/2)2k+ν

k !Γ(k + ν + 1),

where ν is the function order.


We need further only I1(z) to formulate two lemmasproved in Dassios A., Wu Sh..

LemmaThe transition density

p21(t) =√

cα/λ e−(λ+cα)t t−1I1(2t√

cλα).


According to definition of X and UX2,k it is possible to get

the following chain of equalities

P21(β) = E(

e−βUX2,k

)=

∫ ∞0

e−βT−xαe−αxdx

=

∫ ∞0

exp(−υ+

β x)αe−αxdx

= 2cα(√

(cα + β + λ)2 − 4cαλ + (cα + β + λ))−1

due to the fact that every excursion below 0 starts from anovershoot below 0 with length |x | having the exponentialdistribution with parameter α and the excursion length isTx .


Using Bateman H. (1954) it is easy to establish thestatement of lemma. In fact, if the Laplace transformF (β) =

∫∞0 e−βt f (t) dt of a function f (t) has the form

F (β) =((β2 − a2)1/2 + β

)−n

then f (t) = na−nt−1In(at). In our case expression ofP21(β) has such a form (up to additional factor 2cα) withn = 1, a = 2

√cαλ and cα + β + λ instead of β. Using the

properties of inverse Laplace transform we get thedesired result for p21(t).


LemmaThe transition density

p12(t) =√λ/cα e−(λ+cα)t t−1I1(2t

√cλα).


To treat the length of excursions above 0 it isadvantageous to introduce the following stopping-timeT0 = inf{t > 0,Xt < 0 | X0 = 0}. According to GerberH.U., Shiu E.S.W. (1997) and the independence of thetime and the size of the overshoot, i.e. T0 and XT0, wehave

E(e−βT0

)E(exp

(υ−β XT0

))= 1

with

υ−β =−√

(cα + β + λ)2 − 4cαλ− (cα− β − λ)

2c.


Since |XT0 | is exponentially distributed with parameter αone hasP12(β) = E

(e−βUX

1,k

)= E

(e−βT0

)=[∫∞

0 exp(−υ−β x

)αe−αxdx

]−1

= 2λ(√

(β + λ + cα)2 − 4cλα + (β + λ + cα)−1

. (2)

Taking the inverse Laplace transform and using oncemore the results of Bateman H. (1954) we end the proof.


In order to obtain a new measure of insurance companyperformance we introduce

DefinitionDenote by

ηXl = inf{t > 0 | 1{Z X

t =1}(t − gXt ) ≥ l}

the first time when the company surplus stays above zeroduring the interval longer than l .


Theorem

The following result is valid for the process Xt if x = 0

E(

e−βηXl | τX

d > ηXl

)=

e−βl P12(l)1− P21(β)P12(β)

(3)

where

P12(l) = 1−√λ/cα

∫ l

0e−(λ+cα)t t−1I1(2t

√λcα) dt , (4)

P21(β) =√

cα/λ∫ d

0e−(β+λ+cα)t t−1I1(2t

√λcα) dt , (5)

P12(β) =√λ/cα

∫ l

0e−(β+λ+cα)t t−1I1(2t

√λcα) dt (6)

and I1(·) is the modified Bessel function of the first kind.


Let event Bk mean that ηXl hits the level l during the k th

visit of the state 1. Then we can write E(

e−βηXl | τX

d > ηXl

)in the form∑∞

k=1 E(

e−βηXl | Bk ,UX

2,1 < d , . . . ,UX2,k−1 < d

)×

×P(UX2,1 < d , . . . ,UX

2,k−1 < d).Clearly, for k = 1,

E(

e−βηXl | B1, τ

Xd > ηX

l

)= E

(e−βη

Xl | UX

1,1 ≥ l)

= E(e−βl | UX

1,1 ≥ l)

= e−βl .


For k ≥ 2, on the set Bk r.v. ηXl =

∑k−1n=1(UX

1,n + UX2,n) + l

under condition that Z Xt has spent less than l during the

first k − 1 visits of the state 1. Since UX1,n and UX

2,n areindependent and have distribution functions P12(t) andP21(t), respectively, it is possible to rewriteE(

e−βηXl | Bk ,UX

2,1 < d , . . . ,UX2,k−1 < d

)as follows


E(e−β(

k−1∑n=1

(UX1,n+UX

2,n)+l)|UX

1,1 < l , . . . ,UX1,k−1 < l ,

UX1,k ≥ l ,UX

2,1 < d , . . . ,UX2,k−1 < d)

= e−βl{∫ l

0e−βt p12(t)

P12(l)dt}k−1{∫ d

0e−βt p21(t)

P21(d)dt}k−1

.


Moreover, since

P(Bk ,UX2,1 < d , . . . ,UX

2,k−1 < d) = P12(l)k−1P12(l)P21(d)k−1

we get

E(

e−βηXl | τX

d > ηXl

)= E

(e−βη

Xl | UX

1,1 ≥ l)

P(UX1,1 ≥ l)

+∞∑

k=2

E(

e−βηXl | Bk ,UX

2,1 < d , . . . ,UX2,k−1 < d

)×P(UX

2,1 < d , . . . ,UX2,k−1 < d)


= e−βl∞∑

k=2

(∫ l

0e−βt p12(t)

P12(l)dt∫ d

0e−βt p21(t)

P21(d)dt)k−1

[P12(l)P21(d)]k−1P12(l) + e−βlP12(l)

= e−βlP12(l)

[1 +

∫ l0 e−βup12(u) du

∫ d0 e−βup21(u) du

1−∫ l

0 e−βup12(u) du∫ d

0 e−βup21(u) du

]

=e−βlP12(l)

1− P21(β)P12(β).


The functions P12(l), P21(β) and P12(β) are defined by (4),(5) and (6), respectively. Hence, the proof is completed.

CorollaryFor the process Xt with initial state x = 0 the followingrelation is true

P(ηX

l <∞ | ηXl < τX

d

)=

P12(l)1− P21(d)P12(l)

.

Obviously, this result follows from Theorem 1 by puttingβ = 0 in (3) and using the relations P12(0) = P12(l) andP21(0) = P21(d).


LemmaFor any positive d and l

P(ηX

l <∞ | ηXl < τX

d

)≥ P

(ηX

l <∞ | ηXl < τX)

where τX is the first time when the surplus reaches thelevel 0.

proof/ Due to equality P21(0) = 0, we getP(ηX

l <∞ | ηXl < τX

)= P12(l) by putting d = 0 in

expression of P(ηX

l <∞ | ηXl < τX

d

). Since

1− P21(d)P12(l) ≤ 1 the desired inequality holds.


LemmaThe following relation

P(ηX

l < τXd

)=

P12(l)1− P21(d)P12(l)

(7)

is true for the surplus Xt with x = 0.

To get the desired equality we represent the set {ηXl < τX

d }as a sum of disjoint sets and write P

(ηX

l < τXd

)in the form


P(UX1,1 ≥ l)

+∞∑

k=2

P(UX1,1< l , . . . ,UX

1,k−1< l ,UX1,k≥ l ,UX

2,1<d , . . . ,UX2,k−1<d)

= P12(l) +∞∑

k=2

P12(l)[P21(d)P12(l)]k−1

= P12(l)(

1 +P21(d)P12(l)

1− P21(d)P12(l)

)thus obtaining (7).


Sensitivity Analysis

TheoremFor Xt with x = 0 the following relation holds

E(

e−βτXd

)=

e−βd P21(d)P12(β)

1− P12(β)P21(β),

where P12(β) and P21(β) are given by (2) and (5)respectively,

P21(d) = 1−√

cα/λ∫ d

0e−(λ+cα)t t−1I1

(2t√

cλα)

dt ,

and I1(·) is a modified Bessel function of the first kind.


CorollaryFor Xt with X0 = x > 0 one has

h(λ, c, α, x ,d) = P(τX

d <∞)

=λ

cαe(λc −α)x cαP21(d)

cα− λP21(d).


Derivatives of the Ultimate Parisian Ruin Probability versus Scatterplots.

The simplest local measure of parameter importance isthe derivative of the model output with respect to thisparameter.The derivative with respect to parameter x has thefollowing form

∂

∂xh(λ, c, α, x ,d) =

(λ

c− α

)P(τX

d <∞).

Since we assumed the condition of net profit to befulfilled, that is, c > λ

α, and probability takes values on

interval [0,1], the derivative takes always negative values.


Thus, P(τX

d <∞)

decreases as x increases. Moreover,the Parisian ruin probability tends to zero when the initialcapital infinitely grows. It is easy to see that the secondderivative in x is positive, hence, the probability underconsideration is a convex function of this parameter.


Put A = λcαe(λc −α)x , B =

√cαλ

e−(λ+cα)d l−1I1(

2d√

cλα)

,then it is possible to obtain the following expression forderivative in d

∂

∂dh(λ, c, α, x ,d) = A

Bcα(λ− αc)

(cα− λP21(d))2 < 0,

establishing that it is negative, because B > 0,A > 0 andλ− αc < 0.It confirms the intuitive conclusion that the Parisian ruinprobability decreases if the length of negative excursionincreases.


Now turn to ∂∂λ

h(λ, c, α, x ,d) = 1cαe(λc −α)x cαP21(d)

cα−λP21(d)+

λxc2α

e(λc −α)x cαP21(d)cα−λP21(d)

+ λcαe(λc −α)x

−cαP′21λ(d)(cα−λP21(d))+(P21(d)+λP′21λ(d))cαP′21λ(d)

(cα−λP21(d))2 ,


with P ′21λ(d) having the form−(1/2)

√cαλ−3e−(λ+cα)t t−1I1

(2t√

cλα)

-√

cαλ−1e−(λ+cα)t I1(

2t√

cλα)

+cα√λ−1e−(λ+cα)t(

I0(

2t√

cλα)

+ I2(

2t√

cλα))

, where I0(t) and I2(t) aremodified Bessel functions of the first kind.


Take c = 10, α = 10, x = 10,d = 10, then varying λ sothat net profit condition is valid, we get the graphicsdepicted by Fig. 1.

Figure: Derivative in λ for c = 10, α = 10, x = 10,d = 10


Short-term credit

We consider a discrete-time insurance system (or otherorganization) which is interested in short-term credits (orbank loans).

It is supposed that at the beginning of each period (year,month or week) it is possible to apply to a bank in order toobtain a credit card valid for a fixed number of periods.

The card is provided immediately. The upper limit z of thecredit is chosen by the applicant who pays bank at oncethe amount cz where c is the interest rate.

The aim of loan is to satisfy the claims flow described by asequence of nonnegative i.i.d. random variables {ξn}n≥1.


We assume that each claim ξ has a known distributionfunction F (t) possessing a density ϕ(t) > 0 for t > 0 anda finite expectation.

If a claim amount ξ is larger than the cash amount uavailable for payment then another loan is obtained at theinterest rate p, p > c, its size is ξ − u.

The amount u − ξ, not used for payment before the cardexpiration term, is lost. Moreover, in this case the financialloss of applicant is equal to k(u − ξ).

Our aim is to determine the optimal n-period strategy ofapplicant. Optimality means the minimization of expecteddiscounted costs entailed by the n-step credit strategy.


One-period credit

Assume, at first, that the credit is valid for one period only.That means, the money not used for payment during theperiod cannot be used later.

Denote by fn(x) the minimal expected discounted costsincurred by the implementation of n-period credit strategy.Here x is the cash amount available initially for claimspayment if x > 0 and |x | is the debt amount if x < 0.

Put H1(y) = cy + L(y) withL(y) = p

∫∞y (s − y)ϕ(s) ds + k

∫ y0 (y − s)ϕ(s) ds and

y = x + z where z is the credit limit. Then the followingstatements are valid.


One-step case

Lemma

For any x,f1(x) = −cx + min

y≥xH1(y). (8)

If p > c then there exists the critical level x1 defined by therelation

F (x1) = (p − c)(p + k)−1 (9)

such that optimal credit limit is given byz1(x) = max(0, x1 − x). The function f1(x) is twicedifferentiable and convex, whereas

f ′1(x) =

{−c, x ≤ x1,L′(x), x ≥ x1.

(10)


Multi-step case

Let α ∈ (0,1] be the discount factor.

Theorem (1)

The function fn(x) specified by

fn(x) = −cx + miny≥x

Hn(y), (11)

where Hn(y) has the form

Hn(y) = H1(y)+αfn−1(0)F (y)+α

∫ ∞y

fn−1(y−s)ϕ(s) ds, (12)

is twice differentiable and convex for all n > 1. There existsx > x1 such that the optimal credit limit zn(x) = max(0, x − x)for any x and n > 1. The critical level x is defined by therelation F (x) = (p − c(1− α))(p + k + αc)−1.


Dependence of optimal levels on cost parameters

Thus, we have established that for any n > 1 the optimalcredit strategy is determined by a single critical number x ,whereas for n = 1 one has to use x1 < x instead of x . It isnot difficult to prove the following results clarifying thedependence of critical levels on the cost parameters.

Corollary

Critical level x1 is increasing function of p and decreasingfunction of c and k, whereas x increases in p and α anddecreases in c and k.


Asymptotic analysis

First of all we establish the limit behavior of the minimalcosts as the planning horizon tends to infinity.

Theorem (2)If α < 1 the functions fn(x) defined in Theorem convergeuniformly, as n→∞, to a function f (x) satisfying thefollowing equation

f (x) = −cx+miny≥x

[H1(y)+αf (0)F (y)+α

∫ ∞y

f (y−s)ϕ(s) ds].

(13)


System stability

Next, suppose that there exist two different claim distributions,namely, F and G, and we would like to estimate the differencebetween the corresponding optimal cost functions.

Further on, the index F (or G) will be added to all the functionsarising in our study under assumption that claim distribution isF (resp. G). Thus, we are going to deal with xF = F−1(b)where F−1 is inverse to F , that is, F−1(t) = inf{x : F (x) > t}and b = (p − c(1− α))(p + k + αc)−1. Similarly we define xG.

It is clear that LF (y) := L(y), HFk (y) := Hk (y) and

f Fk (x) := fk (x), k ≥ 1, since previously we had only onedistribution F . Definition of analogous functions with index G isobvious, namely, in all the formulas we write G instead of F .


System stability

We need also to introduce some probability metrics.Thus, Kantorovich (Wasserstein) metric is defined as

κ(F ,G) =

∫ ∞−∞|F (x)−G(x)|dx .

Kolmogorov (uniform) metric is given by

γ(F ,G) = sup−∞<t<+∞

|F (t)−G(t)|.


System stability with respect to distributions perturbation

The main result concerning the system stability is givenbelow.

Theorem (3)Let distributions F and G be such that κ(F ,G) < ε then,for any n ≥ 1 and α ∈ (0,1),

γn = sup−∞<x<∞

|f Fn (x)− f G

n (x)| < Dε

where D = (max(k ,p) + αc)(1− α)−1.


Asymptotic optimality

Turning to the case α = 1 we note that minimal n-step coststend to∞, as n→∞, for any initial capital x and it isimpossible to establish the estimate (not depending on n) forthe difference between costs corresponding to claimdistributions F and G. However we can employ anotherobjective function, namely, long-run average costs per period.Furthermore, we introduce the following

DefinitionA policy y(x) = {yn(x),n ≥ 1} is asymptotically optimal if

limn→∞

1n

fn(x) = limn→∞

1n

fn(x)

where fn(x) are the costs obtained by applying the policy y(x)and fn(x) are the minimal n-step costs.


Asymptotic optimality

Theorem (4)

The policy y(x) with yn(x) = max(x , x) for all n ≥ 1 isasymptotically optimal. Moreover,

limn→∞

1n

fn(x) = cxF (x) + c∫ ∞

xs dF (s) + L(x).

Proof is carried out in two steps. Step 1. We establish that

1n

(fn(x)− fn(x))→ 0, as n→∞. (14)


Step 2. Now we have to obtain the explicit form oflimn→∞

1n fn(x).

To this end it is necessary to calculateHn(x) = H1(x) + fn−1(0)F (x) +

∫∞x fn−1(x − s) dF (s).

Since, for x ≤ x , we have fn−1(x) = Hn−1(x)− cx , it is obviousthat

Hn(x) = D(x) + Hn−1(x) = . . . = (n − 1)D(x) + H1(x) (15)

with D(x) = L(x) + cxF (x) + c∫∞

x s dF (s).


Remark

It is possible to strengthen the result of Theorem 4establishing that under the asymptotically optimal policynot only expected average costs tend to limit D(x) but(random) average costs converge with probability one tothe same limit.

For this purpose one has to use Wald’s identity, the stronglaw of large numbers and other properties of renewalprocesses.


New models were developed in actuarial sciences duringthe last two decades. They include different notions ofinsurance company ruin (bankruptcy) and other objectivefunctions evaluating the company performance. Severaltypes of decision (such as dividends payment,reinsurance, investment) are used for optimization ofcompany functioning. Therefore it is necessary to be surethat the model under consideration is stable with respectto parameters fluctuation and perturbation of underlyingstochastic processes. The aim of the talk is description ofmethods for investigation of these problems andpresentation of recent results concerning some insurancemodels.


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