2000 Mathematics Subject Classification: 60K10, 62P05We consider the risk model in which the claim counting process {N(t)} is a modified stationary renewal process. {N(t)} is governed by a sequence of independent and identically distributed inter-occurrence times with a common exponential distribution function with mass at zero equal to ρ>0. The model is called a Polya-Aeppli risk model. The Cramer-Lundberg
approximation and the martingale approach of the model are given.This paper is partially supported by Bulgarian NFSI grant MM-1103/2001

Minkova, Leda D.

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Pliska Stud. Math. Bulgar. 16 (2004), 129-135STUDIA MATHEMATICABULGARICAA MODIFIED MODEL OF RISK BUSINESSLeda D. Minkova1We consider the risk model in which the claim counting process {N(t)} isa modified stationary renewal process. {N(t)} is governed by a sequence ofindependent and identically distributed inter-occurrence times with a com-mon exponential distribution function with mass at zero equal to ρ > 0.The model is called a Po´lya - Aeppli risk model. The Crame´r - Lundbergapproximation and the martingale approach of the model are given.1. IntroductionAssume that the standard model of an insurance company, called risk process{X(t), t ≥ 0} is given byX(t) = ct−N(t)∑k=1Zk, (0∑1= 0).(1)Here c is a positive real constant representing the risk premium rate. Thesequence {Zk}∞k=1 of mutually independent and identically distributed randomvariables with common distribution function F , F (0) = 0, and finite mean valueµ is independent of the counting process N(t), t ≥ 0. The process N(t) isinterpreted as the number of claims on the company during the interval [0, t]. In1This paper is partially supported by Bulgarian NFSI grant MM-1103/2001.2000 Mathematics Subject Classification: 60K10, 62P05Key words: Po´lya - Aeppli risk model, ruin probability, Crame´r - Lundberg approximation130 L. Minkovathe classical risk model the process N(t) is a homogeneous Poisson process, seefor instance [1] and [6].The Po´lya - Aeppli distribution is a generalization of the classical Po(λ)distribution, by adding a new parameter ρ, see [2]. It appears in [4] and [5] as acompound Poisson distribution. The additional parameter ρ is called an inflationparameter. The Po´lya - Aeppli process as a generalization of the Poisson processis defined in [3].We will suppose that N(t) is described by the Po´lya - Aeppli distributionwith mean function λ1−ρ t, i.e.P (N(t) = n) ={e−λt, n = 0e−λt∑ni=1(n−1i−1) [λ(1−ρ)t]ii! ρn−i, n = 1, 2, . . . .(2)In this section we will discus briefly the basic properties of the Po´lya - Aeppliprocess.In [3] is proved that the non-negative random variables T1, T2, . . . , represent-ing the inter- arrival times are mutually independent. The time T1 until the firstepoch is exponentially distributed with parameter λ. T2, T3, . . . are identicallydistributed as a random variable T2. Moreover T2 is zero with probability ρ, andwith probability 1 − ρ exponentially distributed with parameter λ, i.e. T2 is ex-ponentially distributed with mass at zero equal to ρ. The probability distributionfunction is given byFT2(t) = 1− (1− ρ)e−λt, t ≥ 0.The mean value is ET2 =1−ρλ.The process, described above is a delayed renewal process. It is easy to verifythat the probability distribution functions of the delay T1 and the inter - arrivaltimes satisfy the following relationFT1 =1ET2∫ t0[1− FT2(u)]du.In this case the delayed renewal process is the only stationary renewal process.So, the Po´lya - Aeppli process is a homogeneous process and if ρ = 0 it becomesa homogeneous Poisson process.In this paper we need also the probability generating function (pgf) of thePo´lya - Aeppli process. It is given byPN(t)(s) = e−λt 1−s1−ρs .A modified model of risk business 131The Po´lya - Aeppli risk model is defined in [3]. The probability of ruin andthe Crame´r - Lundberg approximation are derived. In this paper the martingaleapproach is given.2. The Po´lya - Aeppli risk modelWe consider the risk process X(t), defined by (1), where N(t) is the Po´lya -Aeppli process, independent of the claim sizes Zk, k = 1, 2, . . . . This process iscalled a Po´lya - Aeppli risk model.The relative safety loading θ is defined byθ =c(1− ρ)− λµλµ=c(1− ρ)λµ− 1,(3)and in the case of positive safety loading θ > 0, c > λµ1−ρ .LetFI(x) =1µ∫ x0[1− F (z)]dzbe the integrated tail distribution of the claim size. Let us define the functionH(z) = ρF (z) +λµcFI(z)(4)and note thatH(∞) = ρF (∞) +λc∫ ∞0[1− F (z)]dz = ρ +λµc< 1.Denote byτ(u) = inf{t > 0, u + X(t) ≤ 0}the time to ruin of a company having initial capital u. We let τ = ∞, if for allt > 0 u + X(t) > 0.The probability of ruin in the infinite horizon case isΨ(u) = P (τ(u) < ∞)and in the finite horizon caseΨ(u, t) = P (τ(u) ≤ t).Assume that there exists a constant R > 0 such that∫ ∞0eRzdH(z) = 1,(5)132 L. Minkovawhere H(z) is given by (4), and denote h(R) =∫∞0 eRzdF (z)− 1.The following proposition, proved in [3] is an analogue of the correspondingapproach to the classical risk model.Proposition 1. Let, for the Po´lya-Aeppli risk model, the Crame´r condition(5) holds and h′(R) < ∞. Thenlimu→∞Ψ(u)eRu =µθA2(µ, θ,R, ρ)h′(R)− µ(1 + θ),(6)whereA(µ, θ,R, ρ) =1− [1− µ(1 + θ)R]ρ1− ρ.The relation (5) is known as the Crame´r condition and the constant R, if itexists, as the adjustment coefficient or Lundberg exponent for the Po´lya-Aepplirisk model.If ρ = 0, A(µ, θ,R, 0) = 1 and (6) coincides with the Crame´r - Lundbergapproximation for the classical risk model [1].3. Martingales for the Po´lya-Aeppli risk modelLet us denote by (FXt ) the natural filtration generated by any stochastic processX(t). (FXt ) is the smallest complete filtration to which X(t) is adapted.Let us denote by LSZ(r) =∫∞0 e−rxdF (x) the Laplace-Stieltjes transform(LS-transform) of any random variable Z with distribution function F (x).Lemma 1. For the Po´lya-Aeppli risk modelEe−rX(t) = eg(r)t,whereg(r) =11− ρLSZ(−r)[ρcrLSZ(−r) + λ(LSZ(−r)− 1)− cr].P r o o f. Let us consider the random sum from the right hand side of (1)St =N(t)∑k=1Zk,A modified model of risk business 133where N(t) is a Po´lya-Aeppli process, independent of Zk, k = 1, 2, . . . . St is acompound Po´lya-Aeppli process and the LS- transform is given byLSSt(r) = PN(t)(LSZ(r)) = e−λt1−LS(r)1−ρLS(r) .For the LS-transform of X(t) we have the followingLSX(t)(r) = Ee−rX(t) = Ee−r[ct−St] = e−rctEerSt == e−rctPN(t)(LSZ(−r)) = e−rcte−λt1−LSZ(−r)1−ρLSZ(−r) = eg(r)t.From the martingale theory we get the followingLemma 2. For all r ∈ R the processMt = e−rX(t)−g(r)t, t ≥ 0is an FXt -martingale, provided that LSZ(−r) < ∞.4. Martingale approach to the Po´lya-Aeppli risk modelUsing the martingale properties of Mt, we will give some useful inequalities forthe ruin probability.Proposition 2. Let r > 0. For the ruin probabilities of the Po´lya-Aeppli riskmodel we have the following resultsi) Ψ(u, t) ≤ e−ru sup0≤s≤t eg(r)s, 0 ≤ t < ∞ii) Ψ(u) ≤ e−ru sups≥0 eg(r)s.iii) If the Lundberg exponent R exists, then R is the unique strictly positivesolution ofρcrLSZ(−r) + λ(LSZ(−r)− 1)− cr = 0(7)and Ψ(u) ≤ e−Ru.P r o o f. i) For any t0 < ∞, the martingale stopping time theorem yields thefollowing1 = M0 = EMt0∧τ = E[Mt0∧τ , τ ≤ t] + E]Mt0∧τ , τ > t] ≥≥ E[Mt0∧τ , τ ≤ t] = E[e−rX(τ)−g(r)τ |τ ≤ t]P (τ ≤ t),134 L. Minkovafrom whichP (τ ≤ t) =e−ruE[e−g(r)τ |τ ≤ t].The statement i) follows from the above relation.ii) follows immediately from i) when t →∞.iii) The Crame´r condition (5) becomesρ∫ ∞0erxdF (x) +λc∫ ∞0erx(1− F (x))dx = 1.Using∫ ∞0erx(1−F (x))dx =∫ ∞0∫ ∞xerxdF (y)dx =∫ ∞0∫ y0erxdxdF (y) =1r[LSZ(−r)−1],it can be written asρLSZ(−r) +λcr(LSZ(−r)− 1) = 1.This is equivalent to the equation (7).Let us denotef(r) = ρcrLSZ(−r) + λ(LSZ(−r)− 1)− cr.So R is a positive solution of f(r) = 0. Because f(0) = 0, f ′(0) = λµ− (1 −ρ)c < 0 and f ′′(r) = (ρcr + λ)LS′′Z(−r) + 2ρcLS′Z(−r) > 0 there is at most onestrictly positive solution. Remark 1. The equation (7) is equivalent to g(r) = 0.Remark 2. The above inequalities are well known for the classical risk mo-del. In the case of ρ = 0 the Po´lya - Aeppli risk model becomes the classical riskmodel. The Crame´r condition (5) and the function g(r) are the same, see [7].REFERE NCES[1] J. Grandell, Aspects of Risk Theory. Springer - Verlag, New York, 1991.[2] N.L. Johnson, S. Kotz, A.W. Kemp, Univariate Discrete Distributions,2nd ed., Wiley Series in Probability and Mathematical Statistics, 1992.A modified model of risk business 135[3] L. Minkova, The Po´lya-Aeppli process and ruin problems, RT-MAE 9926,Sa˜o Paulo, 1999.[4] L. Minkova, A Family of Compound Discrete Distributions, J. Bulg. Acad.Sci. 54(2) (2001), 11–14.[5] L. Minkova, A generalization of the classical discrete distributions, Comm.Statist. Theory and Methods 31(6) (2002), 871–888.[6] T. Rolski, H. Schmidli, V. Schmidt, J. Teugels, Stochastic Processesfor Insurance and Finance,John Wiley & Sons, Chichester, 1999.[7] H. Schmidli Martingales and Insurance Risk, 8th International SummerSchool on Probability Theory and Mathematical Statistics (1996), 155–188.Leda D. MinkovaFaculty of Mathematics and InformaticsSofia University, 5 J. Boucher Str.,1407 Sofia, Bulgariae-mail: leda@fmi.uni-sofia.bg

A Modified Model of Risk Business

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A Modified Model of Risk Business

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