Ruin Probability for Non-standard Poisson Risk Model with Stochastic Returns

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1 Ruin Probability for Non-standard Poisson Risk Model with Stochastic Returns Tao Jiang Abstract This paper investigates the finite time ruin probability in non-homogeneous Poisson risk model, conditional Poisson risk models and renewal risk model with stochastic returns. Under the assumption that the claimsize is subexponentially distributed, a simple asymptotic relation is established when the initial capital tends to infinity. The results obtained extend the corresponding results of constant interest force. Key Words: Keywords: ruin probability, conditional Poisson process, renewal risk model, non-homogeneous Poisson process, subexponential classy, regularly varying tail 1 The model We consider a Sparre Anderson model perturbed by a diffusion. In this model the claim sizes, X n, n 1, 2,...,constitute a sequence of independent, identically distributed i.i.d. and non-negative random variables r.v. s with common distribution function d.f. F 1 F. The claim arrival times, σ n,n1, 2,..., form a renewal counting process Nt max{n 1:σ n t},t>, 1 with a constant intensity λ, where,maxφ byconvention. The total surplus of a company up to time t, with perturbed term σ W t, is denoted by Ut, which satisfies the following equation: Nt Ut u + ct X k + σ B s, 2 k1 where, u> is the initial capital, c> is the constant rate of premium, {B t,t } is a standard Brownian motion and σ > is the volatility coefficient of σ B t. If the inter-arrival times σ 1,σ n σ n 1 for n 2, 3,... have a common exponential distribution, then the model above is called C-L model. IfinthecaseofC-Lmodel,parameterλ is time dependent, then {Nt,t } is called non-homogeneous with This work was supported by the National Natural Science Foundation of China Grant No and Grant No and the planning project of National Educational Bureau of China Grant No. 8JA6378. Manuscript submitted March 15, 29. Postal address: School of Finance Zhejiang Gongshang University Hangzhou 3118, Zhejiang, P. R. China. jtao@263.nettao Jiang intensity function {λt,t }. t, s >, Nt satisfies that P Nt + s Ns k If for arbitrarily fixed λt λtn e G Λ dλ, n! where Λ is a r.v. with d.f. G Λ. Then {Nt,t } is called conditional Poisson process. All limit relationships in this paper, unless otherwise stated, are for u. A B and A > B respectively mean that lim u A B 1 and lim u A B Stochastic Returns If an insurer invests his capital in a risky asset, then its capital value should be specified by a geometric Brownian motion dv t V t μdt + σdbt, 3 where {Bt,t } is a standard Brownian motion and r,σ are respectively called expected rate of return and volatility coefficient. It is well known that stochastic equation 3 has the following solution V t V e μ 1 2 σ2 t+σbt. Therefore, at time t, the surplus with risky investment couldbeexpressedas Ut e Δt u + where, Δt βt + σbt, β μ σ 2 /2. e Δs dus, 4 Through out, {X n,n 1}, {Nt,t }, {Bt,t } and {B t,t } are assumed to be mutually independent. We define ψu, T P inf Us < U u, s T the finite time ruin probability within time T.IfT, we say that ψu, is ultimate ruin probability. This concept illustrates the possibility that the surplus process moves below zero. Under the assumption that the risk models are nonhomogenous, conditional Poisson process and renewal risk model respectively, In this paper will derive some asymptotics of finite time ruin probabilities with stochastic returns. ISBN: WCE 29

2 1.2 Some Related Results Heavy-tailed risk has played an important role in insurance and finance because it can describe large claims; see Embrechts et al. see [6] for a nice review. We give here several important classes of heavy-tailed distributions for further references: class L Long-tailed : a d.f. F belongs to L iff F x + t lim 1 x F x for any t or equivalently, for t 1; class R α : a distribution F belongs to R α iff F x x α Lx, x >, where Lx is a slowly varying function as x and index α <. R α is called regularly varying function class, or Pareto-like function class with index α. class S Subexponential: a d.f. F belongs to S iff F lim n x n x F x for any n or equivalently, for n 2;whereF n denotes the n-fold convolution of F, with convention that F is a d.f. degenerate at. These heavy-tailed classes satisfies R α S Lsee Embrechts et al. [6]. The asymptotic behavior of the ultimate ruin probability ψu is an important topic in risk theory. In the recent literature, the asymptotic behavior of the ruin probability with constant interest force has been extensively investigated. One of the interesting results was obtained by Klüppelberg and Stadtmuller[12], who used a very complicated L p transform method, proved that, in the Cramér-Lundberg risk model, if the claimsize is of regularly varying with index α, then ψu λ F u, 5 αr where r is constant interest force. Asmussen [1] and Asmussen et al. [2] obtained a more general result: ψu λ r u F y dy, 6 y where the claimsize is assumed to be in S,animportant subclass of S. In the case of compound Poisson model with constant interest force and without diffusion term, Tang [16] obtained the asymptotic formula of finite time ruin probability for sub-exponential claims. Tang [17] proved that, in the renewal risk model with constant interest force, if the d.f. of claimsize belongs to regularly varying class with index α, then ultimate ruin probability satisfies that ψu Ee rαθ1 F u, rαθ1 1 Ee which extends 5 essentially. Jiang [1] extended some results to the risky case. See also Jiang [8], [9]. Dufresne and Gerber [4] first researched the ruin probability for jump-diffusion Poisson process. Veraverbeke [2] discussed the asymptotic behavior of ruin with diffusion term. The rest of this paper is organized as follows: In Section 2, main results of this paper are presented. In Section 3, after some necessary lemmas are supplied, the proofs of the main results are completed. 2 Main Results The following theorems are main results of this paper: Theorem 1. Consider non-homogenous Poisson model introduced in Section 1. If F R α, then it holds that ψu; T F u λse αβ 1 2 α2 σ 2 s ds. 7 Notes and Comments. When F R α and the perturbed term disappears, the results of Tang [1] is consistent with this Theorem. In particular, this result is also in consistence with that of Veraverbeke [2], who pointed out that the diffusion term W t does not influence the asymptotic behavior of the ruin probability. We should point out that the diffusion term W t playsan essential role in influencing the interest force. Theorem 2. Consider conditional Poisson process introduced in Section 1. If F R α,then ψu; T F ueλ e αβ 1 2 α2 σ 2 s ds. 8 Notes and Comments. In these two Theorems, if parameter λ is a constant and perturbed term disappears, then 7 and 8 turn to the following: ψu; T λ αr F u1 e αrt, 9 which is in consistence with the result of Klüppelberg and Stadtmuller [12]. Theorem 3. In the renewal model with surplus process 4. Denote q Ee αβ 1 2 α2 σ 2 θ 1. ISBN: WCE 29

3 If F S and ms is the renewal function of Ns, then ψu; T If F R α, then 1 tends to P X 1 e βs σbs udms. 1 ψu; T F u e αβ 1 2 α2 σ 2 s dms. 11 If we denote ψ k u as the ruin probability when ruin happens not later than kth claim, then in the renewal case, we can obtain the following Theorem: Theorem 4. In the renewal model with surplus 4. If F R α,then ψ k u F u q qk Notes and Comments. From Theorem 5, we can get the main result of Tang 25a easily. 3 Proofs of the Main Results 3.1 Several Lemmas The following lemma is well known Ross [19]: Lemma 1. Let {Nt} t be a Poisson process with arrival times {σ k,k 1}. Given NT n for any fixed T>, the random vector σ 1,σ 2,..., σ n is equal in distribution to the random vector TU 1,n,..., T U n,n,where U1,n,..., U n,n are the order statistics of n i.i.d., 1 uniformly distributed random variables U 1,..., U n. The following lemma can be found in many standard textbooks on stochastic process, see, for example, Karatzas and Shreve [11]. Lemma 2. If Bt is a standard Brownian Motion, then the moment of any order of max t T Bt exists. The following Lemma can be found in Tang [15]: Lemma 3. Let {X i, 1 i n} be n i.i.d. subexponential r.v.s, with common distribution F. Then for any fixed <a b<, uniformly for all a c i b, 1 i n n n P c i X i >u P c i X i >u. 3.2 Proofs of Main Results Proof of Theorem 1. By the definition of ruin probability, we have ψu; T P e Δt Ut < for some T t> U u. 13 For each t,t], we have Nt u X i e Δσi + σ e Δs db s e Δt Ut Nt u + c e Δs ds X i e Δσi + σ e Δs db s. 14 Without essential difficulty, one can see that ψu; T satisfies that NT ψu; T P X i e Δσi u + c ξ + ξη 15 β and NT ψu; T P X i e Δσi u + ξη, 16 where ξ e σ max s T Bs and η σ max e βs db s. t T From Ross [19], Nt with intensity function λs can be regarded as a random sampling of some homogenous Poisson process Nt with constant parameter λ, where λs λ. Now we introduce the indicator function f event A i,ia i. We say that A i happens, if at time σ i, with probability λσ i /λ, X i is picked out. From Lemma 3 NT P X i e Δσi u NT P X i e Δσi IA i u P X i e Δσi IA i Iσ i T u F u F u F u E[e αδs λs/λ]df σi s E[e αδs λs/λ]dms λse αβ 1 2 α2 σ 2 s ds, 17 ISBN: WCE 29

4 where, we have used the fact that, the renewal function of Poisson process mt, is just λt. For any fixed ε> NT P X i e Δσi u + c β ξ + ξη NT P X i e Δσi 1 + εu P c ξ + ξη εu β NT P X i e Δσi 1 + εu Eξτ Eη + c β τ εu τ, 18 where we have used Markov inequality. Lemma 2 implies that Eξ τ <. Choosingτ> such that Eξ τ Eη + c β τ εu τ is the higher order infinitesimal of By 17 NT P X i e Δσi 1 + εu. NT P X i e Δσi 1 + εu F u λse αβ 1 2 α2 σ 2 s ds 1 + ε α. 19 By the arbitrariness of ε, we obtain that > NT P X i e Δσi u + c β ξ + ξη F u On the other hand Similarly NT P X i e Δσi u + ξη λse αβ 1 2 α2 σ 2 s ds. 2 NT P X i e Δσi 1 εu+p ξη εu NT P X i e Δσi 1 εu+ Eξτ Eη τ εu τ. 21 < NT P X i e Δσi u + ξη F u λse αβ 1 2 α2 σ 2 s ds. 22 Notes and Comments. max t T max t 1 e 2rT 2r Rewrite e rs dbs 1 2r ln1 2rt e rs dbs. 23 Denote 1 2r ln1 2rt e rs dbs by Mt. We aim to prove that Mt is Brownian motion. From Fima [7], we only need to prove that, the quadratic variation process, [M,M]t, equals to t, because Mt isalocalmartingale. Using the definition of the quadratic variation, we have [M,M]t 1 2r ln1 2rt e 2rs ds t. Hence, the m.g.f. of max t T e rs dbs existsandeη τ exists. It is not difficult to check that the result of Klüppelberg and Stadtmuller [12] is the special case of Theorem 1 if F R α, λ is some constant and T. Proof of Theorem 2. Similar to the proof of Theorem 1, we have NT P X i e Δσi u n P X i e Δσi u NT n, Λλ n1 P NT n Λ λgdλ n P X i e ΔTUi u n1 P NT n Λ λgdλ P X 1 e ΔTU1 u np NT n Λ λgdλ EΛF u n1 wherewehaveusedlemma2. e αβ 1 2 α2 σ 2 s ds, 24 Proof of Theorem 3. We only consider Because while P NT P X i e Δσi u. NT P X i e Δσi u k1 k P X i e Δσi u, NT k, 25 k X i e Δσi u, NT k ISBN: WCE 29

5 hence NT k NT k P k X i e Δσi udf σ 1,..., σ k+1 E[E[P k X i e Δσi u Bσ 1,..., Bσ k ]]df σ 1,..., σ k+1 k E[E[ P X i e Δσi u NT k Bσ 1,..., Bσ k ]]df σ 1,..., σ k+1 k P X i e Δσi udf σ 1,..., σ k+1 NT k k P X i e Δσi u, NT k, 26 NT P X i e Δσi u k1 k P X i e Δσi u, NT k k1 ki k P X i e Δσi u, NT k P X i e Δσi u, NT k P X i e Δσi u, NT i P X i e Δσi u, σ i T Hence Theorem 3 is completed. P X i e Δs udf σi s P X i e Δs udms. 27 Proof of Theorem 4. We deal with the proof by induction. For k 1 ψ 1 u P u + θ1 F θ1 ds e Δy dy X 1 e Δθ1 < P X 1 >ue βs+σt + s e βs+σt e Δy 1 dy e t2 2πs F u e ασs F θ1 ds 2s dt 1 2π e v2 2 e ασv s dv F uq, 28 we have used the fact that X 1 is in L class. Assume that ψ n u F u q qn+1, 29 we should prove that 29 holds for n +1. Denote θ1 ue Δθ1 + c e Δθ1 Δy dy by V θ 1,u. With total probability formula ψ n+1 u ψ 1 u+ψ 1 ue[ψ n V X 1,uIV X 1 ] ψ 1 u+ q qn+1 F θ1 ds Φ,s dt e αrs ασt F X1 dyf u q + q qn+1 e αμs F θ1 ds e 1 2 α2 σ 2s Φ,1 dtf u q qn+2 F u, 3 so by using induced assumption, Theorem 4 is finished. We can see that, when n, this Theorem turns to ψ q F u, which contains the result of Tang [16] as a special case. Acknowledgement: This work was supported by the National Natural Science Foundation of China Grant No and Grant No and the planning project of National Educational Bureau of ChinaGrant No. 8JA6378. References [1] Asmussen, S.: Subexponential asymptotics for stochastic processes: extremal behavior, statonary distribution and first passage probabilities. The Annals of Appl. Prob. 8, [2] Asmussen, S., Kalashnikov, V., Konstantinides, D., Klüppelberg, C., Tsitsiashvili, G.: A local limit theorem for random walk maxima with heavy tails. Statist. Probab. Lett. 56, no. 4, [3] Cai, J.: Discrete time risk models under rates of interest. Prob. Eng. Inf. Sci. 16: [4] Dufrense F., and Gerber, H.: Risk theory for the compound Poisson process that is perturbed by diffusion[j]. Insurance: Mathematics Economics, 1: ISBN: WCE 29

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