Asymptotics of discounted aggregate claims for renewal risk model with risky investment

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1 Appl. Math. J. Chinese Univ. 21, 25(2: Asymptotics of discounted aggregate claims for renewal risk model with risky investment JIANG Tao Abstract. Under the assumption that the claim size is subexponentially distributed and the insurance surplus is totally invested in risky asset, a simple asymptotic relation of tail probability of discounted aggregate claims for renewal risk model within finite horizon is obtained. The result extends the corresponding conclusions of related references. 1 Introduction Consider a renewal risk model, in which the claims X n for n 1 form a sequence of independent identically distributed(i.i.d. non-negative random variables(r.v.s with a common distribution function(d.f. F(x 1 F(x P(X x for x [, and finite mean µ EX 1. The inter-occurrence times θ n for n 1 form another sequence of i.i.d. nonnegative r.v.s with mean Eθ 1 1/λ. The random variables σ k k θ i for k 1, 2,... constitute a renewal counting process N(t sup {n 1, 2,... : σ n t} (1.1 with mean λ(t EN(t. By convention, the cardinality of the empty set is. It becomes a compound Poisson model when θ n is exponentially distributed. The total amount of claims accumulated by time t is represented as a compound sum S(t N(t X k, t, (1.2 where the summation over an empty index set is considered to be. Let x > be the initial surplus and let c > be the rate at which the premium is collected. The total surplus up to time t, denoted by U(t, can be expressed as U(t x + ct S(t. (1.3 Received: MR Subject Classification: 91B3, 6G7, 62P5. Keywords: Discounted aggregate claims, ruin probability within finite horizon, renewal risk model, risky investment, subexponential class. Digital Object Identifier(DOI: 1.17/s Supported by the National Natural Science Foundation of China(787114, the Planning Project of the National Educational Bureau of China(8JA6378 and the Project of Key Research Base of Human and Social Sciences (Finance for Colleges in Zhejiang Province(Grant No. of Academic Education of Zhejiang [28]255.

2 21 Appl. Math. J. Chinese Univ. Vol. 25, No. 2 If an insurance company invests its capital totally in risky asset (for example, stock, its dynamic capital should be specified by a geometric Brownian motion dv t V t (rdt + σdb(t (1.4 where {B(t, t } is a standard Brownian motion, r and σ are expected rate of return and volatility coefficient respectively. By Klebaner[1], (1.4 has the solution V t V e σb(t+βt in which β r σ 2 /2. Therefore, D(t, the present value of discounted aggregate claims from time t, can be expressed as D(t N(t X i e σb(σi βσi (1.5 where {X n, n 1}, {N(t, t } and {B(t, t } are assumed to be mutually independent. In modern actuarial risk theory, some parameters at ruin are both of theoretical interest and of practical value, for examples, the present value of discounted sum of dividend payments until ruin occurs, the deficit at ruin, and the severity of ruin. See, for instance, Gerber[7], Lin and Willmot[15], Lin et al.[16] and other references therein. On the other hand, in the practice of insurance industry, a chief risk officer is quite interested in the present value of discounted aggregate claims at ruin. It is well known that the pricing of premium depends heavily on it. Hence the study of D(t is very significant. If σ, we say that all the capital is non-risky. Then (1.3 becomes U(t xe rt + c t N(t e r(t s ds X k e r(t σk, (1.6 which corresponds to the case of constant interest force. See, for instance, Tang[17], Jiang and Yan[9], and Konstantinides et al.[14]. In modelling extremal event, heavy-tailed risk has played an important role in insurance and finance, because it can describe large claims efficiently; see Embrechts et al.[5], Kong[12,13] and Goldie & Klüppelberg[8] for nice reviews. We give here several important classes of heavy-tailed distributions for further references: (1 Class R α : A distribution F belongs to R α if F(x x α L(x, x >, where L(x is a slowly varying function as x and the index α <. The class R α is called the class of regularly varying functions, or the class of Pareto-like function class with index α. (2 Class ERV ( α, β: A distribution F belongs to ERV ( α, β if for some < α β < and for any y > 1, y β F(uy liminf u F(u limsup F(uy u F(u y α. (3 Class L (Long-tailed: A distribution F belongs to L if for any t (or equivalently, for t 1, F(x + t lim 1. x F(x

3 JIANG Tao. Asymptotics of discounted aggregate claims for renewal risk model with risky investment 211 (4 Class D: A distribution F belongs to D if for any fixed < y < 1 (or, equivalently, for y 1/2, F(xy lim sup x F(x <. (5 Class S (Subexponential: A distribution F belongs to S if for any n (or equivalently, for n 2, F lim n (x n. x F(x Here F n is the n-fold convolution of F, with the convention that F is a d.f. degenerate at. These heavy-tailed classes have the following inclusions[7]: R α ERV ( α, β L D S L. (1.7 All limit processes in this paper, unless otherwise stated, are for x. We write A B for the fact that lim x A(x/B(x 1. We present the main result(theorem 2.1 and some related lemmas in Section 2. Then we give the proof of Theorem 2.1 in Section 3. 2 Main result and some lemmas The following theorem is the main result of this paper. Theorem 2.1 In the renewal risk model introduced in Section 1, if F L D, then the discounted aggregate claims up to time T satisfies P(D(T > x P(X 1 e βs σb(s xdm(s, (2.1 where m(s is the renewal function of the process, i.e., m(s EN(s and β r σ 2 /2. If T, then P(D( > x P(X 1 e βs σb(s xdm(s. (2.2 Remark. This result is closely related to the ruin probability ψ(x, T within time T with ψ(x, T P( inf U(t < U( x tt P(U(t < for some T t. (2.3 Therefore, ψ(x, is called the ultimate ruin probability. The asymptotic behavior of the ultimate ruin probability is an important topic in risk theory. Many results have been obtained, see, for instance, Embrechts and Veraverbeke[12], Klüppelberg and Stadtmuller[13], Asmussen[14], Asmussen et al.[15], Tang[16,17], Jiang and Yan[6], and Bi and Yin[18]. Because of the close relationship between the asymptotic formula of P(D(T > x and the asymptotic formula of ψ(x; T (in fact, in some cases P(D(T > x ψ(x; T, Tang[19] investigated the asymptotic formula of P(D(T > x for the ERV class with constant interest force. He established a simple but nice uniformity relationship.

4 212 Appl. Math. J. Chinese Univ. Vol. 25, No. 2 Theorem 2.1 generalizes some corresponding results in the literature. For example, in the Poisson case with σ, (2.2 becomes xe rt P(D(T > x λ F(y dy, (2.4 r x y which is consistence with the result obtained by Asmussen et al.[15] and Jiang and Yan[6]. Similarly, when F R α and the perturbed term disappears, the result of Tang[2] is consistence with (2.2, i.e., Ee rαθ1 P(D( > x F(x. (2.5 rαθ1 1 Ee In this paper, we extend these results to the case with risky investment and to an important subclass L D of the subexponential distribution class S, which is much larger than the ERV class. To prove Theorem 2.1, we need some lemmas. Lemma 2.1.[8] If F is subexponential, the tail of its n-fold convolution is bounded by the tail of F in the following way. For any ε >, there exists A(ε > such that uniformly for all n 1 and all x, we have F n (x A(ε(1 + ε n F(x. (2.6 Lemma 2.2.[17] 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 we have ( n n P c i X i > x P(c i X i > x for all a c i b with 1 i n. Lemma 2.3.[21] Let X and Y be two independent random variables with distributions F and G and let H be the distribution of the product X and Y. (1 Suppose F L and G does not degenerate to zero. If for any fixed a >, it is true that then XY L. (2 If F D and P(Y > >, then H D. G(x/a lim, x H(x (3 If X S and Y is bounded and does not degenerate to zero, then H S. 3 Proof of Theorem 2.1 Let (t βt σb(t. From Lemma 2.1, we know that there exists a constant C(ε > such that P X i e (t x C(ε(1 + ε k P(X 1 e (t x (3.1

5 JIANG Tao. Asymptotics of discounted aggregate claims for renewal risk model with risky investment 213 holds for all k 1, 2,..., t and x. Notice that ( N(T P X i e (σi x P X i e (σi x, N(T k ( N + kn +1 P X i e (σi x, N(T k. (3.2 Let H(t be the distribution of max tst ( (s for some fixed number t. For any arbitrarily fixed ε > and all integer k N, by Lemma 2.1 and the condition on N(T k, we have P X i e (σi x, N(T k P X i e (θ1 xe max θ 1 st ( (s, N(T k P X i e (t xe max tst ( (s, A(ε CA(ε i2 k+1 θ i T t, θ i > T t df θ1 (t P X i e (t xe v P(N(T t k 1dH(vdF θ1 (t P(X 1 e (t xe v (1 + ε k P(N(T t k 1dH(vdF θ1 (t P(X 1 e (t x(1 + ε k P(N(T t k 1dF θ1 (t. (3.3 In the last step, we have used the property of the D class and Lemma 2.3. Consequently, (3.3 holds for any x x for some sufficiently large x >. Hence, for large x, P X i e (σi x, N(T k kn +1 CA(ε P(X 1 e (t x kn +1 [ ] CA(εE (1 + ε N(T T I(N(T N i2 (1 + ε k P(N(T t kdf θ1 (t P(X 1 e (t xdf θ1 (t. (3.4 For any arbitrarily [ fixed ] η >, if we choose ε and N such that (1 + εee θ1 < 1 and N > log (1 qηe T q CA(ε, where q (1 + εee θ1, then [ ] E (1 + ε N(T I(N(T N (1 + ε k P(N(T k kn (1 + ε k P(σ k T ((1 + εee θ1 k e T kn kn η CA(ε. (3.5

6 214 Appl. Math. J. Chinese Univ. Vol. 25, No. 2 We have used Chebyshev inequality in the third inequality. Hence P X i e (σi x, N(T k η P ( X 1 e (t x df θ1 (t. (3.6 kn +1 By Lemma 2.2 and the dominated convergent theorem, for k 1, 2,..., N we have P X i e (σi x, N(T k (v 1v 2 v k T,v k+1 >T (v 1v 2 v k T,v k+1 >T (v 1v 2 v k T,v k+1 >T P X i e (vi x df(v 1,..., v k+1 E (v 1v 2 v k T,v k+1 >T [ [ E P ]] X i e (vi x B(v 1,..., B(v k df(v 1,..., v k+1 [ [ E E P ( X i e (vi x B(v 1,..., B(v k ]] df(v 1,..., v k+1 P ( X i e (vi x df(v 1,..., v k+1 P ( X i e (σi x, N(T k. (3.7 Similarly, we have N P X i e (σi x, N(T k N P ( X i e (σi x, N(T k. Hence there exists some x 1 x 1 (ε, T > such that for all x > x 1, it is true that N P X i e (σi x, N(T k (1 + η P(X i e (σi x, N(T k (1 + η (1 + η (1 + η P(X i e (σi x, N(T i P(X i e (σi x, σ i T P(X 1 e (s xd F σi (s (1 + η P(X 1 e (s xdm(s. (3.8 Thus, combining (3.6 with (3.9, for x > max(x, x 1, we have P ( N(T X i e (σi x (1 + 2η P(X 1 e (s xdm(s. (3.9

7 JIANG Tao. Asymptotics of discounted aggregate claims for renewal risk model with risky investment 215 On the other hand, for the same η, there similarly exists some x 2 (ε, T > such that when x > x 2, we have ( N(T N P X i e (σi x P X i e (σi x, N(T k N (1 η (1 η 2 (1 2η P(X i e (σi x, N(T k P(X i e (σi x, N(T k P(X 1 e (s xdm(s. (3.1 For x > max (x, x 1, x 2, it is true that P( N(T X i e (σi x T P(X 1e (s xdm(s 1 < 2η. (3.11 By the arbitrariness of η, we have ( N(T P X i e (σi x This ends the proof of Theorem 2.1. P(X 1 e (s xdm(s. (3.12 References [1] S Asmussen. Subexponential asymptotics for stochastic processes: extremal behavior, stationary distributions and first passage probabilities, Ann Appl Probab, 1998, 8(2: [2] S Asmussen, V Kalashnikov, D Konstantinides, C Klüppelberg, G Tsitsiashvili. A local limit theorem for random walk maxima with heavy tails, Statist Probab Lett, 22, 56(4: [3] XCBi, CCYin. A local result on ruin probability in renewal risk model, Appl Math J Chinese Univ, 25, 2(1: [4] D B H Cline, G Samorodnitsky. Subexponentiality of the product of independent random variables, Stochastic Process Appl, 1994, 49: [5] PEmbrechts, C Klüppelberg, TMikosch. Modelling Extremal Events for Insurance and Finance, Springer, [6] P Embrechts, N Veraverbeke. Estimates for the probability of ruin with special emphasis on the possibility of large claims, Insurance: Mathematics and Economics, 1982, 1: [7] HGerber. On the probability of ruin in the presence of a linear dividend barrier, Scand Actuar J, 1981, [8] C M Goldie and C Klüppelberg. Subexponential distributions, In: A practical Guide to Heavy- Tails: Statistical Techniques and Applications (Eds. R Adler, R Feldman, M S Taqqu, Birkhäuser, [9] T Jiang, H Yan. The finite-time ruin probability for the jump-diffusion model with constant interest force, Acta Math Sci Ser B, 26, 22(1:

8 216 Appl. Math. J. Chinese Univ. Vol. 25, No. 2 [1] F Klebaner. Introduction to Stochastic Calculus with Applications. Imperial College Press, [11] C Klüppelberg, U Stadtmuller. Ruin probabilities in the presence of heavy-tails and interest rates, Scand Actuar J, 1998, 1: [12] F C Kong. Large deviation for sums of independent random variables with dominatedly varying tails, Appl Math J Chinese Univ Ser B, 27, 22(1: [13] F C Kong. Large deviation for sums of heavy-tailed random variables, Chinese Quart J Math, 27, 22(2: [14] DKonstantinides, QTang, GTsitsiashvili. Estimates for the ruin probability in the classical risk model with constant interest force in the presence of heavy tails, Insurance Math Econom, 22, 31(3: [15] X D Lin, G E Willmot. The moments of the time of ruin, the surplus before ruin, and the deficit at ruin, Insurance Math Econom, 2, 27(4: [16] XSLin, GEWillmot, SDrekic. The classical risk model with a constant dividend barrier: Analysis of the Gerber-Shiu discounted penalty function, Insurance Math Econom, 23, 33(3: [17] Q Tang. The finite time ruin probability of the compound Poisson model with constant interest force, J Appl Probab, 25, 42(3: [18] Q Tang. The ruin probability of a discrete time risk model under constant interest rate with heavy tails, Scand Actuar J, 24, (3: [19] QTang, GTsitsiashvili. Randomly weighted sums of subexponential random variables with application to ruin theory, Extremes, 23, 6(3: [2] Q Tang. Heavy tails of discounted aggregate claims in the continuous-time renewal model, J Appl Probab, 27, 44(2: [21] Q Tang. Asymptotic ruin probabilities of the renewal model with constant interest force and regular variation, Scand Actuar J, 25, (1: 1-5. School of Finance, Zhejiang Gongshang University, Hangzhou 3118, China

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