A Note on the Ruin Probability in the Delayed Renewal Risk Model

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1 Southeast Asian Bulletin of Mathematics : 1 5 Southeast Asian Bulletin of Mathematics c SEAMS A Note on the Ruin Probability in the Delayed Renewal Risk Model Chun Su Department of Statistics and Finance, University of Science and Technology of China, Hefei , Anhui, China suchun@ustc.edu.cn Qihe Tang Department of Mathematics and Statistics, Concordia University, 7141 Sherbrooke Street West, Montreal, Quebec, H4B 1R6, Canada q.tang@uva.nl AMS Mathematics Subject Classification 2000: 62E20; 62P05 Abstract. Veraverbeke 1977, Stochastic Processes Appl. 5, no. 1, and Embrechts and Veraverbeke 1982, Insurance Math. Econom. 1, no. 1, obtained a simple asymptotic relation for the ruin probability in the renewal risk model under the assumption that the claim size is heavy tailed. This note points out that the relation still holds in the delayed renewal risk model. Keywords: Delayed renewal process; Heavy-tailed distribution; Ruin probability. 1. Introduction Throughout this note, we say that a distribution function d.f. F = 1 F is supported on [0, if F x = 0 for all x < 0 and F x < 1 for all x 0. For a given d.f. F with a finite mean µ, its equilibrium d.f. is denoted by F e x = 1 µ x 0 F udu for x 0. The ordinary renewal model, which is one of the basic risk models in insurance and finance, was introduced early in 1957 by Sparre Anderson. As summarized recently by Embrechts et al. 1997, it has the following structure:

2 2 C. Su and Q.H. Tang a The claim sizes {Z i, i 1}, form a sequence of independent and identically distributed i.i.d., nonnegative random variables r.v. s with common d.f. F and finite mean µ; b The claim inter-arrival times {θ i, i 1}, also form a sequence of i.i.d. nonnegative r.v. s with common d.f. G and finite mean m, independent of {Z i, i 1}; c The number of claims in the time interval [0, t] is denoted by Nt = sup {n 1 : T n t}, t 0, where T n = n i=1 θ i, n 1, are the locations of claims and supφ = 0 by convention; d The net loss process is then defined by St = Nt i=1 Z i ct, t 0, 1 where the constant 0 < c < denotes the premium rate. If in b the inter-arrival times θ i, i 1, are i.i.d. exponentially distributed, the model above is called the Cramér-Lundberg model. If in b the inter-arrival times θ i, i 2, are i.i.d. with common d.f. G and finite mean m, but θ 1 has a possibly different d.f. G 1, it is called the delayed renewal risk model. See Ross 1983 and Grandell 1991 for details. The delayed renewal model is more realistic in the context of insurance and finance. Throughout the paper we assume that ρ = cez 2 Eθ 2 = cm µ > 0, 2 Eθ 2 µ which can be interpreted as the relative safety loading condition. probability ψx can be defined by ψx = P sup St > x for x 0, t 0 The ruin which is a probability to ruin over infinite horizon provided that the initial capital of an insurance company is equal to x 0. Hereafter, all limit relationships are for x unless stated otherwise; for two positive functions a and b, we write ax bx if lim inf ax/bx 1, write ax bx if lim sup ax/bx 1, and write ax bx if both. Veraverbeke 1977 and Embrechts and Veraverbeke 1982 investigated the ruin probability in the ordinary renewal model with condition 2 and proved that the relation ψx ρ 1 F e x 3

3 A Note on the Ruin Probability 3 holds if the equilibrium d.f. F e S. [See below for this definition.] This note extends their result to the delayed renewal model: Theorem 1.1. In the delayed renewal model with condition 2, relation 3 holds if F e S. Note that we do not add any restriction on the d.f. G 1. See Tang and Su 2002 for a related discussion on such an extension. 2. Preliminaries As many researchers in the field of risk theory, we are interested in the case of heavy-tailed claim sizes. The most important class of heavy-tailed d.f. s is the subexponential class. By definition, a d.f. F is subexponential, denoted by F S, if the relation F lim n x = n x F x holds for any n 2 or equivalently, for n = 2, where F n denotes the n-fold convolution of F. It is well known that any subexponential distribution F is long tailed, denoted by F L, in the sense that the relation F x y lim = 1 x F x holds for any y > 0 or equivalently, for y = 1. For more details about heavytailed d.f. s with applications to insurance and finance, the reader is referred to Embrechts et al We introduce a new class of heavy-tailed d.f. s below; for more details see Tang Definition 2.1. Let F be a d.f. supported on [0, with a finite mean µ, we say F M if F satisfies F x lim = 0. x F tdt x Tang and Su 2002, Lemma 2.5 proved the following result: Lemma 2.2. Let F be a d.f. supported on [0, with a finite mean. Then, F M if and only if F e L. Now we show an interesting result on the subexponential class.

4 4 C. Su and Q.H. Tang Lemma 2.3. Let F be a d.f. supported on [0, with a finite mean. If F e S, then F x = o F e x and F F e x F e x. Proof. By Lemma we know that F M, hence that F x = o F e x. Then, using Proposition 1 of Embrechts et al [see also Lemma A3.23 of Embrechts et al. 1997] we complete the proof of Lemma. Let X and Y be two independent and nonnegative r.v. s. By the dominated convergence theorem we can prove the following result [see also Lemma 4.2 of Tang 2004 for a more general discussion]: Lemma 2.4. If X is long tailed, then P X y > x P X > x for any fixed y > Proof of Theorem We still denote by 1 the net loss process of the considered delayed renewal model. Clearly, ψx=p Z 1 cθ 1 > x + P Z 1 cθ 1 x, Z 1 cθ 1 + M > x where =I 1 x + I 2 x, 4 M = max n 2 n Z i cθ i. i=2 According to the works of Veraverbeke 1977 and Embrechts and Veraverbeke 1982, we know that P M > x ρ 1 F e x, hence that M is subexponentially distributed. Applying Lemmas and, we have I 1 x F x = o F e x. 5 Now we deal with I 2 x. On the one hand, for any fixed C > 0 and all large x > 0, I 2 x P C < Z 1 cθ 1 x P M > x + C ρ 1 P C < Z 1 cθ 1 F e x + C ρ 1 P C < Z 1 cθ 1 F e x, which, together with the arbitrariness of C > 0, gives that I 2 x ρ 1 F e x. 6

5 A Note on the Ruin Probability 5 On the other hand, I 2 x P Applying Lemmas, we obtain Z 1 cθ 1 + M > x P Z 1 + M > x. I 2 x ρ 1 F e x. 7 Simply combining 4 7, we complete the proof of Theorem. Acknowledgement. We would like to thank a referee for his/her careful reading. This work was supported by the National Science Foundation of China Project No and the Special Foundation of USTC. References [1] Embrechts, P., Goldie, C. M. and Veraverbeke, N.: Subexponentiality and infinite divisibility. Z. Wahrsch. Verw. Gebiete 493, [2] Embrechts, P., Klüppelberg, C. and Mikosch, T.: Modelling extremal events for insurance and finance, Springer-Verlag, Berlin, [3] Embrechts, P. and Veraverbeke, N.: Estimates for the probability of ruin with special emphasis on the possibility of large claims, Insurance Math. Econom. 11, [4] Grandell, J.: Aspects of risk theory, Springer-Verlag, New York, [5] Ross, S.M.: Stochastic processes, John Wiley & Sons, Inc., New York, [6] Tang, Q.: Extremal values of risk processes for insurance and financial: with special emphasis on the possibility of large claims, Ph.D. Thesis of University of Science and Technology of China, [7] Tang, Q.: The ruin probability of a discrete time risk model under constant interest rate with heavy tails, Scand. Actuar. J., issue 3, [8] Tang, Q. and Su, C.: Ruin probabilities for large claims in delayed renewal risk model, Southeast Asian Bull. Math. 254, [9] Veraverbeke, N.: Asymptotic behaviour of Wiener-Hopf factors of a random walk, Stochastic Processes Appl. 51,