Asymptotics for ruin probabilities in a discrete-time risk model with dependent financial and insurance risks

Transcription

1 1 Asymptotics for ruin probabilities in a discrete-time risk model with dependent financial and insurance risks Yang Yang School of Mathematics and Statistics, Nanjing Audit University School of Economics and Management, Southeast University, China Dimitrios G. Konstantinides Department of Mathematics, University of the Aegean, Greece Samos, May 2014

2 2 Contents 1. Discrete-time insurance risk model. 2. Dependence structure. 3. Asymptotic results. 4. References.

3 3 1. Discrete-time insurance risk model. Let {X i, i 1} and {Y i, i 1} be insurance and financial risks and the aggregate net losses n i S n = Y j, (1) for each positive integer n. The finite and infinite time ruin probabilities are i=1 X i j=1 ( ) ψ(x, n) = P max S k > x 1 k n ψ(x) = lim ψ(x, n) = P n where x 0 is interpreted as the initial capital., (2) ( ) sup S n > x n 1, (3)

4 4 page If we denote the product θ i = i j=1 Y j in (1), the ruin probabilities ψ(x, n) and ψ(x) represent the tail probabilities of the maximum of randomly weighted sums. In case of independence between {X i, i 1} and {θ i, i 1}, under the presence of heavy-tailed insurance risks, was recently established the asymptotic formula ψ(x, n) n P(X i θ i > x), x, (4) i=1 holds for each fixed n, or ψ(x) P(X i θ i > x), x. (5) i=1

5 5 1. Dependence structure. The survival copula C(u, v) is defined by the formula C(u, v) = u + v 1 + C(1 u, 1 v), (u, v) [0, 1] 2, with respect to the given copula C(u, v). Clearly, the survival copula with respect to C(u, v) can be represented as C(F (x), G(y)) = P(X > x, Y > y).

6 6 Assume that the copula function C(u, v) is absolutely continuous. Denote by C 1 (u, v) = u C(u, v), C 2(u, v) = v C(u, v), C 12(u, v) = 2 u v C(u, v). Then C 2 (u, v) := v C(u, v) = 1 C 2(1 u, 1 v), and C 12 (u, v) := 2 u v C(u, v) = C 12(1 u, 1 v). Assumption A 1. (Albrecher et al. 06) There exists a positive constant M such that lim sup v 1 lim sup u 1 C 12 (u, v) = lim sup v 1 lim sup u 1 C 12 (1 u, 1 v) < M.

7 7 Assumption A 2. (Asimit and Badescu 10) The relation holds uniformly on (0, 1). Clearly, Assumption A 2 is equivalent to C 2 (u, v) u C 12 (0+, v), u 0, 1 C 2 (u, v) (1 u) C 12 (1, v), u 1, holds uniformly on (0, 1). Thus, if the copula C(u, v) of the random vector (X, Y ) satisfies Assumptions A 1 and A 2, then the copula of (X +, Y ), denoted by C + (u, v), satisfies these two assumptions as well.

8 8 Assumption A 3. The relation C 2 (u, v) = 1 C 2 (1 u, 1 v) 0, u 0, holds uniformly on [0, 1]. We remark that Assumption A 3 is equivalent to the fact that holds uniformly on R. P(X x Y = y) = C 2 [F (x), G(y)] 0, x, (6)

9 9 We remind the classes of distributions: 1. L = { F lim x F (x y) F (x) } = 1, y R, 2. D = { F lim sup x F (xu) F (x) } <, u (0, 1). 3. C = { F lim u 1 lim sup x } F (xu) F (x) = 1,.

10 10 A distribution F on R belongs to the class R α, if lim F (x y)/f (x) = y α for some α 0 and all y > 0. It is well known that the following inclusion relationships hold: R α C L D L Furthermore, for a distribution F on R, denote its upper and lower Matuszewska indices, respectively, by J + F = lim y log F (y) log y with F (y) := lim inf F (x y) F (x) for y > 1, J F = lim y log F (y) log y with F (y) := lim sup F (x y) F (x) for y > 1. Clearly, F D is equivalent to J + F <. For each i 1, denote by H i the distribution of X i i j=1 Y j, j 1 and let H 1 = H.

11 11 3. Asymptotic results. Theorem 1. In the discrete-time risk model, assume that {(X i, Y i ), i 1} are i.i.d. random vectors with generic random vector (X, Y ) satisfying Assumptions A 1 A 3. If F C and EY p < for some p > J + F, then, for each fixed n 1, it holds that ψ(x, n) n H i (x). (7) i=1

12 12 Corollary 1. (1) Under the conditions of Theorem 1, if F R α for some α 0, then, for each fixed n 1, it holds that ψ(x, n) EY α c 1 (EY α ) n F (x), (8) 1 EY α by convention, (1 (EY α ) n )/(1 EY α ) = n if EY α = 1.

13 13 Theorem 2. Under the conditions of Theorem 1, if J F > 0 and EY p < 1 for some p > J + F, then it holds that ψ(x) H i (x). (9) i=1

14 14 Corollary 2. Under the conditions of Theorem 2, if F R α for some α > 0, then ψ(x) EY α c 1 EY α F (x). (10) Moreover, (8) holds uniformly over the integers {n 1}.

15 15 The last result shows that the asymptotic relation (11) for the finite time probability is uniform over the integers {n 1}. Theorem 3. Under the conditions of Theorem 2, the asymptotic relation ψ(x, n) n H i (x). (11) i=1 holds uniformly over the integers {n 1}.

16 16 References [1] Albrecher, H., Asmussen, S. and Kortschak, D., Tail asymptotics for the sum of two heavy-tailed dependent risks. Extremes 9, [2] Asimit, A. V. and Badescu, A. L., Extremes on the discounted aggregate claims in a time dependent risk model. Scand. Actuar. J. 2, [3] Cai, J. and Tang, Q., On max-sum equivalence and convolution closure of heavy-tailed distributions and their applications. J. Appl. Probab. 41,

17 17 [4] Chen, Y., The finite-time ruin probability with dependent insurance and financial risks. J. Appl. Probab. 48, [5] Chen, Y. and Yuen, K. C., Sums of pairwise quasi-asymptotically independent random variables with consistent variation. Stochastic Models 25, [6] Foss, S. Korshunov D. and Zachary S., An Introduction to Heavy-tailed and Subexponential Distributions. Springer, New York, Heidelberg. [7] Goldie, C. M. and Maller, R. A., Stability of Perpetuities. Ann. Probability, 23, 3,

18 18 [8] Goovaerts, M. J., Kaas, R., Laeven, R. J. A., Tang, Q. and Vernic, R., The tail probability of discounted sums of Pareto-like losses in insurance. Scand. Actuar. J. 6, [9] Hashorva, E., Pakes, A. G. and Tang, Q., Asymptotics of random contractions. Insurance Math. Econom. 47, [10] Ko, B. and Tang, Q., Sums of dependent nonnegative random variables with subexponential tails. J. Appl. Probab. 45, [11] Nyrhinen, H., On the ruin probabilities in a general economic environment. Stoch. Process. Appl. 83,

19 19 [12] Nyrhinen, H., Finite and infinite time ruin probabilities in a stochastic economic environment. Stoch. Process. Appl. 92, [13] Shen, X., Lin, Z. and Zhang, Y., Uniform estimate for maximum of randomly weighted sums with applications to ruin theory. Methodol. Comput. Appl. Probab. 11, [14] Tang, Q. and Tsitsiashvili, G., Randomly weighted sums of subexponential random variables with application to ruin theory. Extremes 6, [15] Tang, Q. and Tsitsiashvili, G., Finite- and infinite-time ruin probabilities in the presence of stochastic returns on investments. Adv. Appl. Probab. 36,

20 20 [16] Wang, D. and Tang, Q., Tail probabilities of randomly weighted sums of random variables with dominated variation. Stoch. Models 22, [17] Weng, C., Zhang, Y. and Tan, K. S., Ruin probabilities in a discrete time risk model with dependent risks of heavy tail. Scand. Actuar. J. 3, [18] Yang, Y., Leipus, R. and Šiaulys J., 2012b. On the ruin probability in a dependent discrete time risk model with insurance and financial risks. J. Comput. Appl. Math. 236, [19] Yang, H. and Sun, S., Subexponentiality of the product of dependent random variables. Stat. Probab. Lett. 83,

21 Thank you! 21