## Uniform asymptotics for discounted aggregate claims in dependent risk models.(English)Zbl 1303.91097

The renewal risk model under consideration involves a sequence of identically distributed claim sizes, which are not necessarily independent; the interarrival times are i.i.d. nonnegative random variables. The claim arrival times constitute a renewal counting process.
Within this context, supposing that the insurer can make risk free and risky investments, the price process of the investment portfolio is depicted as a geometric Lévy process.
Under the hypothesis that the claim size distribution belongs to some specific classes of heavy tailed distributions, asymptotic formulae are obtained for the tail probability of discounted aggregate claims and ruin probabilities.

### MSC:

 91B30 Risk theory, insurance (MSC2010) 60G51 Processes with independent increments; Lévy processes 60K05 Renewal theory 60G70 Extreme value theory; extremal stochastic processes
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### References:

 [1] Asmussen, S. (1998). Subexponential asymptotics for stochastic processes: extremal behavior, stationary distributions and first passage probabilities. Ann. Appl. Prob. 8 , 354-374. · Zbl 0942.60034 · doi:10.1214/aoap/1028903531 [2] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation . Cambridge University Press. · Zbl 0617.26001 [3] Block, H. W., Savits, T. H. and Shaked, M. (1982). Some concepts of negative dependence. Ann. Prob. 10 , 765-772. · Zbl 0501.62037 · doi:10.1214/aop/1176993784 [4] Cai, J. (2004). Ruin probabilities and penalty functions with stochastic rates of interest. Stoch. Process. Appl. 112 , 53-78. · Zbl 1070.60043 · doi:10.1016/j.spa.2004.01.007 [5] Chen, Y. and Ng, K. W. (2007). The ruin probability of the renewal model with constant interest force and negatively dependent heavy-tailed claims. Insurance Math. Econom. 40 , 415-423. · Zbl 1183.60033 · doi:10.1016/j.insmatheco.2006.06.004 [6] Cline, D. B. H. and Samorodnitsky, G. (1994). Subexponentiality of the product of independent random variables. Stoch. Process. Appl. 49 , 75-98. · Zbl 0799.60015 · doi:10.1016/0304-4149(94)90113-9 [7] Cont, R. and Tankov, P. (2004). Financial Modelling with Jump Processes . Chapman & Hall/CRC, Boca Raton, FL. · Zbl 1052.91043 [8] Hao, X. and Tang, Q. (2008). A uniform asymptotic estimate for discounted aggregate claims with subexponential tails. Insurance Math. Econom. 43 , 116-120. · Zbl 1142.62090 · doi:10.1016/j.insmatheco.2008.03.009 [9] Joag-Dev, K. and Proschan, F. (1983). Negative association of random variables, with applications. Ann. Statist. 11 , 286-295. · Zbl 0508.62041 · doi:10.1214/aos/1176346079 [10] Kalashnikov, V. and Konstantinides, D. (2000). Ruin under interest force and subexponential claims: a simple treatment. Insurance Math. Econom. 27 , 145-149. · Zbl 1056.60501 · doi:10.1016/S0167-6687(00)00045-7 [11] Kalashnikov, V. and Norberg, R. (2002). Power tailed ruin probabilities in the presence of risky investments. Stoch. Process. Appl. 98 , 211-228. · Zbl 1058.60095 · doi:10.1016/S0304-4149(01)00148-X [12] Klüppelberg, C. and Stadtmüller, U. (1998). Ruin probabilities in the presence of heavy-tails and interest rates. Scand. Actuarial J. 1998 , 49-58. · Zbl 1022.60083 · doi:10.1080/03461238.1998.10413991 [13] Konstantinides, D., Tang, Q. and Tsitsiashvili, G. (2002). Estimates for the ruin probability in the classical risk model with constant interest force in the presence of heavy tails. Insurance Math. Econom. 31 , 447-460. · Zbl 1074.91029 · doi:10.1016/S0167-6687(02)00189-0 [14] Lehmann, E. L. (1966). Some concepts of dependence. Ann. Math. Statist. 37 , 1137-1153. · Zbl 0146.40601 · doi:10.1214/aoms/1177699260 [15] Li, J. (2012). Asymptotics in a time-dependent renewal risk model with stochastic return. J. Math. Anal. Appl. 387 , 1009-1023. · Zbl 1230.91076 · doi:10.1016/j.jmaa.2011.10.012 [16] Maulik, K. and Resnick, S. (2004). Characterizations and examples of hidden regular variation. Extremes 7 , 31-67. · Zbl 1088.62066 · doi:10.1007/s10687-004-4728-4 [17] Maulik, K. and Zwart, B. (2006). Tail asymptotics for exponential functionals of Lévy processes. Stoch. Process. Appl. 116 , 156-177. · Zbl 1090.60046 · doi:10.1016/j.spa.2005.09.002 [18] Paulsen, J. (1993). Risk theory in a stochastic economic environment. Stoch. Process. Appl. 46 , 327-361. · Zbl 0777.62098 · doi:10.1016/0304-4149(93)90010-2 [19] Paulsen, J. (2002). On Cramér-like asymptotics for risk processes with stochastic return on investments. Ann. Appl. Prob. 12 , 1247-1260. · Zbl 1019.60041 · doi:10.1214/aoap/1037125862 [20] Paulsen, J. and Gjessing, H. K. (1997). Ruin theory with stochastic return on investments. Adv. Appl. Prob. 29 , 965-985. · Zbl 0892.90046 · doi:10.2307/1427849 [21] Tang, Q. (2005). The finite time ruin probability of the compound Poisson model with constant interest force. J. Appl. Prob. 42 , 608-619. · Zbl 1132.91500 · doi:10.1239/jap/1127322015 [22] Tang, Q. (2007). Heavy tails of discounted aggregate claims in the continuous-time renewal model. J. Appl. Prob. 44 , 285-294. · Zbl 1211.91152 · doi:10.1239/jap/1183667401 [23] Tang, Q. and Tsitsiashvili, G. (2003). Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and financial risks. Stoch. Process. Appl. 108 , 299-325. · Zbl 1075.91563 · doi:10.1016/j.spa.2003.07.001 [24] Tang, Q., Wang, G. and Yuen, K. C. (2010). Uniform tail asymptotics for the stochastic present value of aggregate claims in the renewal risk model. Insurance Math. Econom. 46 , 362-370. · Zbl 1231.91414 · doi:10.1016/j.insmatheco.2009.12.002 [25] Yang, Y. and Wang, Y. (2010). Asymptotics for ruin probability of some negatively dependent risk models with a constant interest rate and dominatedly-varying-tailed claims. Statist. Prob. Lett. 80 , 143-154. · Zbl 1180.62154 · doi:10.1016/j.spl.2009.09.023 [26] Yuen, K. C., Wang, G. and Ng, K. W. (2004). Ruin probabilities for a risk process with stochastic return on investments. Stoch. Process. Appl. 110 , 259-274. · Zbl 1075.91029 · doi:10.1016/j.spa.2003.10.007 [27] Yuen, K. C., Wang, G. and Wu, R. (2006). On the renewal risk process with stochastic interest. Stoch. Process. Appl. 116 , 1496-1510. · Zbl 1109.60071 · doi:10.1016/j.spa.2006.04.012
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