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Finite- and infinite-time ruin probabilities in the presence of stochastic returns on investments. (English) Zbl 1095.91040
The finite- and infinite-time ruin probabilities in a discrete-time stochastic economic environment are investigated. Under the assumption that the insurance risk is extended-regularly-varying tailed or rapidly-varying tailed, some precise estimates for the ruin probabilities are derived.

MSC:
91B30 Risk theory, insurance (MSC2010)
62P05 Applications of statistics to actuarial sciences and financial mathematics
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[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] Brandt, A. (1986). The stochastic equation \(Y_n+1=A_nY_n+B_n\) with stationary coefficients. Adv . Appl. Prob. 18 , 211–220. · Zbl 0588.60056 · doi:10.2307/1427243
[4] Breiman, L. (1965). On some limit theorems similar to the arc-sin law. Teor . Veroyat. Primen. 10 , 351–360 (in Russian). English translation: Theory Prob . Appl. 10 , 323–331. · Zbl 0147.37004
[5] Cai, J. and Tang, Q. (2004). \(L_p\) transform and asymptotic ruin probability in a perturbed risk process. To appear in Stoch. Process. Appl. · Zbl 1054.60012 · doi:10.1239/jap/1077134672
[6] Chistyakov, V. P. (1964). A theorem on sums of independent positive random variables and its applications to branching random processes. Teor . Veroyat. Primen. 9 , 710–718 (in Russian). English translation: Theory Prob . Appl. 9 , 640–648. · Zbl 0203.19401
[7] Chover, J., Ney, P. and Wainger, S. (1973a). Functions of probability measures. J . Anal. Math. 26 , 255–302. · Zbl 0276.60018 · doi:10.1007/BF02790433
[8] Chover, J., Ney, P. and Wainger, S. (1973b). Degeneracy properties of subcritical branching processes. Ann . Prob. 1 , 663–673. · Zbl 0387.60097 · doi:10.1214/aop/1176996893
[9] 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
[10] Davis, R. and Resnick, S. (1988). Extremes of moving averages of random variables from the domain of attraction of the double exponential distribution. Stoch . Process. Appl. 30 , 41–68. · Zbl 0657.60028 · doi:10.1016/0304-4149(88)90075-0
[11] De Haan, L. (1970). On Regular Variation and its Application to the Weak Convergence of Sample Extremes (Math. Centre Tracts 32 ). Mathematisch Centrum, Amsterdam. · Zbl 0226.60039
[12] Embrechts, P. (1983). A property of the generalized inverse Gaussian distribution with some applications. J . Appl. Prob. 20 , 537–544. · Zbl 0536.60022 · doi:10.2307/3213890
[13] Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance . Springer, Berlin. · Zbl 0873.62116
[14] Frolova, A., Kabanov, Y. and Pergamenshchikov, S. (2002). In the insurance business risky investments are dangerous. Finance Stoch . 6 , 227–235. · Zbl 1002.91037 · doi:10.1007/s007800100057
[15] Geluk, J. L. and de Haan, L. (1987). Regular Variation , Extensions and Tauberian Theorems (CWI Tract 40 ). CWI, Amsterdam. · Zbl 0624.26003
[16] Goldie, C. M. (1991). Implicit renewal theory and tails of solutions of random equations. Ann . Appl. Prob. 1 , 126–166. JSTOR: · Zbl 0724.60076 · doi:10.1214/aoap/1177005985 · links.jstor.org
[17] 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
[18] 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
[19] Klüppelberg, C. (1989). Subexponential distributions and characterizations of related classes. Prob . Theory Relat. Fields 82 , 259–269. · Zbl 0687.60017 · doi:10.1007/BF00354763
[20] Klüppelberg, C. and Mikosch, T. (1997). Large deviations of heavy-tailed random sums with applications in insurance and finance. J . Appl. Prob. 34 , 293–308. · Zbl 0903.60021 · doi:10.2307/3215371
[21] Klüppelberg, C. and Stadtmüller, U. (1998). Ruin probabilities in the presence of heavy-tails and interest rates. Scand . Actuarial. J. 49–58. · Zbl 1022.60083 · doi:10.1080/03461238.1998.10413991
[22] 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
[23] Ng, K. W., Tang, Q., Yan, J. and Yang, H. (2003). Precise large deviations for the prospective-loss process. J . Appl. Prob. 40 , 391–400. · Zbl 1028.60024 · doi:10.1239/jap/1053003551
[24] Norberg, R. (1999). Ruin problems with assets and liabilities of diffusion type. Stoch . Process. Appl. 81 , 255–269. · Zbl 0962.60075 · doi:10.1016/S0304-4149(98)00103-3
[25] Nyrhinen, H. (1999). On the ruin probabilities in a general economic environment. Stoch . Process. Appl. 83 , 319–330. · Zbl 0997.60041 · doi:10.1016/S0304-4149(99)00030-7
[26] Nyrhinen, H. (2001). Finite and infinite time ruin probabilities in a stochastic economic environment. Stoch . Process. Appl. 92 , 265–285. · Zbl 1047.60040 · doi:10.1016/S0304-4149(00)00083-1
[27] 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
[28] Resnick, S. I. (1987). Extreme Values , Regular Variation, and Point Processes. Springer, New York. · Zbl 0633.60001
[29] Rogozin, B. A. (1999). On the constant in the definition of subexponential distributions. Teor . Veroyat. Primen. 44 , 455–458 (in Russian). English translation: Theory Prob . Appl. 44 (2000) 409–412. · Zbl 0971.60009 · doi:10.1137/S0040585X97977665
[30] Rogozin, B. A. and Sgibnev, M. S. (1999). Banach algebras of measures on the line with given asymptotics of distributions at infinity. Sibirsk . Mat. Zh. 40 , 660–672 (in Russian). English translation: Siberian Math . J. 40 , 565–576. · Zbl 0936.46021 · doi:10.1007/BF02679764
[31] Sgibnev, M. S. (1996). On the distribution of the maxima of partial sums. Statist . Prob. Lett. 28 , 235–238. · Zbl 0862.60040 · doi:10.1016/0167-7152(95)00129-8
[32] Sundt, B. and Teugels, J. L. (1995). Ruin estimates under interest force. Insurance Math . Econom. 16 , 7–22. · Zbl 0838.62098 · doi:10.1016/0167-6687(94)00023-8
[33] Sundt, B. and Teugels, J. L. (1997). The adjustment function in ruin estimates under interest force. Insurance Math . Econom. 19 , 85–94. · Zbl 0910.62107 · doi:10.1016/S0167-6687(96)00012-1
[34] Tang, Q. (2004). The ruin probability of a discrete time risk model under constant interest rate with heavy tails. Scand . Actuarial J. 229–240. · Zbl 1142.62094 · doi:10.1080/03461230310017531
[35] 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
[36] Teugels, J. L. (1975). The class of subexponential distributions. Ann . Prob. 3 , 1000–1011. · Zbl 0374.60022 · doi:10.1214/aop/1176996225
[37] Tsitsiashvili, G. (2002). Quality properties of risk models under stochastic interest force. Inform . Process. 2 , 264–268.
[38] Vervaat, W. (1979). On a stochastic difference equation and a representation of nonnegative infinitely divisible random variables. Adv . Appl. Prob. 11 , 750–783. · Zbl 0417.60073 · doi:10.2307/1426858
[39] Yang, H. (1999). Non-exponential bounds for ruin probability with interest effect included. Scand . Actuarial J. 66–79. \def\pdfinf · Zbl 0922.62113 · doi:10.1080/03461230050131885
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