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The finite-time ruin probability of a risk model with stochastic return and Brownian perturbation. (English) Zbl 1403.62194

Summary: This paper investigates a renewal risk model with stochastic return and Brownian perturbation, where the price process of the investment portfolio is described as a geometric Lévy process. When the claim sizes have a subexponential distribution, we derive the asymptotics for the finite-time ruin probability of the above risk model. The obtained result confirms that the asymptotics for the finite-time ruin probability of the risk model with heavy-tailed claim sizes are insensitive to the Brownian perturbation.

MSC:

62P05 Applications of statistics to actuarial sciences and financial mathematics
62E10 Characterization and structure theory of statistical distributions
91B30 Risk theory, insurance (MSC2010)
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[1] Asmussen, S., Subexponential asymptotics for stochastic processes: extremal behavior, stationary distributions and first passage probabilities, Ann. Appl. Probab., 8, 354-374, (1998) · Zbl 0942.60034 · doi:10.1214/aoap/1028903531
[2] Athreya, K.B., Ney, P.E.: Branching Processes. Springer, Berlin (1972) · doi:10.1007/978-3-642-65371-1
[3] Chen, Y.; Wang, L.; Wang, Y., Uniform asymptotics for the finite-time ruin probabilities of two kinds of nonstandard bidimensional risk models, J. Math. Anal. Appl., 401, 114-129, (2013) · Zbl 1266.91032 · doi:10.1016/j.jmaa.2012.11.046
[4] Chen, Y.; Ng, KW, The ruin probability of the renewal model with constant interest force and negatively dependent heavy-tailed claims, Insur. Math. Econ., 40, 415-423, (2007) · Zbl 1183.60033 · doi:10.1016/j.insmatheco.2006.06.004
[5] Cheng, J.; Wang, D., Ruin probabilities for a two-dimensional perturbed risk model with stochastic premiums, Acta Math. Appl. Sin. Engl. Ser., 32, 1053-1066, (2016) · Zbl 1359.62443 · doi:10.1007/s10255-016-0626-1
[6] Cheng, J.; Gao, Y.; Wang, D., Ruin probabilities for a perturbed risk model with stochastic premiums and constant interest force, J. Inequal. Appl., 2016, 1-13, (2016) · Zbl 1398.91319
[7] Cline, DBH; Samorodnitsky, G., Subexponentiality of the product of independent random variables, Stoch. Process. Appl., 49, 75-98, (1994) · Zbl 0799.60015 · doi:10.1016/0304-4149(94)90113-9
[8] Cont, R., Tankov, P.: Financial Modelling with Jump Processes. Chapman and Hall/CRC, Boca Raton (2004) · Zbl 1052.91043
[9] Embrechts, P., Klüppelberg, C., Mikosch, T.: Modelling Extremal Events for Insurance and Finance. Springer, Berlin (1997) · Zbl 0873.62116 · doi:10.1007/978-3-642-33483-2
[10] Hao, X.; Tang, Q., A uniform asymptotic estimate for discounted aggregate claims with sunexponential tails, Insur. Math. Econ., 43, 116-120, (2008) · Zbl 1142.62090 · doi:10.1016/j.insmatheco.2008.03.009
[11] Jiang, T.; Yan, H., The finite-time ruin probability for the jump-diffusion model with constant interest force, Acta Math. Appl. Sin. Engl. Ser., 22, 171-176, (2006) · Zbl 1153.91595 · doi:10.1007/s10255-005-0295-y
[12] Kalashnikov, V.; Konstantinides, D., Ruin under interest force and subexponential claims: a simple treatment, Insur. Math. Econ., 27, 145-149, (2000) · Zbl 1056.60501 · doi:10.1016/S0167-6687(00)00045-7
[13] Klüppelberg, C.; Stadtmüller, U., Ruin probabilities in the presence of heavy-tails and interest rates, Scand. Actuar. J., 1, 49-58, (1998) · Zbl 1022.60083 · doi:10.1080/03461238.1998.10413991
[14] Konstantinides, D.; Tang, Q.; Tsitsiashvili, G., Estimates for the ruin probability in the classical risk model with constant interest force in the presence of heavy tails, Insur. Math. Econ., 31, 447-460, (2002) · Zbl 1074.91029 · doi:10.1016/S0167-6687(02)00189-0
[15] Li, J., Asymptotics in a time-dependent renewal risk model with stochastic return, J. Math. Anal. Appl., 387, 1009-1023, (2012) · Zbl 1230.91076 · doi:10.1016/j.jmaa.2011.10.012
[16] Li, J., A note on the finite-time ruin probability of a renewal risk model with Brownian perturbation, Stat. Probab. Lett., 127, 49-55, (2017) · Zbl 1391.62195 · doi:10.1016/j.spl.2017.03.028
[17] Li, J.; Liu, Z.; Tang, Q., On the ruin probabilities of a bidimensional perturbed risk model, Insur. Math. Econ., 41, 185-195, (2007) · Zbl 1119.91056 · doi:10.1016/j.insmatheco.2006.10.012
[18] Maulik, K.; Resnick, S., Characterizations and examples of hidden regular variation, Extremes, 7, 31-67, (2004) · Zbl 1088.62066 · doi:10.1007/s10687-004-4728-4
[19] Peng, J.; Wang, D., Asymptotics for ruin probabilities of a non-standard renewal risk model with dependence structures and exponential Lévy process investment returns, J. Ind. Manag. Optim., 13, 155-185, (2017) · Zbl 1367.60106
[20] Peng, J.; Wang, D., Uniform asymptotics for ruin probabilities in a dependent renewal risk model with stochastic return on investments, Stochastics, 90, 432-471, (2018) · doi:10.1080/17442508.2017.1365077
[21] Piterbarg, V.I.: Asymptotic Methods in the Theory of Gaussian Processes and Fields. American Mathematical Society, Providence (1996) · Zbl 0841.60024
[22] Stein, C., A note on cumulative sums, Ann. Math. Stat., 17, 498-499, (1946) · Zbl 0063.07178 · doi:10.1214/aoms/1177730890
[23] Tang, Q., The finite time ruin probability of the compound Poisson model with constant interest force, J. Appl. Probab., 42, 608-619, (2005) · Zbl 1132.91500 · doi:10.1239/jap/1127322015
[24] Tang, Q., On convolution equivalence with applications, Bernoulli, 12, 535-549, (2006) · Zbl 1114.60015 · doi:10.3150/bj/1151525135
[25] Tang, Q., Heavy tails of discounted aggregate claims in the continuous-time renewal model, J. Appl. Probab., 44, 285-294, (2007) · Zbl 1211.91152 · doi:10.1239/jap/1183667401
[26] Tang, Q.; Yuan, Z., Randomly weighted sums of subexponential random variables with application to capital allocation, Extremes, 17, 467-493, (2014) · Zbl 1328.62089 · doi:10.1007/s10687-014-0191-z
[27] Tang, Q.; Wang, G.; Yuen, KC, Uniform tail asymptotics for the stochastic present value of aggregate claims in the renewal risk model, Insur. Math. Econ., 46, 362-370, (2010) · Zbl 1231.91414 · doi:10.1016/j.insmatheco.2009.12.002
[28] Veraverbeke, N., Asymptotic estimates for the probability of ruin in a Poisson model with diffusion, Insur. Math. Econ., 13, 57-62, (1993) · Zbl 0790.62098 · doi:10.1016/0167-6687(93)90535-W
[29] Wang, K.; Wang, Y.; Gao, Q., Uniform asymptotics for the finite-time ruin probability of a dependent risk model with a constant interest rate, Methodol. Comput. Appl. Probab., 15, 109-124, (2013) · Zbl 1263.91027 · doi:10.1007/s11009-011-9226-y
[30] Yang, Y.; Wang, Y., Asymptotics for ruin probability of some negatively dependent risk models with a constant interest rate and dominatedly-varying-tailed claims, Stat. Probab. Lett., 80, 143-154, (2010) · Zbl 1180.62154 · doi:10.1016/j.spl.2009.09.023
[31] Yang, Y.; Wang, K.; Konstantinides, D., Uniform asymptotics for discounted aggregate claims in dependent risk models, J. Appl. Probab., 51, 669-684, (2014) · Zbl 1303.91097 · doi:10.1239/jap/1409932666
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