The finite-time ruin probability of a risk model with stochastic return and Brownian perturbation
Tóm tắt
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.
Từ khóa
Tài liệu tham khảo
Asmussen, S.: Subexponential asymptotics for stochastic processes: extremal behavior, stationary distributions and first passage probabilities. Ann. Appl. Probab. 8, 354–374 (1998)
Athreya, K.B., Ney, P.E.: Branching Processes. Springer, Berlin (1972)
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)
Chen, Y., Ng, K.W.: The ruin probability of the renewal model with constant interest force and negatively dependent heavy-tailed claims. Insur. Math. Econ. 40, 415–423 (2007)
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)
Cheng, J., Gao, Y., Wang, D.: Ruin probabilities for a perturbed risk model with stochastic premiums and constant interest force. J. Inequal. Appl. 2016(214), 1–13 (2016)
Cline, D.B.H., Samorodnitsky, G.: Subexponentiality of the product of independent random variables. Stoch. Process. Appl. 49, 75–98 (1994)
Cont, R., Tankov, P.: Financial Modelling with Jump Processes. Chapman and Hall/CRC, Boca Raton (2004)
Embrechts, P., Klüppelberg, C., Mikosch, T.: Modelling Extremal Events for Insurance and Finance. Springer, Berlin (1997)
Hao, X., Tang, Q.: A uniform asymptotic estimate for discounted aggregate claims with sunexponential tails. Insur. Math. Econ. 43, 116–120 (2008)
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)
Kalashnikov, V., Konstantinides, D.: Ruin under interest force and subexponential claims: a simple treatment. Insur. Math. Econ. 27, 145–149 (2000)
Klüppelberg, C., Stadtmüller, U.: Ruin probabilities in the presence of heavy-tails and interest rates. Scand. Actuar. J. 1, 49–58 (1998)
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)
Li, J.: Asymptotics in a time-dependent renewal risk model with stochastic return. J. Math. Anal. Appl. 387, 1009–1023 (2012)
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)
Li, J., Liu, Z., Tang, Q.: On the ruin probabilities of a bidimensional perturbed risk model. Insur. Math. Econ. 41, 185–195 (2007)
Maulik, K., Resnick, S.: Characterizations and examples of hidden regular variation. Extremes 7, 31–67 (2004)
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)
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)
Piterbarg, V.I.: Asymptotic Methods in the Theory of Gaussian Processes and Fields. American Mathematical Society, Providence (1996)
Stein, C.: A note on cumulative sums. Ann. Math. Stat. 17, 498–499 (1946)
Tang, Q.: The finite time ruin probability of the compound Poisson model with constant interest force. J. Appl. Probab. 42, 608–619 (2005)
Tang, Q.: On convolution equivalence with applications. Bernoulli 12, 535–549 (2006)
Tang, Q.: Heavy tails of discounted aggregate claims in the continuous-time renewal model. J. Appl. Probab. 44, 285–294 (2007)
Tang, Q., Yuan, Z.: Randomly weighted sums of subexponential random variables with application to capital allocation. Extremes 17, 467–493 (2014)
Tang, Q., Wang, G., Yuen, K.C.: Uniform tail asymptotics for the stochastic present value of aggregate claims in the renewal risk model. Insur. Math. Econ. 46, 362–370 (2010)
Veraverbeke, N.: Asymptotic estimates for the probability of ruin in a Poisson model with diffusion. Insur. Math. Econ. 13, 57–62 (1993)
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)
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)
Yang, Y., Wang, K., Konstantinides, D.: Uniform asymptotics for discounted aggregate claims in dependent risk models. J. Appl. Probab. 51, 669–684 (2014)