Abstract Consider a discrete-time risk model with insurance and financial risks in a stochastic economic environment. Assume that the insurance and financial risks form a sequence of independent and identically distributed random vectors with a generic random vector following a wide type of dependence structure. An asymptotic formula for the finite-time ruin probability with subexponential insurance risks is derived. In doing so, the subexponentiality of the product of two dependent random variables is investigated simultaneously.

Fund:Supported in part by the Natural National Science Foundation of China under Grant No. 11671012, the Natural Science Foundation of Anhui Province under Grant No. 1808085MA16, the Provincial Natural Science Research Project of Anhui Colleges under Grant No. KJ2017A024 and KJ2017A028.

Shi-jie WANG,Chuan-wei ZHANG,Xue-jun WANG等. The Finite-time Ruin Probability of a Discrete-time Risk Model with Subexponential and Dependent Insurance and Financial Risks[J]. Acta Mathematicae Applicatae Sinica, English Serie, 2018, 34(3): 553-565.

Chen, Y. The finite-time ruin probability with dependent insurance and financial risks. Journal of Applied Probability, 48:1035-1048(2011)

[5]

Chen, Y., Interplay of subexponential and dependent insurance and financial risks. Insurance:Mathematics and Economics, 77:78-83(2017)

[6]

Chen, Y., Ng, K.W., Yuen, K.C. The maximum of randomly weighted sums with long tails in insurance and finance. Stochastic Analysis and Applications, 41:117-130(2011)

[7]

Chen, Y., Yuen, K. C. Precise large deviations of aggregate claims in a size-dependent renewal risk model. Insurance:Mathematics and Economics, 51:457-461(2012)

[8]

Cline, D.B.H., Samorodnitsky, G. Subexponentiality of the product of independent random variables. Stochastic Processes and their Applications, 48:75-98(1994)

[9]

Embrechts, P., Klüppelberg, C., Mikosch, T. Modelling Extremal Events for Insurance and Finance. Springer, Berlin, Heidelberg, 1997

[10]

Foss, S., Korshunov, D., Zachary, S. An Introduction to Heavy-tailed and Subexponential Distributions. Springer, New York, 2011

[11]

Goovaerts, M.J., Kaas, R., Laeven, R.J.A., Tang, Q., Vernic, R. The tail probability of discounted sums of Pareto-like losses in insurance. Scandinavian Actuarial Journal, 6:446-461(2005)

[12]

Hao, X., Tang, Q. A uniform asymptotic estimate for discounted aggregate claims with subexponential tails. Insurance:Mathematics and Economics, 43:116-120(2008)

[13]

Hashorva, E., Pakes, A.G., Tang, Q. Asymptotics of random contractions. Insurance:Mathematics and Economics, 47:405-414(2010)

[14]

Joe, H. Multivariate models and dependence concepts. Chapman Hall, London, 1997

[15]

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. Insurance:Mathematics and Economics, 31:447-460(2002)

[16]

Laeven, R.J.A., Goovaerts, M.J., Hoedemakers, T. Some asymptotic results for sums of dependent random variables, with actuarial applications. Insurance:Mathematics and Economics, 2:154-172(2005)

[17]

Li, J., Tang, Q. Interplay of insurance and financial risks in a discrete-time model with strongly regular variation. Bernoulli 3:1800-1823(2015)

[18]

Li, J., Tang, Q., Wu, R. Subexponential tails of discounted aggregate claims in a time-dependent renewal risk model. Advances in Applied Probability, 42:1126-1146(2010)

[19]

Liu, R., Wang, D. The ruin probabilities of a discrete-time risk model with dependent insurance and financial risks. Journal of Mathematical Analysis and Applications, 444(1):80-94(2016)

[20]

Nelsen, R.B. An introduction to copulas, 2nd ed., Springer, New York, 2006

[21]

Norberg, R. Ruin problems with assets and liabilities of diffusion type. Stochastic Processes and their Applications, 81:255-269(1999)

[22]

Nyrhinen, H. On the ruin probabilities in a general economic environment. Stochastic Processes and their Applications, 83:319-330(1999)

[23]

Nyrhinen, H. Finite and infinite time probabilities in a stochastic economic environment. Stochastic Processes and their Applications, 92:265-285(2001)

[24]

Shen, X., Lin, Z., Zhang, Y. Uniform estimate for maximum of randomly weighted sums with applications to ruin theory. Methodology and Computing in Applied Probability, 11:669-685(2009)

[25]

Su, C., Chen, Y. On the behavior of the product of independent random variables. Science in China Series A, 49:342-359(2006)

[26]

Sun, Y., Wei, L. The finite-time ruin probability with heavy-tailed and dependent insurance and financial risks. Insurance:Mathematics and Economics, 59:178-183(2014)

[27]

Tang, Q. The subexponentiality of products revisited. Extremes, 9:231-241(2006)

[28]

Tang, Q., Tsitsiashvili, G. Precise estimate for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and financial risks. Stochastic Processes and their Applications, 108:299-325(2003)

[29]

Tang, Q., Tsitsiashvili, G. Finite-and infinite-time ruin probabilities in the presence of stochastic returns on investments. Advances in Applied Probability, 36:1278-1299(2004)

[30]

Tang, Q., Vernic, R. The impact on ruin probabilities of the association structure among financial risks. Statistics and Probability Letters, 77:1522-1525(2007)

[31]

Tang, Q., Yuan, Z. Randomly weighted sums of subexponential random variables with application to capital allocation. Extremes, 17:467-493(2014)

[32]

Weng, C., Zhang, Y., Tan, K. S. Ruin probabilities in a discrete time risk model with dependent risks of heavy tail. Scandinavian Actuarial Journal, 3:205-218(2009)

[33]

Yang, H., Gao, W., Li, J. Asymptotic ruin probabilities for a discrete-time risk model with dependent insurance and financial risks. Scandinavian Actuarial Journal, 1:1-17(2016)

[34]

Yang, H., Sun, S. Subexponentiality of the product of dependent random variables. Statistics and Probability Letters, 83:2039-2044(2013)

[35]

Yang, Y., Konstantinides, D. Asymptotic for ruin probabilities in a discrete-time risk model with dependent financial and insurance risks. Scandinavian Actuarial Journal, 8:641-659(2015)

[36]

Yi, L., Chen, Y., Su, C. Approximation of the tail probability of randomly weighted sums of dependent random variables with dominated variation. Journal of Mathematical Analysis and Applications, 376:365-372(2011)

[37]

Zhang, Y., Shen, X., Weng, C. Approximation of the tail probability of randomly weighted sums and applications. Stochastic Processes and their Applications, 119:655-675(2009)