Abstract
We consider a discrete-time risk model with insurance and financial risks. Within period i ≥ 1, the real-valued net insurance loss caused by claims is the insurance risk, denoted by X i , and the positive stochastic discount factor over the same time period is the financial risk, denoted by Y i . Assume that {(X, Y), (X i , Y i ), i ≥ 1} form a sequence of independent identically distributed random vectors. In this paper, we investigate a discrete-time risk model allowing a dependence structure between the two risks. When (X, Y ) follows a bivariate Sarmanov distribution and the distribution of the insurance risk belongs to the class ℒ(γ) for some γ > 0, we derive the asymptotics for the finite-time ruin probability of this discrete-time risk model.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
N.H. Bingham, C.M. Goldie, and J.L. Teugels, Regular Variation, Cambridge Univ. Press, Cambridge, 1987.
Y. Chen, The finite-time ruin probability with dependent insurance and financial risks, J. Appl. Probab., 48(4):1035–1048, 2011.
Y. Chen and K.C. Yuen, Sums of pairwise quasi-asymptotically independent random variables with consistent variation, Stoch. Models, 25(1):76–89, 2009.
D. Cheng, F. Ni, A.G. Pakes, and Y. Wang, Some properties of the exponential distribution class with applications to risk theory, J. Korean Stat. Soc., 41(4):515–527, 2012.
V.P. Chistyakov, A theorem on sums of independent positive random variables and its applications to branching processes, Theory Probab. Appl., 9(4):640–648, 1964.
J. Chover, P. Ney, and S.Wainger, Degeneracy properties of subcritical branching processes, Ann. Probab., 1(4):663–673, 1973.
J. Chover, P. Ney, and S. Wainger, Functions of probability measures, J. Anal. Math., 26(1):255–302, 1973.
P. Embrechts, C. Klüppelberg, and T. Mikosch, Modelling Extremal Events for Insurance and Finance, Springer, Berlin, 1997.
Q. Gao and Y. Wang, Randomly weighted sums with dominated varying-tailed increments and application to risk theory, J. Korean Stat. Soc., 39(3):305–314, 2010.
M.J. Goovaerts, R. Kaas, R.J.A. Laeven, Q. Tang, and R. Vernic, The tail probability of discounted sums of paretolike losses in insurance, Scand. Actuar. J., 2005(6):446–461, 2005.
E. Hashorva, A.G. Pakes, and Q. Tang, Asymptotics of random contractions, InsuranceMath. Econ., 47(3):405–414, 2010.
S. Kotz, N. Balakrishnan, and N.L. Johnson, ContinuousMultivariate Distributions. Vol. 1: Models and Applications, Wiley, New York, 2000.
M.T. Lee, Properties and applications of the Sarmanov family of bivariate distributions, Commun. Stat., Theory Methods, 25(6):1207–1222, 1996.
J. Li and Q. Tang, Interplay of insurance and financial risks in a discrete-time model with strongly regular variation, Bernoulli, 21(3):1800–1823, 2015.
K.Maulik and M. Podder, Ruin probabilities under Sarmanov dependence structure, Stat. Probab. Lett., 117(3):173–182, 2016.
R. Norberg, Ruin problems with assets and liabilities of diffusion type, Stochastic Processes Appl., 81(2):255–269, 1999.
H. Nyrhinen, On the ruin probabilities in a general economic environment, Stochastic Processes Appl., 83(2):319–330, 1999.
H. Nyrhinen, Finite and infinite time ruin probabilities in a stochastic economic environment, Stochastic Processes Appl., 92(2):265–285, 2001.
A. Pakes, Convolution equivalence and infinite divisibility, J. Appl. Probab., 41(2):407–424, 2004.
X. Shen, Z. Lin, and Y. Zhang, Uniform estimate for maximum of randomly weighted sums with applications to ruin theory, Methodol. Comput. Appl. Probab., 11(4):669–685, 2009.
Q. Tang, Asymptotic ruin probabilities in finite horizon with subexponential losses and associated discount factors, Probab. Eng. Inf. Sci., 20(1):103–113, 2006.
Q. Tang, From light tails to heavy tails through multiplier, Extremes, 11(4):379–391, 2008.
Q. Tang and G. Tsitsiashvili, Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and financial risks, Stochastic Processes Appl., 108(4):299–325, 2003.
Q. Tang and G. Tsitsiashvili, Finite- and infinite-time ruin probabilities in the presence of stochastic returns on investments, Adv. Appl. Probab., 36(4):1278–1299, 2004.
Q. Tang and Z. Yuan, Interplay of insurance and financial risks with bivariate regular variation, in D.K. Dey and J. Yan (Eds.), Extreme ValueModeling and Risk Analysis: Methods and Applications, Chapman and Hall/CRC, Boca Raton, FL, 2015, pp. 419–438.
C. Weng, Y. Zhang, and K.S. Tan, Ruin probabilities in a discrete time risk model with dependent risks of heavy tail, Scand. Actuar. J., 2009(3):205–218, 2009.
Y. Yang and Y. Wang, Tail behavior of the product of two dependent random variables with applications to risk theory, Extremes, 16(1):55–74, 2013.
L. Yi, Y. Chen, and C. Su, Approximation of the tail probability of randomly weighted sums of dependent random variables with dominated variation, J. Math. Anal. Appl., 376(1):365–372, 2011.
Y. Zhang, X. Shen, and C.Weng, Approximation of the tail probability of randomly weighted sums and applications, Stochastic Processes Appl., 119(2):655–675, 2009.
M. Zhou, K. Wang, and Y. Wang, Estimates for the finite-time ruin probability with insurance and financial risks, Acta Math. Appl. Sin., Engl. Ser., 28(4):795–806, 2012.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work is supported by the National Natural Science Foundation of China (Nos. 11401418, 71471090), the 333 Talent Training Project of Jiangsu Province, the Jiangsu Province Key Discipline in the 13th Five-Year Plan and the Graduate Research Innovation Project of SUST (No. SKCX16_056).
Rights and permissions
About this article
Cite this article
Wang, K., Gao, M., Yang, Y. et al. Asymptotics for the finite-time ruin probability in a discrete-time risk model with dependent insurance and financial risks*. Lith Math J 58, 113–125 (2018). https://doi.org/10.1007/s10986-017-9378-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10986-017-9378-8