Journal of Applied Probability

The finite-time ruin probability with dependent insurance and financial risks

Yiqing Chen

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Consider a discrete-time insurance risk model. Within period i, the net insurance loss is denoted by a real-valued random variable Xi. The insurer makes both risk-free and risky investments, leading to an overall stochastic discount factor Yi from time i to time i - 1. Assume that (Xi, Yi), iN, form a sequence of independent and identically distributed random pairs following a common bivariate Farlie-Gumbel-Morgenstern distribution with marginal distribution functions F and G. When F is subexponential and G fulfills some constraints in order for the product convolution of F and G to be subexponential too, we derive a general asymptotic formula for the finite-time ruin probability. Then, for special cases in which F belongs to the Fréchet or Weibull maximum domain of attraction, we improve this general formula to be transparent.

Article information

J. Appl. Probab., Volume 48, Number 4 (2011), 1035-1048.

First available in Project Euclid: 16 December 2011

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Zentralblatt MATH identifier

Primary: 62P05: Applications to actuarial sciences and financial mathematics
Secondary: 62E10: Characterization and structure theory 91B30: Risk theory, insurance

Asymptotics Farlie-Gumbel-Morgenstern distribution maximum domain of attraction finite-time ruin probability subexponential distribution


Chen, Yiqing. The finite-time ruin probability with dependent insurance and financial risks. J. Appl. Probab. 48 (2011), no. 4, 1035--1048. doi:10.1239/jap/1324046017.

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  • Breiman, L. (1965). On some limit theorems similar to the arc-sin law. Theory Prob. Appl. 10, 323–331.
  • Cline, D. B. H. (1986). Convolution tails, product tails and domains of attraction. Prob. Theory Relat. Fields 72, 529–557.
  • Cline, D. B. H. and Samorodnitsky, G. (1994). Subexponentiality of the product of independent random variables. Stoch. Process. Appl. 49, 75–98.
  • Cossette, H., Marceau, E. and Marri, F. (2008). On the compound Poisson risk model with dependence based on a generalized Farlie–Gumbel–Morgenstern copula. Insurance Math. Econom. 43, 444–455.
  • Embrechts, P. and Goldie, C. M. (1980). On closure and factorization properties of subexponential and related distributions. J. Austral. Math. Soc. Ser. A 29, 243–256.
  • Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events. Springer, Berlin.
  • Goovaerts, M. J. et al. (2005). The tail probability of discounted sums of Pareto-like losses in insurance. Scand. Actuarial J. 2005, 446–461.
  • Grey, D. R. (1994). Regular variation in the tail behaviour of solutions of random difference equations. Ann. Appl. Prob. 4, 169–183.
  • Hashorva, E., Pakes, A. G. and Tang, Q. (2010). Asymptotics of random contractions. Insurance Math. Econom. 47, 405–414.
  • Jiang, J. and Tang, Q. (2011). The product of two dependent random variables with regularly varying or rapidly varying tails. Statist. Prob. Lett. 81, 957–961.
  • Klüppelberg, C. (1988). Subexponential distributions and integrated tails. J. Appl. Prob. 25, 132–141.
  • 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.
  • Kotz, S., Balakrishnan, N. and Johnson, N. L. (2000). Continuous Multivariate Distributions, Vol. 1, 2nd edn. Wiley-Interscience, New York.
  • Nyrhinen, H. (1999). On the ruin probabilities in a general economic environment. Stoch. Process. Appl. 83, 319–330.
  • Nyrhinen, H. (2001). Finite and infinite time ruin probabilities in a stochastic economic environment. Stoch. Process. Appl. 92, 265–285.
  • Tang, Q. (2006a). Asymptotic ruin probabilities in finite horizon with subexponential losses and associated discount factors. Prob. Eng. Inf. Sci. 20, 103–113.
  • Tang, Q. (2006b). The subexponentiality of products revisited. Extremes 9, 231–241.
  • 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.
  • Tang, Q. and Tsitsiashvili, G. (2004). Finite- and infinite-time ruin probabilities in the presence of stochastic returns on investments. Adv. Appl. Prob. 36, 1278–1299.
  • Tang, Q. and Vernic, R. (2007). The impact on ruin probabilities of the association structure among financial risks. Statist. Prob. Lett. 77, 1522–1525.
  • Teugels, J. L. (1975). The class of subexponential distributions. Ann. Prob. 3, 1000–1011.
  • Weng, C., Zhang, Y. and Tan, K. S. (2009). Ruin probabilities in a discrete time risk model with dependent risks of heavy tail. Scand. Actuarial J. 2009, 205–218.
  • Yi, L., Chen, Y. and Su, C. (2011). Approximation of the tail probability of randomly weighted sums of dependent random variables with dominated variation. J. Math. Anal. Appl. 376, 365–372.
  • Zhang, Y., Shen, X. and Weng, C. (2009). Approximation of the tail probability of randomly weighted sums and applications. Stoch. Process. Appl. 119, 655–675.
  • Zhou, M., Wang, K. and Wang, Y. (2011). Estimates for the finite-time ruin probability with insurance and financial risks. To appear in Acta Math. Appl. Sinica.