Hostname: page-component-7b7567c8c-kwt9d Total loading time: 0 Render date: 2026-11-14T06:05:21.639Z Has data issue: false hasContentIssue false
  • English
  • Français

The Finite-Time Ruin Probability with Dependent Insurance and Financial Risks

Part of:Distribution theoryMathematical economicsApplications

Published online by Cambridge University Press:  14 July 2016

Yiqing Chen
Yiqing Chen*
Affiliation:
University of Liverpool
*
∗ Postal address: Department of Mathematical Sciences, University of Liverpool, Liverpool, L69 7ZL, UK. Email address: yiqing.chen@liv.ac.uk
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

Consider a discrete-time insurance risk model. Within period i, the net insurance loss is denoted by a real-valued random variable X i . The insurer makes both risk-free and risky investments, leading to an overall stochastic discount factor Y i from time i to time i − 1. Assume that (X i , Y i ), i ∈ N, 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.

Keywords

AsymptoticsFarlie-Gumbel-Morgenstern distributionmaximum domain of attractionfinite-time ruin probabilitysubexponential distribution

MSC classification

Secondary:62P05: Applications to actuarial sciences and financial mathematics62E10: Characterization and structure theory91B30: Risk theory, insurance

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 48 , Issue 4 , December 2011, pp. 1035 - 1048
Copyright
Copyright © Applied Probability Trust 2011 

References

Breiman, L. (1965). On some limit theorems similar to the arc-sin law. Theory Prob. Appl. 10, 323–331.Google Scholar
Cline, D. B. H. (1986). Convolution tails, product tails and domains of attraction. Prob. Theory Relat. Fields 72, 529–557.Google Scholar
Cline, D. B. H. and Samorodnitsky, G. (1994). Subexponentiality of the product of independent random variables. Stoch. Process. Appl. 49, 75–98.Google Scholar
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.Google Scholar
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.Google Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events. Springer, Berlin.Google Scholar
Goovaerts, M. J. {it et al. } (2005). The tail probability of discounted sums of Pareto-like losses in insurance. Scand. Actuarial J. 2005, 446–461.Google Scholar
Grey, D. R. (1994). Regular variation in the tail behaviour of solutions of random difference equations. Ann. Appl. Prob. 4, 169–183.Google Scholar
Hashorva, E., Pakes, A. G. and Tang, Q. (2010). Asymptotics of random contractions. Insurance Math. Econom. 47, 405–414.Google Scholar
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.Google Scholar
Klüppelberg, C. (1988). Subexponential distributions and integrated tails. J. Appl. Prob. 25, 132–141.Google Scholar
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.Google Scholar
Kotz, S., Balakrishnan, N. and Johnson, N. L. (2000). Continuous Multivariate Distributions, Vol. 1, 2nd edn. Wiley-Interscience, New York.Google Scholar
Nyrhinen, H. (1999). On the ruin probabilities in a general economic environment. Stoch. Process. Appl. 83, 319–330.Google Scholar
Nyrhinen, H. (2001). Finite and infinite time ruin probabilities in a stochastic economic environment. Stoch. Process. Appl. 92, 265–285.Google Scholar
Tang, Q. (2006a). Asymptotic ruin probabilities in finite horizon with subexponential losses and associated discount factors. Prob. Eng. Inf. Sci. 20, 103–113.Google Scholar
Tang, Q. (2006b). The subexponentiality of products revisited. Extremes 9, 231–241.Google Scholar
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.Google Scholar
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.Google Scholar
Tang, Q. and Vernic, R. (2007). The impact on ruin probabilities of the association structure among financial risks. Statist. Prob. Lett. 77, 1522–1525.Google Scholar
Teugels, J. L. (1975). The class of subexponential distributions. Ann. Prob. 3, 1000–1011.Google Scholar
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.Google Scholar
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.Google Scholar
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.Google Scholar
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. Google Scholar