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Finite- and Infinite-Time Ruin Probabilities with General Stochastic Investment Return Processes and Bivariate Upper Tail Independent and Heavy-Tailed Claims

Part of: Applications Mathematical economics

Published online by Cambridge University Press:  04 January 2016

Fenglong Guo  and
Dingcheng Wang
Fenglong Guo
Affiliation:
Nanjing Audit University, and University of Electronic Science and Technology of China
Dingcheng Wang*
Affiliation:
Nanjing Audit University, Australian National University, and University of Electronic Science and Technology of China
*
∗ Postal address: Center of Financial Engineering, Nanjing Audit University, Nanjing 211815, China. Email address: wangdc@nau.edu.cn
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Abstract

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In this paper we investigate the asymptotic behaviors of the finite- and infinite-time ruin probabilities for a Poisson risk model with stochastic investment returns which constitute a general adapted càdlàg process and heavy-tailed claim sizes which are bivariate upper tail independent. The results of this paper show that the asymptotic ruin probabilities are dominated by the extreme of insurance risk but not by that of investment risk. As applications of the results, we discuss four special cases when the investment returns are determined by a fractional Brownian motion, an integrated Vasicek model, an integrated Cox–Ingersoll–Ross model, and the Heston model.

Keywords

ruin probabilityheavy tailinvestment return processupper tail dependencefractional Brownian motionVasicek modelCox–Ingersoll–Ross modelHeston model

MSC classification

Primary: 62P05: Applications to actuarial sciences and financial mathematics
Secondary: 62P20: Applications to economics 91B30: Risk theory, insurance
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 45 , Issue 1 , March 2013 , pp. 241 - 273
Copyright
© Applied Probability Trust 

References

Biagini, F., Hu, Y., Øksendal, B. and Zhang, T. (2008). Stochastic Calculus for Fractional Brownian Motion and Applications. Springer, London.CrossRefGoogle Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.CrossRefGoogle Scholar
Breiman, L. (1965). On some limit theorems similar to the arc-sin law. Theoret. Prob. Appl. 10, 323–331.Google Scholar
Brigo, D. and Mercurio, F. (2001). Interest Rate Models—Theory and Practice. Springer, Berlin.Google Scholar
Cai, J. and Tang, Q. (2004). On max-sum equivalence and convolution closure of heavy-tailed distributions and their applications. J. Appl. Prob. 41, 117–130.Google Scholar
Cai, J. and Yang, H. (2005). Ruin in the perturbed compound Poisson risk process under interest force. Adv. Appl. Prob. 37, 819–835.Google Scholar
Chen, Y. and Ng, K. W. (2007). The ruin probability of the renewal model with constant interest force and negatively dependent heavy-tailed claims. Insurance Math. Econom. 40, 415–423.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
Collamore, J. F. (2009). Random recurrence equations and ruin in a Markov-dependent stochastic economic environment. Ann. Appl. Prob. 19, 1404–1458.CrossRefGoogle Scholar
Drăgulescu, A. A. and Yakovenko, V. M. (2002). Probability distribution of returns in the Heston model with stochastic volatility. Quant. Finance 2, 443–453.Google Scholar
Dufresne, D. (2001). The integrated square-root process. Research paper.Google Scholar
Emmer, S. and Klüppelberg, C. (2004). Optimal portfolios when stock prices follow an exponential Lévy process. Finance Stoch. 8, 17–44.CrossRefGoogle Scholar
Frolova, A., Kabanov, Y. and Pergamenshchikov, S. (2002). In the insurance business risky investments are dangerous. Finance Stoch. 6, 227–235.CrossRefGoogle Scholar
Gaier, J. and Grandits, P. (2002). Ruin probabilities in the presence of regularly varying tails and optimal investment. Insurance Math. Econom. 30, 211–217.Google Scholar
Goldie, C. M. (1991). Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Prob. 1, 126–166.CrossRefGoogle Scholar
Heyde, C. C. and Wang, D. (2009). Finite-time ruin probability with an exponential Lévy process investment return and heavy-tailed claims. Adv. Appl. Prob. 41, 206–224.Google Scholar
Kalashnikov, V. and Norberg, R. (2002). Power tailed ruin probabilities in the presence of risky investments. Stoch. Process. Appl. 98, 211–228.Google Scholar
Klüppelberg, C. and Kostadinova, R. (2008). Integrated insurance risk models with exponential Lévy investment. Insurance Math. Econom. 42, 560–577.Google Scholar
Konstantinides, D. G. and Mikosch, T. (2005). Large deviations and ruin probabilities for solutions to stochastic recurrence equations with heavy-tailed innovations. Ann. Prob. 33, 1992–1992.Google Scholar
Mikosch, T. and Samorodnitsky, G. (2000). The supremum of a negative drift random walk with dependent heavy-tailed steps. Ann. Appl. Prob. 10, 1025–1064.Google Scholar
Nelsen, R. B. (2006). An Introduction to Copulas, 2nd edn. Springer, New York.Google Scholar
Paulsen, J. (2002). On Cramér-like asymptotics for risk processes with stochastic return on investments. Ann. Appl. Prob. 12, 1247–1260.CrossRefGoogle Scholar
Paulsen, J. and Gjessing, H. K. (1997). Ruin theory with stochastic return on investments. Adv. Appl. Prob. 29, 965–985.Google Scholar
Protter, P. (2004). Stochastic Integration and Differential Equations, 2nd edn. Springer, Berlin.Google Scholar
Shreve, S. E. (2004). Stochastic Calculus for Finance II. Springer, New York.CrossRefGoogle Scholar
Tang, Q. (2006). Insensitivity to negative dependence of the asymptotic behavior of precise large deviations. Electron. J. Prob. 11, 107–120.CrossRefGoogle Scholar
Wang, D. and Tang, Q. (2004). Maxima of sums and random sums for negatively associated random variables with heavy tails. Stat. Prob. Lett. 68, 287–295.CrossRefGoogle Scholar
Wang, D. and Tang, Q. (2006). Tail probabilities of randomly weighted sums of random variables with dominated variation. Stoch. Models 22, 253–272.Google Scholar
Wang, D., Chen, P. and Su, C. (2007). The supremum of random walk with negatively associated and heavy-tailed steps. Statist. Prob. Lett. 77, 1403–1412.Google Scholar
Wang, D., Su, C. and Zeng, Y. (2005). Uniform estimate for maximum of randomly weighted sums with applications to insurance risk theory. Sci. China Ser. A 48, 1379–1394.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