Abstract
Considering an insurer who is allowed to make risk-free and risky investments, as in Tang et al. (2010), the price process of the investment portfolio is described as a geometric Lévy process. We study the tail probability of the stochastic present value of future aggregate claims. When the claim-size distribution is of extended regular variation, we obtain an asymptotically equivalent formula which holds uniformly for all time horizons, and furthermore, the same asymptotic formula holds for the finite-time ruin probabilities. The results extend the works of Tang et al. (2010).
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References
Applebaum D. Lévy Processes and Stochastic Calculus. Cambridge: Cambridge University Press, 2004
Bingham N H, Goldie C M and Teugels J L. Regular Variation. Cambridge: Cambridge University Press, 1987
Cai J. Ruin probabilities and penalty functions with stochastic rates of interest. Stochastic Process Appl, 2004, 112: 53–78
Cline D B H, Samorodnitsky G. Subexponentiality of the product of independent random variables. Stochastic Process Appl, 1994, 49: 75–98
Cont R, Tankov P. Financial Modelling with Jump Processes. Boca Raton, FL: Chapman & Hall/CRC, 2004
Embrechts P, Klüppelberg C, Mikosch T. Modelling Extremal Events for Insurance and Finance. Berlin: Springer-Verlag, 1997
Hao X, Tang Q. Asymptotic ruin probabilities for a bivariate Lévy-driven risk model with heavy-tailed claims and risky investments. J Appl Probab, 2012, 49: 939–954
Kalashnikov V, Norberg R. Power tailed ruin probabilities in the presence of risky investments. Stochastic Process Appl, 2002, 98: 211–228
Maulik K, Zwart B. Tail asymptotics for exponential functionals of Lévy processes. Stochastic Process Appl, 2006, 116: 156–177
Paulsen J. On Cramér-like asymptotics for risk processes with stochastic return on investments. Ann Appl Probab, 2002, 12: 1247–1260
Paulsen J. Ruin models with investment income. Probab Surv, 2008, 5: 416–434
Resnick S I, Willekens E. Moving averages with random coefficients and random coefficient autoregressive models. Comm Statist Stochastic Models, 1991, 7: 511–525
Sato K. Lévy Processes and Infinitely Divisible Distributions. Cambridge: Cambridge University Press, 1999
Tang Q. Uniform estimates for the tail probability of maxima over finite horizons with subexponential tails. Probab Engrg Inform Sci, 2004, 18: 71–86
Tang Q, Tsitsiashvili G. Randomly weighted sums of subexponential random variables with application to ruin theory. Extremes, 2003, 6: 171–188
Tang Q, Wang G J, Yuen K C. Uniform tail asymptotics for the stochastic present value of aggregate claims in the renewal risk model. Insurance Math Econom, 2010, 46: 362–370
Tang Q, Yuan Z Y. Randomly weighted sums of subexponential random variables in insurance and finance. Extremes, 2014, in press
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Zhu, C., Gao, Q. & Lin, J. Uniform tail asymptotics for the aggregate claims with stochastic discount in the renewal risk models. Sci. China Math. 58, 1079–1090 (2015). https://doi.org/10.1007/s11425-014-4863-6
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DOI: https://doi.org/10.1007/s11425-014-4863-6