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The finite-time ruin probability of the compound Poisson model with constant interest force

Part of: Stochastic processes Mathematical economics Applications

Published online by Cambridge University Press:  14 July 2016

Qihe Tang
Qihe Tang*
Affiliation:
Concordia University
*
∗Postal address: Department of Mathematics and Statistics, Concordia University, 7141 Sherbrooke Street West, Montreal, Quebec H4B 1R6, Canada. Email address: [email protected]
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Abstract

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In this paper, we establish a simple asymptotic formula for the finite-time ruin probability of the compound Poisson model with constant interest force and subexponential claims in the case that the initial surplus is large. The formula is consistent with known results for the ultimate ruin probability and, in particular, is uniform for all time horizons when the claim size distribution is regularly varying tailed.

Keywords

Asymptoticsfinite-time ruin probabilityPoisson processregular variationsubexponentialityuniform asymptoticsuniform convergence

MSC classification

Primary: 91B30: Risk theory, insurance
Secondary: 60G70: Extreme value theory; extremal processes 62P05: Applications to actuarial sciences and financial mathematics
Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 3 , September 2005 , pp. 608 - 619
Copyright
© Applied Probability Trust 2005 

References

Asmussen, S. (1998). Subexponential asymptotics for stochastic processes: extremal behavior, stationary distributions and first passage probabilities. Ann. Appl. Prob. 8, 354–374.Google Scholar
Asmussen, S. et al. (2002). A local limit theorem for random walk maxima with heavy tails. Statist. Prob. Lett. 56, 399–404.CrossRefGoogle Scholar
Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, New York.Google Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.Google Scholar
Chistyakov, V. P. (1964). A theorem on sums of independent positive random variables and its applications to branching random processes. Teor. Veroyat. Primen. 9, 710–718 (in Russian). English translation: Theory Prob. Appl. 9, 640-648.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
Embrechts, P. and Omey, E. (1984). A property of longtailed distributions. J. Appl. Prob. 21, 80–87.Google Scholar
Embrechts, P., Goldie, C. M. and Veraverbeke, N. (1979). Subexponentiality and infinite divisibility. Z. Wahrscheinlichkeitsth. 49, 335–347.Google Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Springer, Berlin.Google Scholar
Kalashnikov, V. and Konstantinides, D. (2000). Ruin under interest force and subexponential claims: a simple treatment. Insurance Math. Econom. 27, 145–149.Google Scholar
Klüppelberg, C. (1988). Subexponential distributions and integrated tails. J. Appl. Prob. 25, 132–141.Google Scholar
Klüppelberg, C. and Stadtmüller, U. (1998). Ruin probabilities in the presence of heavy-tails and interest rates. Scand. Actuarial J. 1998, 49–58.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.CrossRefGoogle Scholar
Nagaev, S. V. (1979). Large deviations of sums of independent random variables. Ann. Prob. 7, 745–789.Google Scholar
Petrov, V. V. (1995). Limit Theorems of Probability Theory. Sequences of Independent Random Variables. Oxford University Press.Google Scholar
Ross, S. M. (1983). Stochastic Processes. John Wiley, New York.Google Scholar
Sundt, B. and Teugels, J. L. (1995). Ruin estimates under interest force. Insurance Math. Econom. 16, 7–22.CrossRefGoogle Scholar
Tang, Q. (2004). The ruin probability of a discrete time risk model under constant interest rate with heavy tails. Scand. Actuarial J. 2004, 229–240.Google Scholar
Tang, Q. (2005). Asymptotic ruin probabilities of the renewal model with constant interest force and regular variation. Scand. Actuarial J. 2005, 1–5.Google Scholar
Tang, Q. and Tsitsiashvili, G. (2003). Randomly weighted sums of subexponential random variables with application to ruin theory. Extremes 6, 171–188.Google Scholar