×

zbMATH — the first resource for mathematics

The finite-time ruin probability of the compound Poisson model with constant interest force. (English) Zbl 1132.91500
Summary: 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.

MSC:
91B30 Risk theory, insurance (MSC2010)
60G70 Extreme value theory; extremal stochastic processes
62P05 Applications of statistics to actuarial sciences and financial mathematics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Asmussen, S. (1998). Subexponential asymptotics for stochastic processes: extremal behavior, stationary distributions and first passage probabilities. Ann. Appl. Prob. 8 , 354–374. · Zbl 0942.60034 · doi:10.1214/aoap/1028903531
[2] Asmussen, S. et al. (2002). A local limit theorem for random walk maxima with heavy tails. Statist. Prob. Lett. 56 , 399–404. · Zbl 0997.60047
[3] Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, New York. · Zbl 0259.60002
[4] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press. · Zbl 0617.26001
[5] 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. · Zbl 0203.19401
[6] Cline, D. B. H. and Samorodnitsky, G. (1994). Subexponentiality of the product of independent random variables. Stoch. Process. Appl. 49 , 75–98. · Zbl 0799.60015 · doi:10.1016/0304-4149(94)90113-9
[7] Embrechts, P. and Omey, E. (1984). A property of longtailed distributions. J. Appl. Prob. 21 , 80–87. · Zbl 0534.60015 · doi:10.2307/3213666
[8] Embrechts, P., Goldie, C. M. and Veraverbeke, N. (1979). Subexponentiality and infinite divisibility. Z. Wahrscheinlichkeitsth. 49 , 335–347. · Zbl 0397.60024 · doi:10.1007/BF00535504
[9] Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Springer, Berlin. · Zbl 0873.62116
[10] Kalashnikov, V. and Konstantinides, D. (2000). Ruin under interest force and subexponential claims: a simple treatment. Insurance Math. Econom. 27 , 145–149. · Zbl 1056.60501 · doi:10.1016/S0167-6687(00)00045-7
[11] Klüppelberg, C. (1988). Subexponential distributions and integrated tails. J. Appl. Prob. 25 , 132–141. · Zbl 0651.60020 · doi:10.2307/3214240
[12] Klüppelberg, C. and Stadtmüller, U. (1998). Ruin probabilities in the presence of heavy-tails and interest rates. Scand. Actuarial J. 1998 , 49–58. · Zbl 1022.60083 · doi:10.1080/03461238.1998.10413991
[13] 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. · Zbl 1074.91029 · doi:10.1016/S0167-6687(02)00189-0
[14] Nagaev, S. V. (1979). Large deviations of sums of independent random variables. Ann. Prob. 7 , 745–789. JSTOR: · Zbl 0418.60033 · doi:10.1214/aop/1176994938 · links.jstor.org
[15] Petrov, V. V. (1995). Limit Theorems of Probability Theory. Sequences of Independent Random Variables. Oxford University Press. · Zbl 0826.60001
[16] Ross, S. M. (1983). Stochastic Processes. John Wiley, New York. · Zbl 0555.60002
[17] Sundt, B. and Teugels, J. L. (1995). Ruin estimates under interest force. Insurance Math. Econom. 16 , 7–22. · Zbl 0838.62098 · doi:10.1016/0167-6687(94)00023-8
[18] 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. · Zbl 1142.62094 · doi:10.1080/03461230310017531
[19] Tang, Q. (2005). Asymptotic ruin probabilities of the renewal model with constant interest force and regular variation. Scand. Actuarial J. 2005 , 1–5. · Zbl 1144.91030 · doi:10.1080/03461230510006982
[20] Tang, Q. and Tsitsiashvili, G. (2003). Randomly weighted sums of subexponential random variables with application to ruin theory. Extremes 6 , 171–188. · Zbl 1049.62017 · doi:10.1023/B:EXTR.0000031178.19509.57
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.