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The perturbed compound Poisson risk model with two-sided jumps. (English) Zbl 1185.91198
In this rather technical paper, the authors study a classical compound Poisson risk model perturbed by a Brownian motion with two-sided jumps. The upward jumps can be interpreted as the random gains of an insurance company, the downward jumps being the random losses. Defective renewal equations, discounted penalty functions (at ruin caused by a jump or caused by oscillation) and their Laplace transforms are derived. Asymptotic behavior for the probability of ruin is studied in the case the loss jumps are heavy-tailed. A numerical example concludes the paper.

##### MSC:
 91G80 Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems) 91B30 Risk theory, insurance
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##### References:
 [1] Gerber, H. U.: An extension of the renewal equation and its application in the collective theory of risk, Skandinavisk aktuarietidskrift, 205-210 (1970) · Zbl 0229.60062 [2] Dufresne, F.; Gerber, H. U.: Risk theory for the compound Poisson process that is perturbed by diffusion, Insurance: mathematics and economics 10, No. 1, 51-59 (1991) · Zbl 0723.62065 · doi:10.1016/0167-6687(91)90023-Q [3] Gerber, H. U.; Landry, B.: On the discounted penalty at ruin in a jump-diffusion and the perpetual put option, Insurance: mathematics and economics 22, No. 3, 263-276 (1998) · Zbl 0924.60075 · doi:10.1016/S0167-6687(98)00014-6 [4] Tsai, C. C. L.: On the discounted distribution functions of the surplus process perturbed by diffusion, Insurance: mathematics and economics 28, No. 3, 401-419 (2001) · Zbl 1074.91562 · doi:10.1016/S0167-6687(01)00067-1 [5] Tsai, C. C. L.; Willmot, G. E.: A generalized defective renewal equation for the surplus process perturbed by diffusion, Insurance: mathematics and economics 30, No. 1, 51-66 (2002) · Zbl 1074.91563 · doi:10.1016/S0167-6687(01)00096-8 [6] Tsai, C. C. L.; Willmot, G. E.: On the moments of the surplus process perturbed by diffusion, Insurance: mathematics and economics 31, No. 3, 327-350 (2002) · Zbl 1063.91051 · doi:10.1016/S0167-6687(02)00159-2 [7] Perry, D.; Stadje, W.; Zacks, S.: First-exit time for the compound Poisson processes for some types of positive and negative jumps, Stochastic models 18, No. 1, 139-157 (2002) · Zbl 0998.60089 · doi:10.1081/STM-120002778 [8] Kou, S. G.; Wang, H.: First passage times of a jump diffusion process, Advances in applied probability 35, No. 2, 504-531 (2003) · Zbl 1037.60073 · doi:10.1239/aap/1051201658 [9] Xing, X.; Zhang, W.; Jiang, Y.: On the time to ruin and the deficit at ruin in a risk model with double-sided jumps, Statistics & probability letters 78, No. 16, 2692-2699 (2008) · Zbl 1153.91024 · doi:10.1016/j.spl.2008.03.034 [10] Jacobsen, M.: The time to ruin for a class of Markov additive risk process with two-sided jumps, Advances in applied probability 37, No. 4, 963-992 (2005) · Zbl 1100.60021 · doi:10.1239/aap/1134587749 [11] H. Yang, Z. Zhang, On a compound Poisson risk model with two-sided jumps, Working paper, 2008 [12] Wang, G.: A decomposition of the ruin probability for the risk process perturbed by diffusion, Insurance: mathematics and economics 28, No. 1, 49-59 (2001) · Zbl 0993.60087 · doi:10.1016/S0167-6687(00)00065-2 [13] Li, S.; Garrido, J.: The gerber--shiu function in a sparre andersen risk process perturbed by diffusion, Scandinavian actuarial journal 3, 161-186 (2005) · Zbl 1092.91049 · doi:10.1080/03461230510006955 [14] Wang, G.; Wu, R.: The expected discounted penalty function for the perturbed compound Poisson risk process with constant interest, Insurance: mathematics and economics 42, No. 1, 59-64 (2008) · Zbl 1141.91551 · doi:10.1016/j.insmatheco.2006.12.003 [15] Dickson, D. C. M.; Hipp, C.: On the time to ruin for $Erlang(2)$ risk processes, Insurance: mathematics and economics 29, No. 3, 333-344 (2001) · Zbl 1074.91549 · doi:10.1016/S0167-6687(01)00091-9 [16] Li, S.; Garrido, J.: On ruin for the $Erlang(n)$ risk process, Insurance: mathematics and economics 34, No. 3, 391-408 (2004) · Zbl 1188.91089 · doi:10.1016/j.insmatheco.2004.01.002 [17] Lin, X. S.; Willmot, G. E.: Analysis of a defective renewal equation arising in in ruin theory, Insurance: mathematics and economics 25, No. 1, 63-84 (1999) · Zbl 1028.91556 · doi:10.1016/S0167-6687(99)00026-8 [18] Rolski, T.; Schmidli, H.; Schmidt, V.; Teugels, J.: Stochastic processes for insurance and finance, (1999) · Zbl 0940.60005 [19] Embrechts, P.; Klüppelberg, C.; Mikosch, T.: Modeling extrement events for insurance and finance, (1997) [20] Embrechts, P.; Villasenor, J. A.: Ruin estimation for large claims, Insurance: mathematics and economics 7, No. 4, 269-274 (1988) · Zbl 0666.62098 · doi:10.1016/0167-6687(88)90084-4 [21] Yin, C.; Zhao, J.: Nonexponential asymptotics for the solutions of renewal equations, with applications, Journal of applied probability 43, No. 3, 815-824 (2006) · Zbl 1125.60090 · doi:10.1239/jap/1158784948 [22] Veraverbeke, N.: Asymptotic estimates for the probability of ruin in a Poisson model with diffusion, Insurance: mathematics and economics 13, No. 1, 57-62 (1993) · Zbl 0790.62098 · doi:10.1016/0167-6687(93)90535-W