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Ruin problems with stochastic premium stochastic return on investments. (English) Zbl 1148.60067
Summary: Ruin problems in the risk model with stochastic premium incomes and stochastic return on investments are studied. The logarithm of the asset price process is assumed to be a Lévy process. An exact expression for expected discounted penalty function is established. Lower bounds and two kinds of upper bounds for expected discounted penalty function are obtained by the inductive method and martingale approach. Integro-differential equations for the expected discounted penalty function are obtained when the Lévy process is a Brownian motion with positive drift and a compound Poisson process, respectively. Some analytical examples and numerical examples are given to illustrate the upper bounds and the applications of the integro-differential equations in this paper.

##### MSC:
 60K10 Applications of renewal theory (reliability, demand theory, etc.) 60G44 Martingales with continuous parameter 60J65 Brownian motion 60K05 Renewal theory 91B28 Finance etc. (MSC2000) 91B30 Risk theory, insurance (MSC2010)
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##### References:
 [1] Cai J. Ruin probabilities and penalty functions with stochastic rates of interest. Stochastic Process Appl, 2004, 112: 53–78 · Zbl 1070.60043 · doi:10.1016/j.spa.2004.01.007 [2] Yang H L, Zhang L H. Optimal investment for insurer with jump-diffusion risk process. Insurance Math Econom, 2005, 37: 615–634 · Zbl 1129.91020 · doi:10.1016/j.insmatheco.2005.06.009 [3] Paulsen J, Gjessing H. Ruin theory with stochastic return on investments. Adv Appl Probab, 1997, 29: 965–985 · Zbl 0892.90046 · doi:10.2307/1427849 [4] Paulsen J. Ruin theory with compounding assets–a survey. Insurance Math Econom, 1998, 22: 3–16 · Zbl 0909.90115 · doi:10.1016/S0167-6687(98)00009-2 [5] Kalashnikov V, Norberg R. Power tailed ruin probabilities in the presence of risky investments. Stochastic Process Appl, 2002, 98: 211–228 · Zbl 1058.60095 · doi:10.1016/S0304-4149(01)00148-X [6] Wang R M, Yang H L, Wang H X. On the distribution of surplus immediately after ruin under interest force and subexponential claims. Insurance Math Econom, 2004, 35: 703–714 · Zbl 1122.91347 · doi:10.1016/j.insmatheco.2004.07.002 [7] Wu R, Wang G J, Zhang C S. On a joint distribution for the risk process with constant interest rate. Insurance Math Econom, 2005, 36: 365–374 · Zbl 1110.62149 · doi:10.1016/j.insmatheco.2005.03.001 [8] Yuen K C, Wang G J, Wu R. On the risk process with stochastic interest. Stochastic Process Appl, 2006, 116: 1496–1510 · Zbl 1109.60071 · doi:10.1016/j.spa.2006.04.012 [9] Melnikov L. Risk analysis in finance and insurance. New York: CRC Press, 2004 · Zbl 1031.91087 [10] Embrechts P, Klüppelberg C, Mikosch T. Modelling Extremal Events. Berlin: Springer, 1997 · Zbl 0873.62116 [11] Gerber H U, Shiu E S W. The joint distribution of the time of ruin, the surplus immediatly before ruin, and the deficit at ruin. Insurance Math Econom, 1997, 21: 129–137 · Zbl 0894.90047 · doi:10.1016/S0167-6687(97)00027-9 [12] Cai J, Dickson D C M. On the expected discounted penalty function at ruin of a surplus process with interest. Insurance Math Econom, 2002, 30: 389–404 · Zbl 1074.91027 · doi:10.1016/S0167-6687(02)00120-8 [13] Grandell J. Aspects of Risk Theory. New York: Springer, 1991 · Zbl 0717.62100 [14] Cai J, Dickson D C M. Upper bounds for ultimate ruin probabilities in the Sparre Andersen model with interest. Insurance Math Econom, 2003, 32: 61–71 · Zbl 1074.91028 · doi:10.1016/S0167-6687(02)00204-4 [15] Dufresne F, Gerber H U. Risk theory for the compound Poisson process that is perturbed by diffusion. Insurance Math Econom, 1991, 10: 51–59 · Zbl 0723.62065 · doi:10.1016/0167-6687(91)90023-Q
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