Ruin problems with stochastic premium stochastic return on investments

Abstract

In this paper, 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 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.

Keywords

Expected discounted penalty function / integro-differential equation / Lévy process / martingale method / ruin probability / stochastic premium income

References

[1.] Cai J. Ruin probabilities and penalty functions with stochastic rates of interest. Stochastic Process Appl, 2004, 112: 53-78.

[2.] Yang H. L., Zhang L. H. Optimal investment for insurer with jump-diffusion risk process. Insurance Math Econom, 2005, 37: 615-634.

[3.] Paulsen J., Gjessing H. Ruin theory with stochastic return on investments. Adv Appl Probab, 1997, 29: 965-985.

[4.] Paulsen J. Ruin theory with compounding assets—a survey. Insurance Math Econom, 1998, 22: 3-16.

[5.] Kalashnikov V., Norberg R. Power tailed ruin probabilities in the presence of risky investments. Stochastic Process Appl, 2002, 98: 211-228.

[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.

[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.

[8.] Yuen K. C., Wang G. J., Wu R. On the risk process with stochastic interest. Stochastic Process Appl, 2006, 116: 1496-1510.

[9.] Melnikov L. Risk analysis in finance and insurance, 2004, New York: CRC Press.

[10.] Embrechts P., Klüppelberg C., Mikosch T. Modelling Extremal Events, 1997, Berlin: Springer.

[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.

[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.

[13.] Grandell J. Aspects of Risk Theory, 1991, New York: Springer.

[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.

[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.