The Study of the Discrete Risk Model with Random Income
Authors: Peichen Zhao
In this paper, we extend the compound binomial model to the case where the premium income process, based on a binomial process, is no longer a linear function. First, a mathematically recursive formula is derived for non ruin probability, and then, we examine the expected discounted penalty function, satisfy a defect renewal equation. Third, the asymptotic estimate for the expected discounted penalty function is then given. Finally, we give two examples of ruin quantities to illustrate applications of the recursive formula and the asymptotic estimate for penalty function.
Keywords: Discounted penalty function, compound binomial process, recursive formula, discrete renewal equation, asymptotic estimate.
Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1080239Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1316
 Gerber,H. U. Mathematical fun with the compound binomial process. Astin Bulletin 18(1988)161-168.
 Shiu,E. S. W. Probability of eventual ruin in the compound binomial model. Astin Bulletin 19(1989)179-190.
 Dickson,D.C.M. Some comments on the compound binomial model. Astin Bulletin 24(1994) 33-45.
 Cheng,S., Zhu,R. The asymptotic formulas and Lundberg upper bound in fully discrete risk model. Applied Mathematics. A Journal of Chinese Universities Series A 16(3)(2001)348-358 (in Chinese).
 Cheng,S., Gerber,H.U., Shiu, E.S.W, Discounted probabilities and ruin theory in the compound binomial model. Insurance: Mathematics and Economics 26(2000) 239-250.
 Temnov,G. Risk process with random income. Journal of Mathematical Sciences 123(2004) 3780-3794.
 Gerber,H.U., Shiu,E.S.W. On the time value of ruin. North American Actuarial Journal 2 (1998) 48-78.
 Bao,Zhen-hua. The expected discounted penalty at ruin in the risk process with random income. Applied mathematics and computation 179(2006) 559-566.
 Pavlova,K. P., Willmot,G. E. The discrete stationary renewal risk model and the Gerber-Shiu discounted penalty function. Insurance: Mathematics and Economics 35(2004) 267-277.
 Cai,J., Dickson,D.C.M. On the expected discounted penalty function at ruin of a surplus process with interest. Insurance: Mathematics and Economics 30(2002) 389-404.
 Willmot,G. E., Dickson,D.C.M. The Gerber-Shiu discounted penalty function in the stationary renewal risk model. Insurance: Mathematics and Economics 32 (2003) 403-411.
 Karlin,S., Taylor,H.M. A first course in stochastic processes. Academic press,New York 1975.
 Xiao, Y.T., Guo, J.Y., The compound binomial risk model with timecorrelated claims. Insurance: Mathematics and Economics 41(2007)124- 133.
 De Vylder, F., Marceau, E., Classical numerical ruin probabilities. Scandination Actuarial Journal(1996)109-123.