Some Ruin Problems for a Risk Process with Stochastic Interest

Abstract As investment plays an increasingly important role in the insurance business, ruin analysis in the presence of stochastic interest (or stochastic return on investments) has become a key issue in modern risk theory, and the related results should be of interest to actuaries. Although the study of insurance risk models with stochastic interest has attracted a fair amount of attention in recent years, many significant ruin problems associated with these models remain to be investigated. In this paper we consider a risk process with stochastic interest in which the basic risk process is the classical risk process and the stochastic interest process (or the stochastic return-on-investmentgenerating process) is a compound Poisson process with positive drift. Within this framework, we first derive an integro-differential equation for the Gerber-Shiu expected discounted penalty function, and then obtain an exact solution to the equation. We also obtain closed-form expressions for the expected discounted penalty function in some special cases. Finally, we examine a lower bound for the ruin probability of the risk process.

[1]  Guojing Wang,et al.  On a joint distribution for the risk process with constant interest force , 2005 .

[2]  Jozef L. Teugels,et al.  Ruin estimates under interest force , 1995 .

[3]  Kai W. Ng,et al.  Ruin probabilities for a risk process with stochastic return on investments , 2004 .

[4]  J. Michael Harrison,et al.  Ruin problems with compounding assets , 1977 .

[5]  Hailiang Yang,et al.  On the distribution of surplus immediately before ruin under interest force , 2001 .

[6]  Rong Wu,et al.  Distributions for the risk process with a stochastic return on investments , 2001 .

[7]  P. Protter.  Stochastic integration and differential equations : a new approach , 1990 .

[8]  David C. M. Dickson,et al.  On the distribution of the surplus prior to ruin , 1992 .

[9]  Jun Cai,et al.  On the expected discounted penalty function at ruin of a surplus process with interest , 2002 .

[10]  H. Gerber,et al.  On the Time Value of Ruin , 1997 .

[11]  F. Olver.  Asymptotics and Special Functions , 1974 .

[12]  Jostein Paulsen,et al.  Ruin theory with stochastic return on investments , 1997, Advances in Applied Probability.

[13]  R. Friedland.  Life Expectancy in the Future , 1998 .

[14]  Jun Cai,et al.  Ruin Probabilities and Penalty Functions with Stochastic Rates of Interest , 2004 .

[15]  Ragnar Norberg,et al.  Power tailed ruin probabilities in the presence of risky investments , 2002 .

[16]  K. D. Tocher,et al.  Confluent Hypergeometric Functions , 1960 .

[17]  Jozef L. Teugels,et al.  The adjustment function in ruin estimates under interest force , 1997 .

[18]  F. Delbaen,et al.  Classical risk theory in an economic environment , 1987 .

[19]  Jostein Paulsen,et al.  Risk theory in a stochastic economic environment , 1993 .

[20]  Jostein Paulsen.  On Cramér-like asymptotics for risk processes with stochastic return on investments , 2002 .