A generalized defective renewal equation for the surplus process perturbed by diffusion

Abstract In this paper, we consider the surplus process of the classical continuous time risk model containing an independent diffusion (Wiener) process. We generalize the defective renewal equation for the expected discounted function of a penalty at the time of ruin in Garber and Landry [Insurance: Math. Econ. 22 (1998) 263]. Then an asymptotic formula for the expected discounted penalty function is proposed. In addition, the associated claim size distribution is studied, and reliability-based class implications for the distribution are given.

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

[2]  B. Everitt,et al.  Finite Mixture Distributions , 1981 .

[3]  Hans U. Gerber,et al.  Risk theory for the compound Poisson process that is perturbed by diffusion , 1991 .

[4]  Hans U. Gerber,et al.  An extension of the renewal equation and its application in the collective theory of risk , 1970 .

[5]  Bounds for compound distributions based on mean residual lifetimes and equilibrium distributions , 1997 .

[6]  Gordon E. Willmot,et al.  Analysis of a defective renewal equation arising in ruin theory , 1999 .

[7]  A decomposition of the ruin probability for the risk process perturbed by diffusion , 2001 .

[8]  G. Willmot,et al.  ON CLASSES OF LIFETIME DISTRIBUTIONS WITH UNKNOWN AGE , 2000, Probability in the Engineering and Informational Sciences.

[9]  W. D. Ray,et al.  Stochastic Models: An Algorithmic Approach , 1995 .

[10]  Lennart Bondesson.  On preservation of classes of life distributions under reliability operations: Some complementary results , 1983 .

[11]  Hans U. Gerber,et al.  On the discounted penalty at ruin in a jump-diffusion and the perpetual put option , 1998 .

[12]  G. Willmot,et al.  Exact and approximate properties of the distribution of surplus before and after ruin , 1998 .

[13]  Rong Wu,et al.  Some distributions for classical risk process that is perturbed by diffusion , 2000 .

[14]  F. Pellerey,et al.  New partial orderings and applications , 1993 .

[15]  Enrico Fagiuoli,et al.  Preservation of certain classes of life distributions under Poisson shock models , 1994, Journal of Applied Probability.

[16]  A. Cohen,et al.  Finite Mixture Distributions , 1982 .

[17]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[18]  Gordon E. Willmot,et al.  Lundberg approximations for compound distributions with insurance applications , 2001 .

[19]  Aging concepts for items of unknown age , 1994 .