Abstract
We construct a discrete-time ruin model with general premium rate and dependent setting, where the time between two occurrences depends on the previous claim size. The generating function and defective renewal equation satisfied by the Gerber-Shiu expected discounted penalty function are derived by using the roots of a generalized Lundberg’s equation. Explicit expressions for the Gerber-Shiu function are obtained with discrete $K_{m}$ -family claim sizes and geometric thresholds. Numerical illustration is then examined.
Highlights
In ruin theory, the compound binomial model and the risk model based on a discretetime renewal process have been extensively analyzed by [, ], among many others
We mention that the dependence among claim sizes and interclaim arrivals through bivariate geometric distributions and copula functions have been investigated by Marceau [ ], where explicit expressions for the Gerber-Shiu expected penalty function are derived
Inspired by the works [ ] and [ ] in a continuous-time risk process with dependence, we consider a fully discrete risk model, in which the distribution of the time until the claim depends on the amount of the previous claim
Summary
The compound binomial model and the risk model based on a discretetime renewal process have been extensively analyzed by [ , ], among many others. Woo [ ] analyzes a generalized Gerber-Shiu function in a discrete-time renewal risk model with an arbitrary dependence structure. Liu and Bao [ ] consider a particular dependence structure among the interclaim time and the subsequent claim size and derive defective renewal equation satisfied by the Gerber-Shiu expected discounted penalty function. In Section , we obtain the explicit expressions for the Gerber-Shiu function when the claim sizes have discrete Km distributions and the random thresholds follow geometric distributions. We only consider the case where the roots of Lundberg’s equation ) eventually lead to the defective renewal equation for the Gerber-Shiu discounted penalty function mi(u), as presented in the following theorem. Differentiating ( . ) with respect to z c and taking the limit v → , we eventually find z c ( – p )( – p ) c( – p )ξ ( ) +
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