On a Risk Model With Delayed Claims Under Stochastic Interest Rates

Abstract

This article studies a compound binomial risk model where the delayed claims are considered, and defines two types of individual claims, main claims and by-claims, respectively. Each by-claim is induced by the main claim and may be delayed for one time period with a certain probability. An extended definition of the Gerber-Shiu discounted penalty function is proposed to analyze this risk model in the framework of stochastic interest rates which follow a Markov chain with finite state space. By applying generating function and generalized Rouch’s theorem, we derive an explicit expression for this generalized Gerber-Shiu discounted penalty function in terms of the zeros of a determinant. Furthermore, we examine the original Gerber-Shiu discounted penalty function in the compound binomial model with delayed claims. In addition, we prove that the original Gerber-Shiu discounted penalty function satisfies a defective renewal equation, and derive the exact solution of this equation via an associated compound geometric distribution. Moreover, the closed form expressions of ruin probability and the distribution function of the surplus before ruin are obtained for some special cases. Finally, numerical results are provided to illustrate the applicability of our main result and the impact of the delay of by-claims on the Gerber-Shiu discounted penalty functions.