On a perturbed by diffusion compound Poisson risk model with delayed claims and multi-layer dividend strategy

In this paper we consider an extension to the compound Poisson risk process perturbed by diffusion in which two types of dependent claims, main claims and by-claims, are incorporated. Every by-claim is induced by the main claim and may be delayed for one time period with a certain probability. An integro-differential equation system for the Gerber-Shiu expected discounted penalty functions is derived and solved by proving that the Gerber-Shiu function satisfies some defective renewal equation. An exact representation for the solution of this equation is derived through an associated compound geometric distribution and an analytic expression for this quantity is given when both the claim and the by-claim amounts belong to the rational family of distributions. Further, the same risk model is considered in the presence of a multi-layer dividend strategy. A system of integro-differential equations for the expected discounted penalty functions depending on the current surplus level, with certain initial and boundary conditions, is obtained. To solve this, we derive a general solution to a certain second order integro-differential equation system. This solution is obtained by transforming this system to a Volterra-type system of integral equations of second kind, which is solved by using Laplace transforms provided an explicit expression for the Gerber-Shiu functions depending on the current surplus level. Finally, numerical results for the ruin probability are given to illustrate the applicability of our results. (C) 2013 Elsevier B.V. All rights reserved.