On the moments of the surplus process perturbed by diffusion

In this paper we extend the results of Lin and Willmot [Insurance: Mathematics and Economics 27 (2000) 19–44] to those based on the surplus process perturbed by diffusion. We first derive the expression for the (discounted) moments of deficit at the time of ruin. An upper bound is also given if the claim size distribution function satisfies a certain condition. Next, we show that the joint moment of the penalty function and the time of ruin due to a claim satisfies a defective renewal equation and has an explicit expression; then the joint moment of the deficit at ruin and the time of ruin is just a special case by an appropriate choice of the penalty function. Finally, the moments of the time of ruin due to oscillation and caused by a claim, respectively, are studied. We also find that these two kinds of moments of the time of ruin have the same recursive expressions.