Simple approximation for the ruin probability in renewal risk model under interest force via Laguerre series expansion

Although the ruin probability in a renewal insurance risk model with credit interest may be viewed as a classical research problem, exact solutions are only available in the literature in very special cases when both the claim amounts and the interclaim times follow distributions such as the exponential. This is a long standing problem especially from a computational point of view, and the difficulty lies in the fact that the ruin probability usually satisfies a higher order integro-differential equation and/or an ordinary differential equation with non-constant coefficients. In this paper, for a large class of interclaim time distributions (including a combination of exponentials), we shall develop an approximation for the ruin probability using Laguerre series expansion as a function of the initial surplus level. It is shown that the (approximated) Laguerre coefficients can be solved from a system of linear equations, a procedure that is very easy to implement. A main advantage of our approach is that no specific distributional assumption on the claim amounts is required, apart from some mild differentiability and integrability conditions that can be verified. Numerical examples are provided to illustrate the very good performance of our approximation including both light-tailed and heavy-tailed claims.