Computing Gerber-Shiu function in the classical risk model with interest using collocation method

The Gerber-Shiu function is a classical research topic in actuarial science. However, exact solutions are only available in the literature for very specific cases where the claim amounts follow distributions such as the exponential distribution. This presents a longstanding challenge, particularly from a computational perspective. For the classical risk process in continuous time, the Gerber-Shiu discounted penalty function satisfies a class of Volterra integral equations. In this article, we use the collocation method to compute the Gerber-Shiu function for risk model with interest. Our methodology demonstrates that the function can be expressed as a linear algebraic system, which is straightforward to implement. One major advantage of our approach is that it does not require any specific distributional assumptions on the claim amounts, except for mild differentiability and continuity conditions that can be easily verified. We also examine the convergence orders of the collocation method. Finally, we present several numerical examples to illustrate the desirable performance of our proposed method.