A bivariate Laguerre expansions approach for joint ruin probabilities in a two-dimensional insurance risk process

Abstract

In this paper, we consider a two-dimensional insurance risk model where each business line faces not only stand-alone claims but also common shocks that induce dependent losses to both lines simultaneously. The joint ruin probability is analyzed, and it is shown that under some model assumptions it can be expressed in terms of a bivariate Laguerre series with the initial surplus levels of the two business lines as arguments. Our approach is based on utilizing various attractive properties of Laguerre functions to solve a partial-integro differential equation satisfied by the joint ruin probability, so that continuum operations such as convolutions and partial differentiation are translated to lattice operations on the Laguerre coefficients. For computational purposes, the bivariate Laguerre series needs to be truncated, and the corresponding Laguerre coefficients can be obtained through a system of linear equations. The computational procedure is easy to implement, and a numerical example is provided that illustrates its excellent performance. Finally, the results are also applied to address a related capital allocation problem.