Moments of discounted aggregate claims with dependence based on Spearman copula

Abstract

This paper considers an extension to the classical compound Poisson risk process by introducing a dependence structure between the inter-claim time and the claim size. We adopt the Spearman copula for constructing the dependence with the purpose of covering a wide range of positive dependence and developing a convex approximation to some bivariate copulas. We study the Laplace transform of the moments of the discounted aggregate claims in this framework. Then we derive the explicit expressions for the first three moments of the discounted aggregate claims in the case of the exponential and Pareto-distributed claim sizes. Numerical examples are provided to measure the impact of dependence on the discounted aggregate claims and to illustrate the efficiency of the proposed approximation method.