Joint moments of the total discounted gains and losses in the renewal risk model with two-sided jumps

This paper considers a renewal insurance risk model with two-sided jumps (e.g. Labbé et al., 2011), where downward and upward jumps typically represent claim amounts and random gains respectively. A generalization of the Gerber-Shiu expected discounted penalty function (Gerber and Shiu, 1998) is proposed and analyzed for sample paths leading to ruin. In particular, we shall incorporate the joint moments of the total discounted costs associated with claims and gains until ruin into the Gerber-Shiu function. Because ruin may not occur, the joint moments of the total discounted claim costs and gain costs are also studied upon ultimate survival of the process. General recursive integral equations satisfied by these functions are derived, and our analysis relies on the concept of 'moment-based discounted densities' introduced by Cheung (2013). Some explicit solutions are obtained in two examples under different cost functions when the distribution of each claim is exponential or a combination of exponentials (while keeping the distributions of the gains and the inter-arrival times between successive jumps arbitrary). The first example looks at the joint moments of the total discounted amounts of claims and gains whereas the second focuses on the joint moments of the numbers of downward and upward jumps until ruin. Numerical examples including the calculations of covariances between the afore-mentioned quantities are given at the end along with some interpretations.