Approximations for the Gerber-Shiu expected discounted penalty function in the compound poisson risk model

In the classical risk model with initial capital u, let .(u) be the time of ruin, X+(u) be the risk reserve just before ruin, and Y+(u) be the deficit at ruin. Gerber and Shiu (1998) defined the function m.(u) =E[e.. .(u)w(X+(u), Y+(u)) 1 (.(u) < .)], where . . 0 can be interpreted as a force of interest and w(r,s) as a penalty function, meaning that m.(u) is the expected discounted penalty payable at ruin. This function is known to satisfy a defective renewal equation, but easy explicit formulae for m.(u) are only available for certain special cases for the claim size distribution. Approximations thus arise by approximating the desired m.(u) by that associated with one of these special cases. In this paper a functional approach is taken, giving rise to first-order correction terms for the above approximations.