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 tau(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 Shin (1998) defined the function m delta(u) = E[e(-delta tau)(u)w(X+(u), Y+(u)) 1(tau(u) < infinity)], where delta >= 0 can be interpreted as a force of interest and w (r, s) as a penalty function, meaning that m delta(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 delta(u) are only available for certain special cases for the claim size distribution. Approximations thus arise by approximating the desired m delta(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.