Asymptotic aspects of the Gerber-Shiu function in the renewal risk model using Wiener-Hopf factorization and convolution equivalence

We study the asymptotic behavior of the Gerber-Shiu expected discounted penalty function in the renewal risk model. Under the assumption that the claim-size distribution has a convolution-equivalent density function, which allows both heavy-tailed and light-tailed cases, we establish some asymptotic formulas for the Gerber-Shiu function with a fairly general penalty function. These formulas become completely transparent in the compound Poisson risk model or for certain choices of the penalty function in the renewal risk model. A by-product of this work is an extension of the Wiener-Hopf factorization to include the times of ascending and descending ladders in the continuous-time renewal risk model.