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.