Estimating the Gerber–Shiu function in a Lévy risk model by Laguerre series expansion

Abstract

In this paper, we provide a new method for estimating the Gerber–Shiu function in a pure jump Lévy risk model. First, we show that the Gerber–Shiu function can be expressed on the Laguerre basis and the Laguerre coefficients can be easily obtained by solving a linear system. Next, based on a high-frequency observation of the aggregate claims process, we estimate the Laguerre coefficients and this leads to a new estimator of the Gerber–Shiu function. We derive the consistency property of this estimator when the sample size is large. Finally, we do some simulation studies to illustrate the finite sample size performance.