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

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.