Estimating the Gerber-Shiu Function in Lévy Insurance Risk Model by Fourier-Cosine Series Expansion

Wen Su,Yunyun Wang

doi:10.3390/math9121402

Abstract

In this paper, we propose an estimator for the Gerber–Shiu function in a pure-jump Lévy risk model when the surplus process is observed at a high frequency. The estimator is constructed based on the Fourier–Cosine series expansion and its consistency property is thoroughly studied. Simulation examples reveal that our estimator performs better than the Fourier transform method estimator when the sample size is finite.

Highlights

The classical compound Poisson risk model, known as the Cramér-Lundberg model, was first proposed by Lundberg [1]
Given that the classical compound Poisson risk model is very limited, many scholars have devoted themselves to generalizing it with various stochastic surplus models, see, e.g., Gerber [4], Tsai [5], Li and Garrido [6], who considered the Cramér–Lundberg risk model perturbed by Brownian motion
We note that the aforementioned papers have focused on the explicit solutions of ruin probability and ruin-related quantities based on some specific assumptions regarding the claim size distributions

Summary

Introduction

The classical compound Poisson risk model, known as the Cramér-Lundberg model, was first proposed by Lundberg [1]. We note that the aforementioned papers have focused on the explicit solutions of ruin probability and ruin-related quantities based on some specific assumptions regarding the claim size distributions. Their probabilistic characteristics are usually unknown to the insurer. Chau et al [22] studied the ultimate ruin probability and Gerber–Shiu function by Fourier Cosine method in the Lévy risk model. The main goal of this paper is to estimate the Gerber–Shiu function by Fourier–Cosine series expansion based on a discretely observed sample of the aggregate claims process.

The Estimator

Simulations

Conclusions