Estimating the Gerber-Shiu Expected Discounted Penalty Function for Lévy Risk Model

Yujuan Huang,Chaoran Cui,Yu Pan,Wenguang Yu

doi:10.1155/2019/3607201

Abstract

This paper studies the statistical estimation of the Gerber-Shiu discounted penalty functions in a general spectrally negative Lévy risk model. Suppose that the claims process and the surplus process can be observed at a sequence of discrete time points. Using the observed data, the Gerber-Shiu functions are estimated by the Laguerre series expansion method. Consistent properties are studied under the large sample setting, and simulation results are also presented when the sample size is finite.

Highlights

In this paper, the cash flow of an insurance company is described by the following spectrally negative Levy process: Ut = u + ct + σBt − Xt, t ≥ 0. (1)Here, u ≥ 0 denotes the initial reserve and c > 0 is the rate of premium
This paper studies the statistical estimation of the Gerber-Shiu discounted penalty functions in a general spectrally negative Levy risk model
As a special type of Gerber-Shiu functions, the ruin probability is estimated by Mnatsakanov et al [31], Masiello [32], and Zhang et al [33] under the classical compound Poisson risk model

Summary

Introduction

We use the Gerber-Shiu expected discounted penalty function to discuss the ruin problems. This function is defined by φ (u) = E [e−δτw (󵄨󵄨󵄨󵄨Uτ󵄨󵄨󵄨󵄨) 1(τ

Preliminaries on Gerber-Shiu Functions

Estimating the Gerber-Shiu Function

Consistent Properties

Simulation Studies