The Gerber-Shiu Expected Penalty Function for the Risk Model with Dependence and a Constant Dividend Barrier

Donghai Liu,Zaiming Liu,Dan Peng

doi:10.1155/2014/730174

Abstract

We consider a compound Poisson risk model with dependence and a constant dividend barrier. A dependence structure between the claim amount and the interclaim time is introduced through a Farlie-Gumbel-Morgenstern copula. An integrodifferential equation for the Gerber-Shiu discounted penalty function is derived. We also solve the integrodifferential equation and show that the solution is a linear combination of the Gerber-Shiu function with no barrier and the solution of an associated homogeneous integrodifferential equation.

Highlights

In the classical compound Poisson risk model, the surplus process has the form N(t)U (t) = u + ct − ∑ Xi, t ≥ 0, (1)i=1 where u ≥ 0 is the initial surplus, c ≥ 0 is the premium income rate, and {Xi}∞ i=1 are i.i.d. random variables representing the individual claim amounts with probability density (c.d.f.)function FX, and (Lpa.pdl.af.c)efXtr,acnusmfourmlati(vLeTd)isftrX∗ib. uTthioencfouunncttiionng process {N(t); t ≥ 0} denotes the number of claims up to time t and is defined as N(t) = max{k : W1 + W2 + ⋅ ⋅ ⋅ + Wk ≤ t}, where the interclaim times {Wi, i = 1, 2, . . .} form a sequence of independent and strictly positive realvalued random variables (r.v.s.)
We assume that θ ≠ 0; otherwise our model reduces to the constant dividend barrier in the classical risk model
The main purpose of this section is to derive an integrodifferential equation for the expected discounted penalty function mb,δ(u), This equation will be useful to derive an explicit solution for mb,δ(u)

Summary

Introduction

I=1 where u ≥ 0 is the initial surplus, c ≥ 0 is the premium income rate, and {Xi}∞ i=1 are i.i.d. random variables representing the individual claim amounts with probability density (c.d.f.). Cossette et al [7] use the Farlie-Gumbel-Morgenstern (FGM) copula to define the dependence structure between the claim size and the interclaim time; they derive the integrodifferential equation and the Laplace transform (LT) of the Gerber-Shiu discounted penalty function. The joint distribution of (X, W) is defined with a FGM copula; we consider the same dependence risk model with the presence of a constant dividend barrier. Liu [17], Bratiichuk [18], Chadjiconstantinidis and Papaioannou [19], and Wang [20] As we know, this is the first time to consider the classic risk model with dependence structure based on FGM copula and a constant dividend barrier.

Dependence Structure and Risk Model

Gerber-Shiu Discounted Penalty Function

A Representation of the Discounted Penalty Function