A Note on a Generalized Gerber–Shiu Discounted Penalty Function for a Compound Poisson Risk Model

Jiechang Ruan,Yujuan Huang,Yihan Sun,Wenguang Yu,Xinliang Yu,Ke Song

doi:10.3390/math7100891

Jiechang Ruan, Yujuan Huang + Show 4 more

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https://doi.org/10.3390/math7100891

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Abstract

In this paper, we propose a new generalized Gerber–Shiu discounted penalty function for a compound Poisson risk model, which can be used to study the moments of the ruin time. First, by taking derivatives with respect to the original Gerber–Shiu discounted penalty function, we construct a relation between the original Gerber–Shiu discounted penalty function and our new generalized Gerber–Shiu discounted penalty function. Next, we use Laplace transform to derive a defective renewal equation for the generalized Gerber–Shiu discounted penalty function, and give a recursive method for solving the equation. Finally, when the claim amounts obey the exponential distribution, we give some explicit expressions for the generalized Gerber–Shiu discounted penalty function. Numerical illustrations are also given to study the effect of the parameters on the generalized Gerber–Shiu discounted penalty function.

Highlights

The classical compound Poisson risk process {U (t)}t≥0 is defined by N (t) U (t) = u + ct − ∑ Xi, t ≥ 0, (1)i =1 where u is the non-negative amount of initial reserves, and c > 0 denotes the constant premium rate per unit time
We discuss a generalized Gerber–Shiu discounted penalty function, which relies on the moment of the time to ruin under the compound Poisson risk model
We present a recursion algorithm for calculating the generalized Gerber–Shiu discounted penalty function by Laplace transform and renewal theory when the claim amounts are subject to an exponential distribution

Summary

Introduction

Given the initial surplus U (0) = u, we define the classical Gerber–Shiu discounted penalty function as follows: Φ(u, δ) = E[e−δτ W (U (τ −), |U (τ )|) I (τ < ∞)|U (0) = u], u ≥ 0, (4) It is clear that the Gerber–Shiu discounted penalty function becomes the probability of ruin when δ = 0, W ( x, y) = 1. Gerber–Shiu discounted penalty function has been extended by many actuarial scholars, so that the new risk measures can be used to study more related quantities.

Results

Conclusion