On the expected discounted penalty function at ruin of a surplus process with interest

Abstract

In this paper, we study the expected value of a discounted penalty function at ruin of the classical surplus process modified by the inclusion of interest on the surplus. The ‘penalty’ is simply a function of the surplus immediately prior to ruin and the deficit at ruin. An integral equation for the expected value is derived, while the exact solution is given when the initial surplus is zero. Dickson’s [Insurance: Mathematics and Economics 11 (1992) 191] formulae for the distribution of the surplus immediately prior to ruin in the classical surplus process are generalised to our modified surplus process.

Introduction

Consider a compound Poisson risk model. Assume that T_n=∑_k=1ⁿY_k is the time of the nth claim and X_n is the amount of the nth claim. Suppose that {X_n,n≥1} and {Y_n,n≥1} are two independent sequences of i.i.d. positive random variables, where {X_n,n≥1} have common distribution F(x)=Pr{X₁≤x} with mean μ>0, and {Y_n,n≥1} have common exponential distribution Pr{Y₁≤x}=1−exp{−λx}, x≥0, where λ>0.

The number of claims up to time t is denoted by N(t)=sup{n:T_n≤t}. The claim number process is a Poisson process with rate λ, and the aggregate claim amount up to time t is

$$\text{Z(t)=∑}{}_{\mathrm{n=1}}{}^{\mathrm{N(t)}}\text{X}{}_{\mathrm{n}}\text{.}$$

Assume that the insurer receives interest on its surplus at a constant force δ per unit time. Let U_δ(t) denote the surplus at time t. Then

$$\text{U}{}_{\mathrm{\delta }}\text{(t)=u}\text{e}{}^{\mathrm{\delta t}}\text{+c}\text{s}\text{̄}{}^{\mathrm{(\delta )}}{}_{\mathrm{}}\text{−∫}{}_0{}^{\mathrm{t}}\text{e}{}^{\mathrm{\delta (t-x)}}\text{d}\text{Z(x),}$$

where u is the initial surplus and c>λμ is the rate of premium income per unit time.

Let the time of ruin be

$$\text{T}{}_{\mathrm{\delta }}\text{=}\text{inf}\text{{t:U}{}_{\mathrm{\delta }}\text{(t)<0},}\text{∞}\text{if}\text{U}{}_{\mathrm{\delta }}\text{(t)≥0}\text{for}\text{all}\text{t>0.}$$

Denote by ψ_δ(u) the ruin probability for the surplus process given by Eq. (1.1). Then

$$\text{ψ}{}_{\mathrm{\delta }}\text{(u)=}\text{Pr}\text{{T}{}_{\mathrm{\delta }}\text{<∞}=}\text{Pr}\text{{∪}{}_{\mathrm{t\ge 0}}\text{(U}{}_{\mathrm{\delta }}\text{(t)<0)}.}$$

The following notation applies throughout this paper:

$$\text{f(x)=}\text{d}\text{d}\text{x}\text{F(x),}\text{F}{}_1\text{(x)=1−}\text{F}\text{̄}{}_1\text{(x)=}\text{1}\text{μ}\text{∫}{}_0{}^{\mathrm{x}}\text{F}\text{̄}\text{(t)}\text{d}\text{t,}\text{ψ}\text{̄}{}_{\mathrm{\delta }}\text{(u)=1−ψ}{}_{\mathrm{\delta }}\text{(u),}\text{ψ(u)=ψ}{}_0\text{(u),}\text{ψ(0)=}\text{λμ}\text{c}\text{,}\text{U(T}{}_{\mathrm{\delta }}{}^{\mathrm{-}}\text{)=}\text{the}\text{surplus}\text{immediately}\text{prior}\text{to}\text{ruin}\text{,}\text{|U(T}{}_{\mathrm{\delta }}\text{)|=}\text{the}\text{deficit}\text{at}\text{ruin}\text{,}\text{F}{}_{\mathrm{\delta }}\text{(u,x)=}\text{Pr}\text{{U(T}{}_{\mathrm{\delta }}{}^{\mathrm{-}}\text{)≤x,T}{}_{\mathrm{\delta }}\text{<∞},}\text{f}{}_{\mathrm{\delta }}\text{(u,x)=}\text{d}\text{d}\text{x}\text{F}{}_{\mathrm{\delta }}\text{(u,x),}\text{H}{}_{\mathrm{\delta }}\text{(u,x,y)=}\text{Pr}\text{{U(T}{}_{\mathrm{\delta }}{}^{\mathrm{-}}\text{)≤x,|U(T}{}_{\mathrm{\delta }}\text{)|≤y,T}{}_{\mathrm{\delta }}\text{<∞},}\text{h}{}_{\mathrm{\delta }}\text{(u,x,y)=}\text{∂}{}^2\text{∂x∂y}\text{H}{}_{\mathrm{\delta }}\text{(u,x,y).}$$

We consider the expected value of a discounted function of the surplus immediately prior to ruin and the deficit at ruin when ruin occurs as a function of the initial surplus u, namely,

where I(A) is the indicator function of a set A, w is a non-negative function, and α is a non-negative valued parameter. We can interpret exp{−αT_δ} as the ‘discounting factor’.

The function Φ_δ,α(u) provides a unified means of studying the joint distribution of the surplus immediately prior to ruin and the deficit at ruin. The distributions of these quantities, both joint and marginal, have been studied by many authors including Dickson, 1992, Dufresne and Gerber, 1988, Gerber et al., 1987, Gerber and Shiu, 1997, Gerber and Shiu, 1998 and Lin and Willmot (1999). In particular, Gerber and Shiu (1998) studied the function Φ_0,α(u) in detail, but they did not consider the case when δ>0.

In this paper, we will follow ideas in Sundt and Teugels (1995). In particular, we will consider the function Φ_δ,0(u)=Φ_δ(u). We will also derive an integral equation for Φ_δ,α(u) and find the Laplace transform of an auxiliary function of Φ_δ,α(u). We then find an exact solution for Φ_δ(0) and generalise Dickson’s (1992) formulae for the distribution of the surplus prior to ruin when δ=0 to the situation when δ>0. Applications of the results will be illustrated by a variety of examples.