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On the expected discounted penalty function

associated with the time of ruin for a risk

model with random income

∗

Wang Rongming

a†

Yao Dingjun

Department of Statistics, East China Normal University, Shanghai, 200062

rmwang@stat.ecnu.edu.cn(Wang R. M. ), yaodingjun@yahoo.com.cn(Yao D.J.)

Abstract

This paper studies the expected discounted penalty function associated with the time of ruin for a risk model

with stochastic premium. The premium process is no longer a linear function of time in contrast with the

classical Cram´er-Lundb erg model. The aggregate premiums constitute a compound Poisson process which is

also independent of the claim process. Integral equation for the penalty function is derived, which provides

a uniﬁed treatment to the ruin quantities. Applications of the integral equation are given to the Laplace

transform of the time of ruin, the deﬁcit at ruin, the surplus immediately before ruin occurs. In some

special cases with exponential distributions, closed form expressions for these quantities are obtained, which

generalize some known results about the problems of ruin in Boikov(2003).

Key words: stochastic premium, integral equation, penalty function, the time of ruin, the deﬁcit at ruin,

the surplus immediately before ruin occurs.

∗

This work was supported by a grant from National Natural Science Foundation of China (10671072), by

Doctoral Program Foundation of the Ministry of Education of China (20060269016), and by ”Shu Guang”

project (04SG27) of Shanghai Municipal Education Commission and Shanghai Education Development Foun-

dation

†

Corresponding author

http://www.paper.edu.cn

1. Introduction and the Risk Model

In the classical Cram´er-Lundberg model, the premium process is a linear function of

time. Boikov(2003) generalized the classical risk model to the case where the premium was

modeled as a compound Poisson process. The exponential lower bounds were given, and

closed formulas in some cases were derived for nonruin probability in Boikov(2003).

Except for the ruin probabilities, there are many other important ruin quantities in ruin

theory which we are also interested in, for example, the Laplace transform of the time of

ruin, E(e

−δT

) = E[e

−δT

I(T < ∞)]; the surplus immediately before ruin, denoted by U (T −);

the deﬁcit at ruin, |U(T )|; etc. The distributions of these quantities, both joint and mar-

ginal, have been studied by many authors including Dickson (1992), Dufresne and Gerber

(1988), Gerber et al. (1987), Gerber and Shiu (1997, 1998), Lin and Willmot (1999), Cai and

Dickson(2002) and Cai(2004). A uniﬁed method to study these ruin quantities was provided

by Gerber and Shiu(1998) for the ﬁrst time. They deﬁned the penalty function associated

with the time of ruin as following

(u) = E[ω(U(T −), |U(T )|)e

−δT

I(T < ∞)|U(0) = u], (1.1)

where ω(x, y), x ≥ 0, y ≥ 0, is a non-negative real function such that φ

(u) exists; δ ≥ 0 and

I(C) is the indicator function of a set C; U(0) = u is the insurance’s initial capital.

Obviously, if ω(x, y) = 1 and δ = 0 in (1.1), then φ

(u) = φ(u) = P(T < ∞|U(0) = u)

is the ruin probability; if ω(x, y) = 1 and δ > 0, then, E(e

−δT

) = E[e

−δT

I(T < ∞)]

is the Laplace transform of the time of ruin; if ω(x, y) = x and δ > 0, then φ

(u) =

E[e

−δT

U(T −)I(T < ∞)|U(0) = u], we can interpret δ as a force of interest, then this func-

tion denotes the expected present value of the surplus immediately before ruin.

The discounted penalty function is an eﬀective tool to deal with some ruin quantities,

but unfortunately, its explicit solution is rarely available. One of the common research

methods used in ruin theory is ﬁrst to derive integral equations for the discounted penalty

function, and then try to solve these equations under speciﬁed circumstances.

In this paper, we ﬁrstly introduce the risk model considered as in Boikov(2003), then de-

rive integral equation satisﬁed by the discounted penalty function. In the case of exponential

http://www.paper.edu.cn

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