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On the expected discounted penalty function associated with the ...
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具有随机保费的风险模型下的平均折现罚金函数 ,汪荣明,姚定俊,本文考虑随机保费的风险模型下平均折现罚金函数。与经典的风险模型相比较保费过程不再是线性过程,总保费过程构成一复合Poisson过�
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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
a
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
1
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
φ
δ
(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
δ
(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
δ
(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
δ
(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
2
http://www.paper.edu.cn
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