Insurance: Mathematics and Economics 26 2000 15–24
Some distributions for classical risk process that is perturbed by diffusion
q
Guojing Wang
a,b,∗
, Rong Wu
b
a
Department of Physics, Hebei University, Baoding 071002, China
b
Department of Mathematics, Nankai University, Tianjin 300071, China Received December 1998; received in revised form July 1999
Abstract
In this paper we discuss the classical risk process that is perturbed by diffusion. We prove some properties of the supremum distribution of the risk process before ruin when ruin occurs and the surplus distribution at the time of ruin. We present the
simple and explicit expression for these distributions when the claims are exponentially distributed. ©2000 Published by Elsevier Science B.V. All rights reserved.
Keywords: Risk process; Ruin probability; Supremum distribution; Surplus distribution at the time of ruin; Integro-differential equation
1. Introduction
Let , J , P be a complete probability space containing all objects defined in the following. We consider the classical risk process that is perturbed by diffusion
R
t
= u + ct + σ W
t
−
N
t
X
k=1
Z
k
, 1.1
where u denotes the initial capital, c the premium income, {W
t
: t ≥ 0} is a standard Brownian motion, {N
t
: t ≥ 0} is a Poisson process with parameter λ 0, it counts the number of the claims in the interval 0, t ], {Z
k
, k ≥ 1} is a non-negative sequence of i.i.d. random variables, Z
k
denotes the amount of the kth claim. R
t
is the surplus of an insurance company at time t. We assume throughout the paper that {N
t
}, {W
t
} and {Z
k
} are independent. Denote the distribution function of the claims by F and their mean value by µ. Let T
1
, T
2
, . . . be the occurrence times of the claims and set T
= 0. The model 1.1 is first introduced by Gerber 1970 and further studied by many authors during the last few
years, such as Dufresne and Gerber 1991, Veraverbeke 1993, Furrer and Schmidli 1994, Schmidli 1995, Zhang
q
Supported by NNSF grant no. 16971047 and DSF in China.
∗
Corresponding author. Address: Department of Mathematics, Nankai University, Tianjin 300071, China. 0167-668700 – see front matter ©2000 Published by Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 7 - 6 6 8 7 9 9 0 0 0 3 5 - 9
16 G. Wang, R. Wu Insurance: Mathematics and Economics 26 2000 15–24
1997 and so on. Many results on ruin probability and other ruin problems have been obtained by above-mentioned works. The purpose of this paper will be to discuss the supremum distribution of the risk process before ruin
when ruin occurs and the surplus distribution at the same time of ruin. We first prove some properties of these two distributions and then give some special examples for those distributions we consider. The ideas and methods we
used in this paper are motivated by Grandell 1991, Paulsen 1993 and Paulsen and Gjessing 1997, can be used to the risk model with return on investment, see Wang and Wu 1998.
In Section 2, we introduce the supremum distribution of the risk process before ruin when ruin occurs. We first derive the integral equation satisfied by it. Then by using the integral equation we prove its differentiability. Finally
we give the integro-differential equation satisfied by it. In the last part of this section we define the supremum distribution of the risk process before ruin.
In Section 3, we discuss the surplus distribution at the time of ruin. We also derive the integral equation and the integro-differential equation satisfied by it. The contents in this section are parallel those in Section 2.
In Section 4, we get the simple and explicit expression for the ruin probability, the supremum distribution of the risk process before ruin when ruin occurs and the surplus distribution at the time of ruin by solving boundary value
problems with same differential equation of third order but with different boundary values with each other when the claims are exponentially distributed. For general expressions of these distributions, we can refer to Dufresne
and Gerber 1991 for the series expression of ruin probability and to Zhang 1997 for the series expression of the surplus distribution at the time of ruin.
2. Supremum distribution before ruin
