Insurance: Mathematics and Economics 28 2001 49–59
A decomposition of the ruin probability for the risk process perturbed by diffusion
q
Guojing Wang
Department of Mathematics, Suzhou University, Suzhou 215006, People’s Republic of China Received 1 November 1999; received in revised form 1 July 2000; accepted 19 September 2000
Abstract
In this paper, we consider the ruin probabilities caused by oscillation or by a claim of the classical risk process perturbed by diffusion and the risk process with return on investments. We will prove their twice continuous differentiability and derive
the integro-differential equations satisfied by them. We will present the explicit expressions for them when the claims are exponentially distributed. © 2001 Elsevier Science B.V. All rights reserved.
Keywords: Risk process; Ruin probability; Integro-differential equation
1. Introduction
Let Ω, F, P be a complete probability space containing all the random variables we meet in the following. The classical risk process perturbed by diffusion is
R
t
= u + ct + σ W
t
−
N
t
X
k=1
Z
k
. 1.1
In 1.1, u, c and σ are all positive constants, u denotes the initial capital of an insurance company and c the rate of premium income; {N
t
} is a Poisson process with parameter λ 0, it counts the total numbers 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
the amount of the kth claim, {W
t
} the standard Brownian motion, it stands for the uncertainty associated with the income of the insurance company at time t , R
t
the surplus of the insurance company at time t , {N
t
}, {Z
k
} and {W
t
} are mutually independent. The process 1.1 is a homogenous strong Markov process.
We now follow the steps in Paulsen and Gjessing 1997 to introduce the risk process with deterministic return on investments. Our risk process is, in fact, a special case of 2.4 in Paulsen and Gjessing 1997.
The basic process is the surplus generating process 1.1. The deterministic return on investments generating process is
I
t
= rt, 1.2
q
Supported by NNSF Grant No. 19971047 in China and Doctoral Foundation of Suzhou University. E-mail address: rgwangsuda.edu.cn G. Wang.
0167-668701 – see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 6 6 8 7 0 0 0 0 0 6 5 - 2
50 G. Wang Insurance: Mathematics and Economics 28 2001 49–59
where r is a constant, it stands for a constant rate of return on investments. The risk process with deterministic return on investments is then the solution of the linear stochastic differential equation
R
t
= R
t
+ Z
t
R
s
−
dI
s
. 1.3
By Paulsen and Gjessing 1997, the solution of 1.3 is R
t
= exp{rt} u +
Z
t
exp{−rs} dR
s
, 1.4
and 1.4 is also a homogenous strong Markov process. Relation 1.4 can be rewritten as R
t
= exp{rt}
u +
c r
1 − exp{−rt} + σ Z
t
exp{−rs} dW
s
−
N
t
X
k+1
exp{−rT
k
}Z
k
, 1.5
where {T
k
}, k ≥ 1, is the sequence of jump times of {N
t
}. Dufresne and Gerber 1991 decompose the ruin probability of the risk process 1.1 into two parts: the ruin
probability caused by oscillation and the ruin probability caused by a claim. They obtain explicit expressions for these two different kinds of ruin probability in the form of series, making the assumption that the ruin probabilities are
twice differentiable. Motivated by this idea in Dufresne and Gerber 1991, we will decompose the ruin probability of the process in 1.4 correspondingly into two parts. We will prove the twice continuous differentiability of these
two different kinds of ruin probability and present their explicit expressions when the claims are exponentially distributed. Note that the twice continuous differentiability ensures that the solutions of the integro-differential
equations satisfied by the ruin probabilities respectively exist.
2. The classical risk process perturbed by diffusion
