Total Duration of Negative Surplus for a Diffusion Surplus Process with Stochastic Return on Investments ()
1. Introduction
Assume that the insurance business is described by the risk process
(1.1)
Here, 
is the initial capital; 
is the fixed rate of premium income; 
is a standard Brownian motion; and 
is a constant, representing the diffusion volatility.
Suppose that the insurer is allowed to invest in an asset or investment portfolio. Following Paulsen and Gjessing [1], we model the stochastic return as a Brownian motion with positive drift. Specifically, the return on the investment generating process is
(1.2)
where r and 
are positive constants. In (1.2), r is a fixed interest rate; 
is another standard Brownian motion independent of
, standing for the uncertainty associated with the return on investments at time
.
Let the risk process 
denote the surplus of the insurer at time 
under this investments assumption. Thus, 
associated with (1.1) and (1.2) is then the solution of the following linear stochastic integral equation:
(1.3)
By Paulsen [2] the solution of (1.3) is given by
(1.4)
where
Note that 
is a homogeneous strong Markov process, see e.g. Paulsen and Gjessing [1].
The risk process (1.4) can be rewritten as
Because the quadratic variational processes of
and
are the same, where 
is a standard Brownian motion, by Ikeda and Watanabe [3, p. 185] they have the same distribution. Thus, in distribution, we have
(1.5)
There are many papers concerning occupation times for different risk models. For example, for the classical surplus process with positive safety loading, Egdio dos Reis [4] derived the moment generating function of the total duration of the negative surplus by martingale methods, which was extended in Zhang and Wu [5] to the classical surplus process perturbed by diffusion. Chiu and Yin [6] derived explicit formula for the double Laplace-Stieltjes Transform (LST) of the occupation time in the exponential case for the compound Poisson model with a constant interest. He et al. [7] gived the LST of the total duration of negative surplus for the classical risk model with debit interest. More recently, Wang and He [8] considered the Brownian motion risk model with interest and derived the LST of total duration of negative surplus. In this paper, we consider a Brownian motion risk model with stochastic return on investments. We will use the limitation idea to obtain the LST of total duration of negative surplus.
The remainder of the paper is organized as follows. In Section 2, we give some preliminary results. In Section 3, by exploiting the limitation idea together with the results obtained in Section 2, we obtain the LST of the total duration of negative surplus. In the last section, we present two examples.
2. Preliminary Results
Given
, where
, define
and if the set is empty
,
and if the set is empty
,
and if the set is empty
,
.
Lemma 2.1 The risk process (1.5) has the strong Markov property: for any finite stopping time T the regular conditional expectation of 
given 
is
, that is
where 
is the information about the process up to time
, and the equality holds almost surely.
Lemma 2.2 For
, the following ordinary differential equation
(2.1)
has two independent solutions
(2.2)
and
(2.3)
where
Proof. From Example 2.2 of Paulsen and Gjessing [1], we get the result.
Lemma 2.3 For
, 
and 
, 
, define
then
where 
and 
are given by (2.2) and (2.3).
Proof. The result can be found in Chapter 16 of Breiman [9].
Lemma 2.4 For any
, then
(2.4)
where 
is a solution of the equation
Proof. By Dynkin’s formula,
where 
is the generator of diffusion (1.5). It follows that
Therefore
Since 
is finite, it takes values 
with probability 
and 
with the complimentary probability. Letting
, we can assert, by dominated convergence, that
Expanding the expectation on the left, we have
This, together with
, gives the result (2.4).
Lemma 2.5 For
, the ruin probability for the risk model (1.5) is given by
(2.5)
The probability that the surplus process 
hit the level 
is given by
(2.6)
where
Proof. By Lemma 2.4, one can derive (2.5) and (2.6).
3. Total Duration of Negative Surplus
In this section, we will derive the main result of this paper. We assume that the risk process (1.5) does not attain the critical level
. For convenience, we assume that the initial surplus 
is positive.
Let the total duration of negative surplus be
For
, define two sequences of stopping times of the process (1.5):
(
if the set is empty),
(
if the set is empty)in general, for 
recursively define
(
if the set is empty),
(
if the set is empty).
Let
. Given 
for some
, from the strong Markov property of the surplus process, we obtain that the periods 
are mutually independent and have a common distribution. Let 
denote the number of
.
Set
. By the monotone convergence theorem, we have
(3.1)
First we give the expression for 
in the following Theorem 3.1.
Theorem 3.1 For 
and
, the LST of 
is given by
(3.2)
where 
and 
are given by Lemmas 2.3 and 2.5.
Proof. From Lemma 2.1, we can get
(3.3)
and
(3.4)
From strong Markov property of the surplus process, we get
This, together with (3.3) and (3.4), gives (3.2).
Theorem 3.2 For 
and
, the LST of total duration of negative surplus is given by
(3.5)
where
, 
and 
are given by Lemmas 2.3 and 2.5.
Proof. It follows from (3.1) and (3.2) that
(3.6)
From Lemma 2.5, it follows that
(3.7)
By 
Hospital’s rule, we get
This, together with (3.6) and (3.7), gives (3.5).
4. Examples
In this section we consider two examples.
Example 4.1. Letting 
in (1.5), we get the risk model
(4.1)
From Cai et al. [10], we know that the two independent solutions of the differential equation
are
(4.2)
and
(4.3)
where M and U are called the confluent hypergeometric functions of the first and second kind respectively. More detail on confluent hypergeometric functions can be found in Abramowitz and Stegun [11].
By Lemmas 2.3 and 2.5, we get
(4.4)
(4.5)
(4.6)
where
According to Theorems 3.1 and 3.2, we get
(4.7)
(4.8)
where
, 
and 
are given by (4.2), (4.5) and (4.6).
Remark 4.1 The results (4.7) and (4.8) coincide with the main results in Wang and He [7].
Example 4.2. Letting 
and 
in (1.5), we get the risk model
(4.9)
It is easy to obtain that the two independent solutions of the ordinary differential equation
are
and
By Lemmas 2.3 and 2.5, we get
and
According to Theorems 3.1 and 3.2, we have
and
5. Conclusion
In this paper, we have studied the diffusion model incorporating stochastic return on investments. We find the LST of the total duration of negative surplus of this process. However, if the risk model (1.1) is extended to a compound Poisson surplus process perturbed by a diffusion, it is difficult to make out. We leave this problem for further research.