ABSTRACT

In this paper, we consider a Brownian motion risk model with stochastic return on investments. Using the strong Markov property and exploiting the limitation idea, we derive the Laplace-Stieltjes Transform (LST) of the total duration of negative surplus. In addition, two examples are also present.

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.

REFERENCES