_{1}

The classical Poisson risk model in ruin theory assumed that the interarrival times between two successive claims are mutually independent, and the claim sizes and claim intervals are also mutually independent. In this paper, we modify the classical Poisson risk model to describe the surplus process of an insurance portfolio. We consider a jump-diffusion risk process compounded by a geometric Brownian motion, and assume that the claim sizes and claim intervals are dependent. Using the properties of conditional expectation, we establish integro-differential equations for the Gerber-Shiu function and the ultimate ruin probability.

Various papers in ruin theory modify the classical Poisson risk model to describe the surplus process of an insurance portfolio. An extension of the classical model is that the risk process perturbed by a diffusion was first introduced by Gerber [

In the risk process that is perturbed by diffusion, the surplus process

where

rate,

is the claim number process (denoting the number of claims up to time t), and the interarrival times

It is explicitly assumed in these papers that the interarrival times

In this paper, it is assumed that the claim occurrence process to be of the following type: If a claim

which is the net profit condition.

In the daily operation of insurance company, in addition to the premium income and claim to the operation of spending has a great influence on the outside, and there is also a factor that interest rates should not be neglected. As in [_{2} are positive constants, and

Denote T to be the ruin time (the first time that the surplus becomes negative), i.e.,

and

This article is interested in the expected discounted penalty (Gerber-Shiu) function:

where

Furthermore, let

For

such that

then,

In this section, a system of integro-differential equations with initial value conditions satisfied by the Gerber-Shiu function

Define

Lemma 3.1 Let

Proof

acting on functions satisfying the reflecting boundary condition

If

and

tingale. Using the separation variable technique, we find that

is a solution, where

Here

Using the initial condition

Applying the Optional Stopping Theorem, it follows that

and thus

This ends the proof of Lemma 3.1.

Similarly, the following lemma can also be obtained.

Lemma 3.2 Let

Theorem 3.1 Assuming that

with the initial value conditions

Proof Let

1) There are no claims in

2) There is exactly one claim in

a) The amount of the claim

b) The amount of the claim

c) The amount of the claim

3) There is more than one claim in

Thus, considering the three cases above and noting that

By Taylor expansion, we have

Then, by Itô’s formula we have

Therefore, by dividing t on both sides of Equation (12), letting

The condition

For all

According to Lemma 3.1, it can be concluded that

Thus,

And for all

This ends the proof of Theorem 3.1.

Let

Obviously,

and

Suppose that

and

Theorem 4.1 Assuming that

with the initial value conditions

Proof According to Equation (9), it can be concluded that

By taking the derivative with respect to u on both sides of the above formula, and after some careful calculations, we obtain Equation (14). And similarly we can prove that Equation (15) holds. This ends the proof of Theorem 4.1.

In this paper, we consider a jump-diffusion risk process compounded by a geometric Brownian motion with dependence between claim sizes and claim intervals. We derive the integro-differential equations for the Gerber-Shiu functions and the ultimate ruin probability by using the martingale measure. Further studies are needed for the numerical solution of Equations (9), (10), (14) and (15). The results derived in this paper can be generalized to similar dependence ruin models.

This research was supported by the National Natural Science Foundation of China (No. 11601036), the Natural Science Foundation of Shandong (No. ZR2014GQ005) and the Natural Science Foundation of Binzhou University (No. 2016Y14).

Gao, H.L. (2016) Integro-Differential Equations for a Jump- Diffusion Risk Process with Dependence between Claim Sizes and Claim Intervals. Journal of Applied Mathematics and Physics, 4, 2061-2068. http://dx.doi.org/10.4236/jamp.2016.411205