^{1}

^{1}

^{1}

The paper deals with the optimal proportional reinsurance in a collective risk theory model involving two classes of insurance business. These classes are dependent through the number of claims. The objective of the insurer is to choose an optimal reinsurance strategy that maximizes the expected exponential utility of terminal wealth. We are able to derive the evolution of the insurer surplus process under the assumption that the number of claims of the two classes of the insurance business has a Poisson bivariate distribution. We face the problem of finding the optimal strategy using the dynamic programming approach. Therefore, we determine the infinitesimal generator for the surplus process and for the value function, and we give the Hamilton Jacobi Bellmann (HJB) equation. Under particular assumptions, we obtain explicit form of the optimal reinsurance strategy on correspondent value function.

The classical Cramer-Lundberg risk model assumes that the stochastic process _{j}, the claim size of the j-th claim. In this model,

In this paper, we consider an optimal proportional reinsurance problem of an insurer whose surplus process is generated by two dependent classes of insurance business. The objective is to choose an optimal reinsurance strategy; in order to maximize the insurer’s expected exponential utility of terminal wealth we use the dynamic programming approach.

The paper is organized as follows. In Section 2, we present the risk model. In Section 3, we find the surplus evolution and the conditional expected utility of the insurer’s terminal surplus, define the problem and give the corresponding value function. In Section 4, using the infinitesimal generator, we derive the HJB equation and justify the form of the value function. Finally, in Section 5, we discuss the solution giving an explicit solution in a particular framework.

In the finite time horizon_{i}, with

where

In the following, we will use the variables X_{i}, i = 1, 2, identically distributed to

We denote by c, i = 1, 2 the premium rate, for the time unit, assuming that the premium calculation principle is the expected value principle with loading coefficient

We introduce a proportional reinsurance: the reinsurer pays

for which it is

Note that the condition

We denote by

We denote by

We recall that

From previous result it follows

where X_{i} are identically distributed to X_{ij},

We consider an utility function

As previously stated, the insurer’s goal is to determine an optimal reinsurance strategy

It follows that the insurer has to find the optimal strategy

with the usual boundary condition (see [

We can find the infinitesimal generator for the process

Theorem 1. Let

Proof. We derive the following infinitesimal generator for the process

it allows us to write the HJB Equation (12).

We recall that, by (6) and (7) it results in

therefore we have, remembering the independence between X_{i} and N_{i}, i = 1, 2:

Therefore V must satisfy Equation (12). ■

We introduce the following utility function

With the purpose to write (9) we observe that:

1) from (6) and remembering that Poisson processes have stationary increments, we obtain:

with, as previously stated,

2) in Section 2 we have assumed that the moment generating functions of random variables

We denote those functions by

3) according to [

4) from the previous considerations, we have:

Because of these considerations, we assume that the value function V, defined by (10) with the condition (11) has the form

with the condition

We consider the assumptions (18); it results in:

Therefore, (12) becomes

with

with

Assuming the particular case where the insurer’s risk exposure is the same for the two classes of the insurance business; that is

with

For simplicity, we write (21) as follows

observing that

from which we obtain:

1)

2)

3)

From the previous results it follows that

if

if

if

being

We therefore obtain the following results.

If

with

then the resulting value function (18) is:

If

and by (21):

with

We thank the Editor and the Referees for their comments.