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In this paper, we study an optimal reinsurance strategy combining a proportional and an excess of loss reinsurance. We refer to a collective risk theory model with two classes of dependent risks; particularly, the claim number of the two classes of insurance business has a bivariate Poisson distribution. In this contest, our aim is to maximize the expected utility of the terminal wealth. Using the control technique, we write the Hamilton-Jacobi-Bellman equation and, in the special case of the only excess of loss reinsurance, we obtain the optimal strategy in a closed form, and the corresponding value function.

In the last two decades the optimal reinsurance problem has had an important impact in the actuarial literature. Several authors have studied this problem with different purposes and referring to different surplus processes. Starting from the classical model where the process of the total claim amount has a Poisson compound distribution or follows a diffusion process, the adjustment coefficient, or the expected utility of the terminal wealth are been optimized (see, for example, [

With similar optimization aims, a more realistic model has been often considered, with two or more dependent classes of insurance business. Similar approaches are, for example: in [

We consider the finite time horizon

where Q_{1}, Q_{2} and Q_{12} are Poisson random variables with positive parameters θ_{1}, θ_{2 }and θ_{12} respectively.

We furthermore assume that_{1j} and X_{2j} have the same distribution functions F_{1} and F_{2 }with F_{i }(x) = 0, for x ≤ 0, and expected value

exist. As usually stated, the random variables

Let_{12} resulting:

We consider the random variables

assume that the random variables X_{i} are upper limited or that

We denote by

We assume that the principal insurer can implement both a proportional and an excess of loss reinsurance referred to both classes of insurance risks, with the respective retention levels

We therefore denote by

The reinsurer, because of the proportional reinsurance, would pay

that is:

We assume that all the premiums are paid using the expected value principle. Therefore, the reinsurance premium rate at time t is, for each class of risk:

where we have assumed the safety loading coefficients

Therefore, after the reinsurances, the premium rate for the insurer is:

We assume that the insurer can choose, for every time

The main goal for the insurer is to choose an optimal reinsurance strategy that maximize the expected exponential utility of terminal wealth. To solve this problem, we will use a dynamic programming approach.

After the reinsurance, remembering (4), referring to the j-th claim of type i, the insurer pays

Hence, the total claim amount charged to the insurer at time t, referred to the i-type claim is:

It follows that the surplus process

We recall that the process

We assume an insurer’s utility function

The insurer looks for an optimal control strategy so as to maximize the expected utility of the terminal surplus under the initial condition regarding the x state at time t. We consider the following value function:

with the boundary condition

We are able to find the infinitesimal generator for the process

Theorem 1. Let V be defined by (10) and let

Proof. We derive the following infinitesimal generator

Remembering (7) and (8), it results:

where we have:

and therefore we find:

the Equation (12) is therefore fulfilled by V.■

As we specified before, we assume the utility function (9), inspired by [

with

We note that:

Therefore, (12) can be written as follows:

Observing that:

it follows that Equation (13), dividing by

In the particular case where

Therefore, (13) becomes:

It is obviously that Equation (15) is the same equation found in [

In the particular case where

In the following section we consider this case.

We face the problem (16), with condition (11), that is

with conditions:

and

We have:

and

from which we deduce that, at the points where the gradient of g is zero, the Hessian matrix of

that is, letting

The solutions can be of the following four kinds:

We observe that the solution

that is impossible.

According to results in [

from which, if:

we have:

we have:

In [

and

the optimal strategy

We observe that, from (28), being true also (27) and since

Finally, we recall that (23) and (24) are incompatible (see [

We are so able to find the value function, substituting the optimal strategy in (17), that is in (16), and obtaining

if (24) is fulfilled, it results:

if (26) and (27) are at the same time fulfilled, we have:

The results obtained in this section are collected within the following theorem.

Theorem 2. The optimal strategy

・ if

it is

and

where

・ if

it is

and

where

・ if

and

it is

and

where

We thank the Editor and the Referees for their comments.

Cristina Gosio,Ester C. Lari,Marina Ravera, (2016) Optimal Dynamic Proportional and Excess of Loss Reinsurance under Dependent Risks. Modern Economy,07,715-724. doi: 10.4236/me.2016.76075