A Reinsurance Approach in a Two-Dimensional Model with Dependent Risks

We consider an insurer having 
two classes of insurance risks dependent through the number of claims of each risk in a given period of time. We assume 
that the insurer chooses a reinsurance strategy related to the first class of 
risk by means a proportional reinsurance contract; we also assume that the 
reinsurance strategy related to the second class of risk is of Excess of Loss 
reinsurance type. Within this paper, we study the possible optimal couples of 
proportional retention level and Excess of Loss retention limit.


Introduction
In recent years, several authors have studied two-dimensional risk processes where an insurer has two classes of business or insurance risks and in each of these two classes there is a surplus process similar to that one of the classical model of risk theory. In [1] [2], and [3], it is possible to find results relating to the definition and to the determination of the ruin probability. In [4] and [5], two-dimensional risk models are considered in the presence of two insurers and under these assumptions, models where the two insurers can be considered as an insurer and a reinsurer are studied. In the literature, further relevant models are: the model presented in [6] where the authors consider a two dimensional ruin problem for two insurance companies that divide between them both claims and premia in some specified proportions and the model presented in [7] where the In this paper, we always consider only one insurer who has two classes of insurance risks; we therefore consider a two-dimensional risk model process where we assume that an insurer has two classes of insurance business. We assume that these two classes are dependent through the number of claims. The risks of the two dependent classes of insurance risks are sometimes of very different kinds (for example the risk class connected to the insurance of the current damages that can happen to the building and the class of risk connected to the insurance of damages related to an earthquake) therefore, in the insurance practice, it may be more suitable the proportional reinsurance for the first class and the Excess of Loss reinsurance for the second. Hence, we present a new model involving these two different kinds of reinsurance: we assume that the insurer reinsures each of the classes with a quota-share and with an Excess of Loss reinsurance, respectively. We address the problem of determining the proportional retention level (a) and the optimal Excess of Loss retention limit (b) that in a given period of time maximize the expected utility of the total wealth of the insurer; that is, our aim is to calculate the couple (a, b) in order to maximize the expected utility of the total terminal wealth after reinsurance. Therefore, inspired by [8], we face in the paper a new problem. This paper can improve the reinsurance policies of the company which could thus improve its insurance offer.
The paper is organized as follows: Section 2 is devoted to the presentation of the model. Section 3 contains the optimization problem and Section 4 presents the Theorem and resumes the results; Section 5 concludes.

The Model
We consider only one insurer having two classes of insurance risks dependent through the number of claims. Let { } , 1, 2, ij X j = be the claim size random variable for risk i, We assume that the , 1, 2, ij X j = have the same distribution function We therefore assume that: [2], we consider a given period of time. Let i N be the number of claims, for classes i, , in the given period of time considered; we assume: where 1 2 , Q Q and 0 Q are independent Poisson random variables with positive parameters 1 2 , θ θ and 0 θ , respectively. Hence (N 1 , N 2 ) is a bivariate Poisson distribution (see [9]). As usually stated, the random variables ij X are mutually independent for each 1, 2, j = , and are independent of i N , After the reinsurances, the insurer is required to pay 1 for each claim of the first class and for each claim of the second class, 1, 2, j = .
Therefore, the aggregate claims paid by the insurer are: ( ) To obtain the above reinsurance policies, the insurer pays the corresponding insurance premiums, and ( ) ( Therefore, after reinsurances, the value of each reinsurer surplus, that we de- For which, the total surplus is:

The Problem
As we have stated before, our goal is to maximize the expected utility of the total wealth of the insurer. According to several Authors, we assume that the insurer's We therefore aim to find the  that is, remembering (2)- (7) and (8): Avoiding the constant terms respect to a and b, our objective function is: In the following, we will consider the variables , 1, 2 , respectively. Remembering that, for assumption (1), ( ) 1 2 , N N has the bivariate Poisson distribution and therefore the moment generating function of [8] and [9]): and therefore, if it is: it results: It follows that the problem (9) can be written as: Remembering (4), (5) and (10), it results: Finally, we observe that minimizing , k a b . We therefore have the problem: with the constraints : We consider the problem (13) using Kuhn-Tucker conditions and we state: We observe that it results: We observe that (16) and it is decreasing when α increases.
We consider the following cases of which ( ) and we observe that both (I 1 ) and (I 2 ) are impossible.
• Case II 0 a = and The following conditions must be fulfilled: that is it must exist 1 0 b > satisfying the conditions: (II 2 ) We observe that it exists 1 0 b > satisfying condition (II 2 ); it is: • Case III 1 a = and The following conditions must be fulfilled: The following conditions must be fulfilled: ( ) The following conditions must be fulfilled:

The Results
The results obtained allow us to show the following: Theorem Let ( ) , a b a Kuhn-Tucker point, solution of the system (14).
1) The following statements are hold: In the following we suppose that condition (II 1 ) is not satisfied, that is (22) holds and that moreover it results:

Final Conclusions
In this paper, we presented a two dimensional risk model where the claim counting processes of the two classes of insurance business are dependent through the [ ] 0,1 a ∈ and a retention limit b, ) 0, b ∈ +∞   , respectively.
Fixed a given period of time, we have considered the expected utility of the insurer's wealth and we have assumed that the insurer looks for the pair (a, b) that maximizes this utility. In the paper, we have constructed the model that describes the above problem and we have assumed, according with several authors, an exponential utility function and the expected value principle for the computation of the insurance premiums. We therefore have considered the possible pairs (a, b) candidates to solve the problem, deriving the conditions under which each pair exists.
We are currently dealing with the possible application of the model presented in this article to a particular case of social significance. Furthermore, as well as other authors have done in models different from the model presented in this paper, a natural development of our study could be consider more than two classes of insurance business with two or more reinsurance types.