Optimal Proportional Reinsurance in a Bivariate Risk Model

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.


Introduction
The classical Cramer-Lundberg risk model assumes that the stochastic process ( ) N t denotes the number of claims up to time t and the random variables X j , the claim size of the j-th claim.In this model, ( ) N t is independent of the claim sizes and the claim sizes are independent and identically distributed; however, this assumption is too restrictive at times.Several authors have proposed models with dependence between the risks.Among the various types of dependence models proposed, in this paper we refer to the case where the dependency is obtained assuming that the insurer has 2 n ≥ correlated classes of insurance business, where ( ), 1, 2, , ,  is the number of claims of the i-th class; , 1, 2, ij X j =  , is the claim sizes of the j-th claim of the i-th class and the numbers ( ) i N t are dependent claim count processes.Models of this type are proposed in [1]- [6].In [3] [4], a risk model involving two dependent classes of insurance business is considered in a given period of time and the expected utility of the insurer wealth is maximized by the determination of optimal retention limits of Unlimited or Limited Excess of Loss reinsurance.
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.

The Model
In the finite time horizon [ ]  , are mutually independent, and independent of , the aggregate claims amounts of the risk i, i = 1, 2. We assume that the processes , are Poisson processes defined as follows: ( where ( ) ( ) ( ) , .
N t N t j j j j

S t X S t X
In the following, we will use the variables X i , i = 1, 2, identically distributed to , 1, 2, .ij X j =  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 i η , that is: We introduce a proportional reinsurance: the reinsurer pays ( ) , of each claim of the type i, i = 1, 2 and he receives from the insurer the reinsurance premium.We denote by ( ) P a , i = 1, 2, the reinsurance premium and we get Note that the condition = linked to the positivity of the security loading is not assumed, as it may or may not be satisfied.

The Problem
We denote by , where ( ) i a t is therefore the risk exposure of insurer at time t.We assume that, at every time t, [ ] , the insurer can choose the risk exposure , according to the observable information about the insurance risk processes up to time t.There-fore ( ) i a t , i = 1, 2, are the insurer's control parameters; let be the set of all admissible policies.The objective for the insurer is to choose an optimal reinsurance strategy that maximize the expected exponential utility of terminal wealth.We will make use of the HJB theory to face the problem.After the reinsurance the total claim amount charged to the insurer is We denote by ( ) , R t α the surplus process of the insurer adopting the reinsurance strategy process The surplus evolves over time as: We recall that ( ) ( ) N t are defined by ( 1), then the process has a bivariate Poisson distribution.This fact, noting that the process has stationary increments (see [7]) and using results in [8], allows us to determine the following joint probabilities: ( ) where  .We consider an utility function : , strictly increasing and concave (that is 0 u′ > and 0 u′′ < ).For each control strategy ( ) , we define the following conditional expected utility of the insurer's terminal surplus: As previously stated, the insurer's goal is to determine an optimal reinsurance strategy A α ∈ so as to max- imize the expected utility of the terminal surplus (8).We therefore consider the following problem It follows that the insurer has to find the optimal strategy * A α ∈ and the corresponding value function ( ) with the usual boundary condition (see [9] [10])

The HJB Equation and the Value Function
We can find the infinitesimal generator for the process ( ) , R t α and for the function V.The procedure is similar to that one used in [11] and in [12].
Theorem 1.Let ( ) , V t x be defined by ( 9) and (10) Therefore, V satisfies the following HJB equation: We derive the following infinitesimal generator for the process ( ) , R t α and for the function V: ( 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, ( ) ( ) ( ) 3) according to [7] and from the probability generating function of the bivariate Poisson distribution (see [8], p. 126), it results in: | and e e e : e a X a X a X a X

S T t S T t S T t S T t N T t N T t
4) from the previous considerations, we x T t P S T t S T t Because of these considerations, we assume that the value function V, defined by (10) with the condition (11) has the form with the condition

Possible Solutions
We consider the assumptions (18); it results in: Therefore, (12) becomes with ( ) 0 B T = .Assuming the particular case where the insurer's risk exposure is the same for the two classes of the insurance business; that is , 1, 2, 0 1; ), and (20) becomes: with ( ) 0 B T = .The problem of the determination of the optimal control * a can be easily solved, as we will see in the following.
For simplicity, we write (21) as follows [ ] ( ) , where is given by (24).with ( ) 0 B T = .From the previous, with a procedure analogous to that followed in the case * 1 a = , it is possi- ble to find ( ) B t and the resulting value function.
G′ ≥ it results in 1 a * = and by (21) we have:

(
value function (18) is:  , the random variable claim size of the risk i, i = 1, 2, assuming that these random variables have respectively the same distribution function F i , with ij X j =  exists.Finally, we assume that the random variables