Dividends and Dynamic Solvency Insurance in Two-Dimensional Risk Models

In this paper we consider two-dimensional risk models where the claim counting processes of the two classes of business are assumed to be Poisson processes. We assume that the dividends are paid because of the presence of a reflecting upper barrier. Furthermore, in order to avoid ruin, we consider dynamic solvency insurance contracts that depend on two different definitions of time of ruin. We present a rather general model and, under different assumptions, we obtain the equations fulfilled by the discounted dividend payments and by the net single premium of dynamic solvency insurance. We also derive some boundary conditions and provide explicit solutions for some special cases.


Introduction
The classical model of collective risk theory has been modified and extended over the years in several ways. Some authors [1] [2] [3] [4] have introduced a two-dimensional risk process where the insurer has two classes of business and in each of the two classes he has a surplus process similar to the one of the classical model. The two-dimensional risk process is defined by a two-dimensional vector having as components the above surplus processes. In this two-dimensional case, different interpretations of the concept of ruin have been proposed, and different definitions of time of ruin have been introduced. As it is well known, classical models of collective risk theory can forecast the dividends payment (for instance, see [5]) and, as proposed in [6], they can also have a dynamic solvency insurance. Under these assumptions we have models with two barriers. The two-dimensional models and some definitions of time of ruin encouraged us to investigate about dividends and the net single premium of a dynamic insurance contract in the two-dimensional models.
The paper is organized as follows: Section 2 is devoted to the presentation of the two-dimensional models under different claim arrival processes; Section 3 considers barriers referring to two definitions of time of ruin. Under these two different assumptions of time of ruin, Sections 4 and 5 contain the detailed computations to get the integro-differential equations, both of the net single premium of the dynamic solvency insurance and of the present value of dividends. In Sections 6 and 7 some boundary conditions for the above equations with some explicit solutions are given.

The Model
We assume that an insurer has two classes of business or insurance risks. Let As usually stated, the random variables ij X are mutually independent for each 1, 2, j =  , and are independent of Let 1, 2 i = ; the aggregate claim amount of each class of insurance risk is: with different but independent counting processes for each class of insurance risk (this assumption has been made, for instance, in [4]).

The Ruin and the Barriers
It is well known that, in two-dimensional risk models, the time of ruin can be defined in different ways (see for instance [1] [2] and [3]): and In this paper we consider the time of ruin (7) or (8). Indeed, we observe that under the definition of max T the ruin would not occur also in the case that one of the two surplus is deeply negative, provided that the other one is even only slightly positive: this scenario seems to be too risky for the insurance company. Under the definition of sum T U t are greater than or equal to zero. Therefore, the dynamic solvency insurance (see [6]) under the assumption (7) refers to the sum process and its payments are made immediately as soon as the surplus ( ) U t falls below zero; instead, under the assumption (8), the dynamic solvency insurance refers separately to ( ) 1 U t and ( ) 2 U t and its payments are made immediately as soon as the surplus ( ) This means that, in the first scenario, we have a reflecting inferior barrier at zero for the sum process, whereas in the second scenario we have reflecting inferior barriers for each process with surplus Even for the dividends payment it is necessary to introduce barriers. In particular, we assume to have the constant upper barrier for the the -th class of the two-dimensional risk process, and we consider to have the constant upper barrier for the sum process. We recall that dividends are paid whenever the corresponding surplus reaches its upper barrier.
Because of all the above assumptions, models in (2), (4) and (6) need to be consequently and coherently modified. This will be achieved in the following sections, where we address the problems of the net single premiums and the discounted dividend payments in the case of two barriers.

The Net Single Premium of Dynamic Solvency Insurance
This section is devoted to determine the equations fulfilled by the net single premium of the dynamic solvency insurance under the assumptions made in Sections 2 and 3. We use a procedure similar to the one presented in [7] and in [8] to determine the ruin probability. Hence, we introduce the force of interest 0 δ > .
We first consider the case in which the time of ruin is sum T defined in (7).
We therefore consider the sum process having a surplus with initial value (in

U t U t U t = +
is the surplus at time t, whose value is obviously dependent both on the dividends payment and on the assumption of the dynamic solvency insurance. We denote by ( ) , A u b the net single premium of the dynamic solvency insurance. Let T be the length of the interval time passing before the occurrence of some first claim for the sum process and let X be the amount of the claim occurred at time T. If no claim occurs, we have that ( ) U t u ct = + and ( ) U t crosses the barrier b at time * t defined by: Following the approach of [7] and [8], we compute ( ) Since we have assumed that the arrival claims processes have the bivariate Poisson distribution defined in (1), the event T t = happens when no claim occurs during the time interval going from 0 to t and the first arrival of (any kind of) claim arrives at the end of this period of time.
Using the results of [9], the above events depend on the following probabilities (neglecting contributions of order higher than dt): Recalling that T t = is the moment when some first claims occur, if we assume that at time t it is 11 1 X x = and/or 21 we have: We denote by F the distribution function of the random variable 11 Because of the assumptions on the distribution functions i F , Hence, by substituting u ct + with z, and subsequently b x − with y, we obtain: By deriving (12) with respect to u, we obtain the following integro-differential equation: insurance contract for the model given in (6).
We now consider the case in which the time of ruin is min T defined by (8). We recall that, using this definition, the ruin occurs when at least one of the surplus 1, 2 i = , becomes negative. Thus, the dynamic solvency insurance has to intervene separately in the two classes, and it has to avoid that both surplus become negative. Let . In the following, using a procedure similar to the one adopted in the previous discussion, we deduce the equations fulfilled by ( ) To this end, we let i T be the length of the interval time passing before the occurrence of some first claim for the -th process and  (10) at in (14). We recall that the arrival claims process has the bivariate Poisson distribution defined in (1). Since the two processes are considered separately, for any 1, 2 i = it results (neglecting contributions of order higher than dt): We are now able to write the following integral equation for the net single premium of dynamic solvency insurance for the i-th claim, Hence, by substituting i i u c t + with z, and subsequently i b x − with y, we obtain: By deriving (16) with respect to i u , we obtain the following integro-differential equation: Obviously, if we restrict ourselves to the case of model given in (4), that is if we let

Discounted Value of the Dividend Payments
In this section we determine the equations fulfilled by the discounted value of the dividend payments under the different assumptions made in Section 4. To get our results, we follow a similar approach to the one used for the net single premium of dynamic solvency insurance.
We first consider the case where the time of ruin is sum T defined in (7).
Hence, we consider the sum process having a surplus ( ) U t with initial value (in 0 t = )  As in Section 4, we assume that T t = is the instant when claims occur. If at time t we have 11 1 X x = and/or 21 2 X x = , then formula (11) still holds. Therefore, we are able to write the following integral equation for the discounted value of the dividend payments: where, by using (19) and with some computations, the last integral can be rewritten as follows: From the previous expression, deriving with respect to u, we obtain the following integro-differential equation:  of the dividend payments for the model given in (6).
We now consider the case in which the time of ruin is min T defined by (8). We note that the observations pointed out in Section 4 still hold and we determine the equations fulfilled by the present value of the dividend payments. We where, by using (19) and with some computations, the last integral can be rewritten as follows: Hence, by substituting in (24) the expression (25) and the new variables   From the previous expression, deriving with respect to i u , we obtain the following integro-differential equation: Obviously, if we restrict ourselves to the case of model given in (4)

Boundary Conditions
In this section we compute some boundary conditions satisfied by the net single premium ( )  23) and (26), (27)), which will be used in Section 7 to provide some explicit solutions.
By simply substituting 0 u = in Equation (13) we get: whereas, by substituting u b = in Equations (12) and (13), and combining them we obtain: In the same way we get the boundary conditions for ( )

Explicit Solutions for Some Special Cases
In this section we provide explicit solutions for the net single premium and the discounted value of dividend payments in some special cases.
We first consider the net single premium ( ) In the same way, using our hypotheses, the integro-differential Equation (23) can be rewritten as follows: In the same way, using our hypotheses, the integro-differential Equation (27) can be rewritten as follows: