Optimal Dividend Policies for PiecewiseDeterministic Compound Poisson Risk Models^{†}^{†}thanks: This research was supported in part by National Science Foundation under grant DMS1108782 and from the UWM Research Growth Initiative, and City University of Hong Kong (SRG) 7002677.
Abstract
This paper considers the optimal dividend payment problem in piecewisedeterministic compound Poisson risk models. The objective is to maximize the expected discounted dividend payout up to the time of ruin. We provide a comparative study in this general framework of both restricted and unrestricted payment schemes, which were only previously treated separately in certain special cases of risk models in the literature. In the case of restricted payment scheme, the value function is shown to be a classical solution of the corresponding HJB equation, which in turn leads to an optimal restricted payment policy known as the threshold strategy. In the case of unrestricted payment scheme, by solving the associated integrodifferential quasivariational inequality, we obtain the value function as well as an optimal unrestricted dividend payment scheme known as the barrier strategy. When claim sizes are exponentially distributed, we provide easily verifiable conditions under which the threshold and barrier strategies are optimal restricted and unrestricted dividend payment policies, respectively. The main results are illustrated with several examples, including a new example concerning regressive growth rates.
Key Words. Piecewisedeterministic compound Poisson model, HJB equation, quasivariational inequality, threshold strategy, barrier strategy.
AMS subject classifications. 93E20, 60J75
1 Introduction
The dividend problem in classical insurance risk models was originated in de Finetti (1957), and drew revived interests in recent literature focusing on optimization of dividend payment strategies. The optimality is often considered to be a strategy which maximizes the expected present value of dividends received by the shareholders. JeanblancPicqué and Shiryaev (1995) and Asmussen and Taksar (1997) investigated in diffusion models the dividend problems where the dividends are permitted to be paid out up to a maximal constant rate or a ceiling. We shall refer to such a type of dividend problem as restricted payment scheme. It was shown in their papers that the dividends should be paid out at the maximal admissible rate as soon as the surplus exceeds a certain threshold. Interestingly, it turns out that such a threshold strategy is the optimal restricted payment scheme in a variety of other risk models. For example, Gerber and Shiu (2006) discussed the threshold strategy in the compound Poisson model and solved the problem explicitly when the claim size is exponentially distributed. Fang and Wu (2007) studied a similar problem in the compound Poisson risk model with constant interest and showed the optimal dividend strategy is a threshold strategy for the case of an exponential claim distribution. See also Asmussen et al. (2000), Bai and Paulsen (2010), Choulli et al. (2003), Hunting and Paulsen (2013) Schmidli (2002), and references therein for some important developments on optimal dividend policies in the setting of controlled diffusions.
On the other hand, there was also a significant amount of literature in which no such restriction of maximal rate is imposed on dividend payment strategies. We shall refer to this type of dividend strategy as the unrestricted payment scheme. Such schemes are motivated by the fact that dividends are not usually paid out in a continuous fashion in practice. For instance, insurance companies may distribute dividends on discrete time intervals, in theory resulting in unbounded payment rate. In such a scenario, the surplus level changes drastically on a dividend payday. In other words, abrupt or discontinuous changes occur due to “singular” dividend distribution policy. This gives rise to a singular stochastic control problem. Such problems are studied in Choulli et al. (2003), Paulsen (2008), Shreve et al. (1984), and the references therein when the surplus dynamics is modeled by a controlled diffusion. But to the best of our knowledge, related work in the setting of piecewisedeterministic compound Poisson risk model is relatively scarce. The most notable includes (Schmidli, 2008, Section 2.4) and Albrecher and Thonhauser (2008), which investigate the optimal unrestricted dividend payment problem when the surplus process follows a classical CramerLundberg risk model without or with the force of interest, respectively.
As pointed out in Cai et al. (2009b), the classical CramérLundberg risk model and the compound Poisson risk model with interest, and the compound Poisson risk model with absolute ruin are all special cases of piecewisedeterministic compound Poisson (PDCP) risk model. One naturally asks whether there exist unifying optimal solutions to both dividend payment schemes in PDCP risk models. Moreover, can we find the most general conditions under which the threshold strategy is the optimal restricted dividend policy whereas the barrier strategy is the optimal unrestricted dividend policy? We formulate and solve the problems within the framework of stochastic control theory in the specific setting of PDCP risk model. Compared with the aforementioned work in the setup of controlled diffusions, the associated HamiltonJacobiBellman (HJB) equation in our work contains a nonlocal term (the integral term with respect to the claim size distribution), resulting in substantial difficulty and technicality in the analysis.
The contribution and novelty of this work also arise from several different aspects.

A salient feature of our model is the generality of pure jump models in which both restricted and unrestricted payment schemes are presented and directly compared. Although special cases of the PDCP risk model have been treated in the literature, this paper extends enormously the spectrum of risk models which exhibit common optimality.

We obtain general optimal solutions in the case of exponential claim size distribution. Moreover, we provide sufficient conditions to guarantee the optimality of the threshold and barrier strategies for the restricted and unrestricted dividend payment schemes in a general PDCP risk model. To the best of our knowledge, these conditions were unknown previously in the literature. Note that the analysis of solutions in the general case (Theorems 2.5 and 3.7) in this paper is entirely based on qualitative study of ordinary differential equation (ODE) and integrodifferential equation (IDE).

It is also worth mentioning that the solution methods presented in this paper are shown with examples to be more efficient alternatives to the known methods in the existing literature. For example, we propose simple procedures (Theorems 2.4 and 3.6) to identify the optimal threshold and barrier levels, rather than the optimization procedure on value functions which would have to be first determined explicitly.
The rest of the paper is organized as follows. After recalling the notion of a PDCP model, we formulate the optimality of dividend strategies as a stochastic control problem in Section 1.1. We consider in Section 2 the restricted dividend payment schemes. Some properties of the value function are derived and the value function is shown to be a classical solution to the HJB equation (2.3). In Section 3, we formulate the optimal unrestricted payment scheme as a singular stochastic control problem and establish a verification theorem of the quasivariational inequality (3.1). When the claims are exponentially distributed, with complete generality of the PDCP risk model, we provide easily verifiable sufficient conditions for the optimality of threshold and barrier dividend payment schemes and obtain explicit solutions for both the restricted and unrestricted dividend payment schemes in Sections 2 and 3, respectively. Three examples are provided for illustrative purpose in Section 4. Finally, the paper is concluded with several remarks in Section 5. Two technical results on the qualitative analysis of the solution to an ODE and several proofs are placed in the Appendix.
A preliminary version of the paper was announced in Feng et al. (2012) without proofs. In addition, the current version contains new results on sufficient conditions for the optimality of the threshold and barrier strategies (Theorems 2.5 and 3.7).
To facilitate later presentations, we introduce some notations here. We use to denote the indicator function of a set . When , and . Throughout the paper, we use the notations and interchangeably. As usual, and .
1.1 Problem Formulation
To give a rigorous formulation of the optimization problem, we start with a filtered probability space satisfying the usual condition. We assume that the surplus level is modeled by a piecewisedeterministic compound Poisson process. Note that the jump points represent the arrivals of insurance claims and the downward jumps are determined by claim sizes.
Suppose the surplus level of an insurance company at time is modeled by a piecewisedeterministic compound Poisson process
(1.1) 
where is the initial surplus, is a Poisson process with rate , are independent and identically distributed nonnegative random variables, and is a Lipschitz continuous function, taking values in , and satisfies the linear growth condition. Denote the common distribution function of by .
Denote by the sequence of jump points of the process , then for . Also, the surplus process between any two consecutive jumps is deterministic and given by with determined by and .
We give the corresponding expressions for and for three special cases in which optimal dividend policies will be developed later as examples.

(CramérLundberg model) The deterministic growth in surplus between any two consecutive claims is defined by the influx of premium at a constant rate per time unit, i.e., . Hence, .

(Constant interest model) All positive surplus earns interest at the constant rate per time unit, i.e., . Hence, .

(Regressive growth model) The surplus growth rate tends to regress to a mean premium rate according to with and Hence, and for .
More PDCP models can be found in Cai et al. (2009a, b, c), and Albrecher and Hartinger (2007).
We now enrich the model by considering dividend payout. Denote by the aggregate dividends by time . Assume that is càdlàg, nondecreasing, and adapted with . Moreover, we require that at any time , the dividend payment should not exceed the current surplus level, i.e., Any dividend payment scheme satisfying the above conditions is called an admissible control and the collection of all admissible controls is denoted by . The dynamics of the controlled surplus process under the admissible control is
(1.2) 
where is the initial surplus. The time of ruin is denoted by
The expected present value (EPV) of dividends up to ruin is defined as
(1.3) 
where is the force of interest. The objective is to find an admissible control that maximizes the EPV. That is, we seek
(1.4) 
Note that for all . Depending on the parameters of the model, can be . In the rest of the paper, to work with a wellformulated maximization problem, we assume that for all . See Section 3 for a sufficient condition for the finiteness of the value function.
2 Restricted Payment Scheme
We first consider problem (1.4) for the case when the dividend payment scheme is absolutely continuous with respect to time. That is, there exists some such that Moreover, we assume that is adapted and that there exists some positive constant such that , for all Denote the collection of all such dividend payment schemes by . The EPV corresponding to the initial surplus under the dividend payment policy is given by
(2.1) 
The goal is to find an admissible policy such that
(2.2) 
Apparently, we have for all , where is the value function in (1.4).
2.1 The HJB Equation and Optimal Strategies
We first derive some elementary properties of the value function (2.2), which will enable us to establish the HJB equation in Theorem 2.2. The proofs can be found in the Appendix.
Lemma 2.1.
The function is bounded by , increasing, and Lipschitz continuous on , and therefore absolutely continuous, and converges to as .
Theorem 2.2.
The function is differentiable and fulfills the HJB equation
(2.3) 
Moreover, the strategy with
(2.4) 
is optimal in the sense that where is the corresponding surplus process under the strategy .
2.2 Exponential Claims
We consider a simple yet thoughtprovoking case where an explicit general solution to the HJB equation (2.3) and an optimal dividend payment policy can be obtained. Assume that the common claim size distribution is given by for some
Lemma 2.3.
The IDE
(2.5) 
has a positive and strictly increasing solution .
Proof.
It is well known that the IDE (2.5) has a unique solution determined up to a multiplicative constant. Without loss of generality, we choose . Then (2.5) implies that . Now the desired assertion will follow if we can prove that for all . Suppose this was not the case, then there would exist some such that for all but . Then by virtue of (2.5),
This is a contradiction and therefore we must have for all .
Theorem 2.4.
Let be a positive and strictly increasing solution to the IDE (2.5). Suppose that is a strictly increasing and concave solution to the ODE
(2.6) 
and that there exists a number such that is concave on and
(2.7) 
Then the value function is given by
(2.8) 
Moreover, the optimal dividend payment policy is the threshold strategy
(2.9) 
where is the corresponding controlled surplus process.
The interpretation of such an optimal strategy is as follows. First, a threshold is determined so that dividend payments start immediately when the threshold is attained. Second, as long as the surplus process remains above the threshold, dividends are paid out continuously at the maximal rate per time unit. No dividend payment is allowed when the surplus drops below the threshold.
Proof.
Denote by the function defined on the righthand side of (2.8). Note that is continuously differentiable with . Since both and are concave functions and , we must have for and for all Hence by virtue of Theorem 2.2, it only remains to show that satisfies the HJB equation (2.3).
It is clear by definition that
Therefore solves the HJB equation (2.3) if we can show that
(2.10) 
To this end, we define for It is straightforward to verify that Consequently, (2.6) can be rewritten as
Denote the LHS of (2.10) by . The equation above shows that for all . Letting in (2.5) and using , we see that . Thus, for all and the claim (2.10) is proved.
Finally the verification of optimality for the strategy defined in (2.9) is straightforward and we shall omit the details here.
Theorem 2.4 establishes the optimality of the threshold strategy and provides an easy procedure to identify the threshold level . These results are based on the existence of solutions to the IDE (2.5) and the ODE (2.6) with certain properties. In particular, the smooth pasting condition (2.7) must hold. One may naturally ask under what conditions these solutions and the threshold level exist. The following theorem gives a minimal set of easily verifiable conditions in the PDCP model.
Theorem 2.5.
Suppose that and satisfies
(2.11)  
(2.12) 
Then (2.6) admits a solution that is negative, strictly increasing and concave.

Otherwise, if
(2.14) then the optimal restricted payment scheme is with for all and the value function is given by
(2.15) where
Proof.
Take , , and in (A.1). Note that and by virtue of (2.12), for all , where . It then follows from Lemma A.2 that there exists a negative, strictly increasing and concave solution to (2.6) on
(i) In this case, by virtue of Theorem 2.4, it remains to prove that there is a such that is concave on and that (2.7) is satisfied, where is a solution to the IDE (2.5). As argued in Lemma 2.3, we can take and . Next, applying the operator to (2.5) gives the ODE
(2.16) 
In particular, due to (2.11), letting in (2.16) yields that
Therefore, With and , as argued before, Now it follows from Lemma A.1 that there exists a such that , for and for .
Using (2.6) and (2.16), we obtain
In particular, noting , the above equation yields
where the last inequality follows from the fact that is increasing and concave. On the other hand, it follows from (2.13) that
By the intermediate value theorem, the solution of (2.7) exists and .
The uniqueness of follows immediately from the mean value theorem. Suppose on the contrary that there were both satisfying (2.7). Then there would exist some with
But this is impossible, since
and
(ii) Note that the condition (2.14) is equivalent to Since for all , we have and hence for all . Moreover, as satisfies the boundary condition
using an argument similar to that in the proof of Theorem 2.4, it is easy to verify that given in (2.15) is indeed a solution to the IDE (2.10) and hence a solution to the HJB equation (2.3).
Remark 2.6.
With the complete generality of the PDCP model, we have shown that under assumptions (2.11) and (2.12), the optimal restricted dividend payment scheme is always the threshold strategy, which is to pay dividends at the maximal rate as long as the surplus is above a threshold level. In the case of (2.13), the threshold level is a unique positive number. Otherwise, if (2.14) holds, the threshold level is set to be zero.
3 Unrestricted Payment Scheme
In Section 2, we considered the case where the dividend payment rate is bounded. Consequently, the surplus level changes continuously in time in response to the dividend payment policy. However, in many applications, the boundedness of the dividend payment rate seems rather restrictive. For instance, insurance companies are more likely to distribute the dividend at discrete time points rather than with a continuous stream of dividend payments. Thus we remove the restriction on the maximal dividend rate and consider the (singular) optimal dividend policy for the PDCP risk model. In this case, , the total amount of dividends paid up to time , is not necessarily absolutely continuous with respect to .
Recall that for a given admissible dividend strategy , the associated EPV is given by (1.3) and the goal is to find an admissible dividend strategy that achieves the value function given by (1.4). The following proposition indicates that is nondecreasing. It can be proved using exactly the same arguments as those used in Song et al. (2011).
Proposition 3.1.
For any , we have
Standard arguments using the dynamic programming principle and Itô’s formula lead to the following verification theorem, which enables us to identify the value function and an optimal dividend policy later.
Theorem 3.2.
Suppose there exists a function with , for all , where is a countable set of points. Suppose satisfies for and that it solves the following quasivariational inequality:
(3.1) 
where is the infinitesimal generator defined by
(3.2) 

Then for every .

Define the continuation region Assume there exists a dividend payment scheme and corresponding process satisfying (1.2) such that,
(3.3) (3.4) (3.5) where denotes the continuous part of , and if , then
(3.6) Then for every and is an optimal dividend payment strategy.
Using exactly the same arguments as those in the proof of Theorem 3.2, part (a), we obtain the following proposition.
Proposition 3.3.
Suppose that there is function , where is a countable set, satisfying for and , for all . Then for any ,
(3.7) 
Remark 3.4.
It follows that if there is a function satisfying the conditions of Proposition 3.3 and also , then for all . For example, if for positive constants and , then for all . In fact, the function , satisfies the conditions of Proposition 3.3. Moreover, we compute for
Then it follows that for any , , we have Thus Proposition 3.3 implies that for all .
Remark 3.5.
3.1 Exponential Claims
In order to obtain an explicit solution to the quasivariational inequality (3.1) and an optimal dividend payment policy, as in Section 2.2, we again assume that the claim sizes are exponentially distributed with mean for some . In such a case, we first construct an explicit solution to (3.1), which is exactly the value function defined in (1.4). Then we provide easily verifiable sufficient conditions for the optimality of the barrier strategy.
Theorem 3.6.
Suppose that is a positive and strictly increasing solution of the IDE (2.5) and achieves its minimum value at and is nondecreasing on . Then
Proof.
(a) Note that . Obviously, if , satisfies (3.1). If , . Therefore it remains to show that
(3.11) 
To this end, we claim that
(3.12) 
By assumption, for and for . Hence it follows that for
(3.13)  
But is nondecreasing on , thus we have
(3.14) 
Since is a solution to (2.5), by applying the operator to (2.5), we see by straightforward calculations that
(3.15) 
A combination of (3.13)–(3.15) leads to
using the fact that is nondecreasing on . Equation (3.12) is therefore established.
As in Section 2.2, the next theorem provides easily verifiable conditions under which the optimality of the barrier strategy is established. Our result is applicable in a general PDCP risk model with minimal assumptions.
Theorem 3.7.
Assume (2.12). Then the following assertions are valid.