This paper deals with optimal dividend payment problem in the general setup of a piecewise-deterministic compound Poisson risk model . The objective of an insurance business under consideration is to maximize the expected discounted dividend payout up to the time of ruin. Both restricted and unrestricted payment schemes are considered . In the case of restricted payment scheme, the value function is shown to be a classical solution of the corresponding Hamilton-Jacobi-Bellman equation, which , in turn , leads to an optimal restricted dividend payment policy . When the claims are exponentially distributed, the value function and an optimal dividend payment policy of the threshold type are determined in closed forms under certain conditions. The case of unrestricted payment scheme gives rise to a singular stochastic control problem. By solving the associated integro-differential quasi-variational inequality,  the value function and an optimal barrier strategy are determined explicitly in exponential claim size distributions. Two examples are demonstrated and compared to illustrate the main results .
This paper considers the optimal dividend payment problem in piecewise-deterministic 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 integro-differential quasi-variational 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 .
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