Optimal Dividend Strategy for a Compound Poisson Risk Model-Measure-valued Dynamic Programming Equation Approach

- This paper is intended to develop a new approach of deriving the measure-valued dynamic programming equation (DPE) to study the optimal dividend distribution problem for compound Poisson risk models. If the dividend payments are not considered, the risk process is a PDMP process, and the DPE is measure-valued. We introduced the characteristics of the Markov controls in the optimal dividend problem firstly, and the dividend strategy is an additive functional of the controlled company's surplus process. Furthermore, we demonstrate that the optimal dividend strategy is a surplus-dependent Markov control-band strategy, in the set of Markov control strategies. Then we identified the verification and demonstrate that the optimal dividend strategy is a stationary Markov strategy, in the set of all admissible strategie. As a description of the measure-valued DPE, when the claim sizes are mixed exponentially distributed, we briefly discuss the optimal dividend problem for a insurance risk model.