The Optimal Dividend With The Expected Discounted Penalty Function In The Continous-time Compound Binomial Model

In this paper,we use the stochastic control theory to research the optimal dividend problem with the expected discounted penalty function in the continous-time compound binomial model.The purpose of this paper is to get the optimal dividend policy which maximizes the accumulated expected discounted dividend with the expected discounted penalty function.First of all,we modify the common continuous-time compound binomial model and establish the dual risk surplus process by the supplementary variable method.In order to get the optimal dividend policy,we discuss the properties of value function in the optimal dividend problem.Then we derive the dynamic programming principle,and according to the dynamic programming principle,the corresponding HJB equation is given.Last,we discuss the solution of the HJB equation with the verification theorem,then get the optimal dividend policy.