Optimal Dividend Problem For A Generalized Risk Model With Investment Incomes And Debit Interest

It is one of the stochastic control problems in financial insurance to maximize shareholders' interests by using dividend process control the reserve process of insurance companies.The goal of optimal dividend problem is to find the optimal dividend strategy,i.e.choosing the times and amounts of dividend payments to maximize the expected total discounted dividends until the time of ruin.There has been a lot of research on this issue through decades of continuous effort.In the process of research,people realized that: from the practical point of view,ruin(negative reserve)does not mean the end of game but only the necessity of borrowing money so that the company can continuous its business.In order to further make shareholders more profitable,we consider appropriate investment.In summary,we consider the optimal dividend problem for a generalized model with credit interest for positive reserve and debit interest for negative reserve.A ceiling is imposed on the dividend rates.The company controls the dividend pay-outs dynamically with the objective to maximize the expected total discounted dividends until the absolute ruin time.We use a new method,i.e.the measure-valued generator theory,to study this problem.Firstly,we characterize admissible control strategies and the Markov control strategies,we show the necessary and sufficient condition for a strategy to be a stationary Markov control strategy.And we derive the dynamic programming principle and dynamic programming equation.We prove the verification theorem in content of all admissible control strategies,even though the DPP and DPE are derived under Markov control strategies.Finally,we prove that the optimal control strategy is a stationary Markov control strategy with a band structure.