Optimal Dividend Distribution,capital Injection And Reinsurance Strategy In The Diffusion Risk Model

Insurance company devote themselves to reduce its risk exposure.An effective way is reinsurance.The company chooses reinsurance to reduce its risk exposure,which implies less income.In recent years,it is a hot issue that to strike a balance between the risk and profit of a company by controlling reinsurance and dividends.This paper studies the optimal reinsurance,dividend payments and capital injections strategies,using the stochastic theory,the optimal strategies theory and HJB equation.We consider proportional reinsurance,bounded dividend and capital injections in general diffusion model and diffusion model with debt liabilities.The aim is to maximise the expected total discounted dividend payments minus the expected total discounted cost of capital injections by the optimal strategy.We derive the closed form of the optimal strategy.On the basic of diffusion model,the model can reduce and increase its risk and profit by controlling the proportional reinsurance,bounded dividend and capital injections,and the earning will be paid once it reaches the standard for dividends,in order to maximize the expected dividends minus the expected capital injections when bankruptcy,namely,get the optimal value function.We derive the closed form of the optimal value function and the optimal strategy by solving corresponding HJB equation.On the basic of diffusion model with a debt liability,the model can reduce and increase its risk and profit by controlling the proportional reinsurance,bounded dividend and compulsory capital injections,and the earning will be paid once it reaches the standard for dividends,in order to maximize the expected dividends minus the expected capital injections when bankruptcy,namely,get the optimal value function.We consider two distinct optimization problems.The first optimization problems is finding corresponding the value function and the optimal strategy under which the controlled surplus process never becomes negative.The second optimization problems is finding corresponding the value function and the optimal strategy under no capital injections.At last,we derive the solution to the general problem and in which situations should choose capital injections or not.