Optimal Ratcheting Of Dividends With Capital Injection

With the improvement of people’s living standards,more and more people pay attention to insurance,and more and more insurance companies open up.How to control the insurance surplus process,so that the insurance company’s bankruptcy probability is the smallest,and the investor’s income is the largest,has been widely concerned by scholars.If only considering dividends,it will lead to the bankruptcy of the insurance company;if only considering capital injection,it will also sacrifice the interests of investors.In order to ensure the healthy operation of the insurance company,we consider the non negative dividend.In this paper,we introduce dividend and capital injection into the classical risk model.Dividend is paid from the insurance surplus.We assume that the dividend is ratchet dividend,that is,the dividend rate is non decreasing.When the surplus is less than 0,we immediately inject capital to make the surplus reach 0.We establish the risk model of capital injection and ratchet dividend strategy payment.The goal of this paper is to maximize the expectation of the difference between the discounted dividend and the penalised discounted capital injection,and take it as a value function to derive some properties of the corresponding optimal value function,that is,the optimal value function does not decrease in the first parameter x and has Lipschitz continuity;the optimal value function does not increase in the second parameter c and also has Lipschitz continuity;the optimal value function is bounded and when x approaches infinity,the value of the optimal value function tends to be a constant;the optimal value function is not negative.Then,the HJB equation of the corresponding optimal value function is derived,and it is proved that the optimal value function is the only viscous solution of the corresponding HJB equation by using the viscosity solution theory and the verification lemma.