The Optimal Dividend And Injection Strategy In The Risk Model With Constant Interest

With the development of the time, insurance companies get a lot proceeds by kinds ofinvesting, i.e. to win more total dividends before the time of ruin. However, how to make surethe dividend strategy, so that the shareholders and policy-holders can win most, is the keystoneof all research, now lots of academicians acquire prodigious achievement about the disquisitionof the dividend strategy. But in real life, only care about the dividend profit is not enough; weshould attention the managing of the companies. As the shareholders get the dividend profit, theyshould be obliged to help the company when it encounters difficulties, the shareholders shouldcover the deficit at ruin, i.e. inject capital, so the surplus at the time of ruin is then0and thecompany survives. Capital injection strategy provides an approach to keep the company runsmoothly. So, to study about the dividend and injection strategy is very important for insurancecompanies.Base on upwards analyses, the note fist calculate the optimal dividend strategy in a riskmodel with constant interest but no capital injections, indicate the optimal dividend strategy is abarrier strategy, then we get the integral equation contented the dividend gross value, and we alsoget the expectation accurately result of the dividend gross value and the definite result underindex claim, when the initial capital u of the insurance companies is equal or greater than theextra dividend bounds b.Secondly, we consider the optimal dividend payments and injection strategy in a risk modelwith capital injections and constant interest. Based on the risk model with constant interest, weinject capital; get a HJB equation and optimal stopping time by using random control theory, atlast we get some properties that the value function is contented.