Optimal Dividend And Capital Injection Problem For Two Kinds Of Duality Models

Recently, in the basic of classical model, most of researchers are interested in the dual risk model at the same time. The application of dual risk model can be seen as the oil company, research company and so on. Since 1957, the dividend strategy was ?rst introduced and discussed by De. Finetti in the discrete time risk model. From then on,the dividend problem has become one of the important issues which insurance companies and actuarial science must face. In recent years, a barrier dividend strategy or a threshold dividend strategy has attracted a lot of attention in the dual risk model. But in the fact,we can not ignore the in?uence of the capital injection to the operation of assets. the optimization problem which through dividends and capital injection to control the assets has become a more and more heated topic. Literature [23] and [29] introduced the optimal asset control problem in the research status of ?nancial insurance. In this paper we consider the optimal dividend and capital injection problems under two kinds of the dual models, one is under the dual model with force of interest and another is under the two-dimensional dual model. We de?ne a performance function V(x, π) between each initial surplus x and each admissible strategy π. Our aim is to ?nd the best admissible strategy π?which can make maximize the expected present value of the dividends minus the discounted cost of capital injections. In fact V(x) = V(x, π?) and V(x) is the value function.In Chapter 1, we introduces some essential essential background and main results.In Chapter 2, we consider the optimal dividend and capital injection problems under the dual model with force of interest. Our aim is to ?nd the best admissible strategy π?that V(x) = V(x, π?) with the restriction on the dividend stream. At ?rst, we give some properties of the value function, then, derive the HJB equation: Last, we ?nd the the optimal capital injection strategy and the the optimal dividend strategy with the help of the HJB function. The optimal dividend strategy is threshold strategy.In Chapter 3, we consider the optimal dividend and capital injection problems under the two-dimensional dual model. Without the restriction on the dividend stream. At?rst, we give some properties of the value function, then, derive the HJB equation:We ?nd the the optimal capital injection strategy and the the optimal dividend strategy with the help of the HJB function. The optimal dividend strategy is Barrier strategy.