Optimal Dividend And Capital Injection Strategy And Optimal Reinsurance Strategy In The Classical Risk Model

How to take some reasonable strategies (for example dividend, capital injection, investment or insurance) to minimize the company's risk or maximize the shareholder's profit is belong to the problem of stochastic control in insurance in insurance.Dividends mean the company pays certain surplus to the shareholders. The aggregate dividend payment may reflect the company's benefit. The optimal dividend problem was first discussed by De Finetti at the 15th International Congress of Actuaries in 1957. He proposed that the company should maximize the expected discounted dividends before ruin. So when to dividend and the amount of dividend payments are hot topics in the stochastic control field. Since 1957, the research on optimal dividend strategy has yield substantial results. We refer the reader to see Avanzi(2009), which is a review on dividend strategy.With the development of dividend strategy, people begin to consider such a problem:as the shareholders get the dividend profit, whether they should be obliged to help the company when it encounter difficulties. Dickson & Waters (2004) pointed that the shareholders should cover the deficit at ruin, i.e. inject capital. So the surplus at the time of ruin is then 0 and the company survives. Capital injection strategy provides an approach to keep the company run smoothly. Gerber et al. (2006), Kulenko & Schmidli (2008), He & Liang (2009) had discussed such strategy.Based on the analysis above, the optimal strategy should combine the dividend and capital injection strategies together. Thus, the company's risk can be weakened and the shareholder's profit can be enhanced. However, in most references dealing with capital injection strategy, the shareholders are required to pay the deficit whatever the severity is. Of course, this requirement has a very important academic role, but it will not satisfy the shareholder's optimal interests. The injected capitals are equal to new investment, so they should receive reasonable rewards. Then what is the real optimal capital injection strategy is in presence of us. In this dissertation, I will discuss the optimal dividend and capital injection strategy to maximize the net profit of the shareholders.Except for the two strategies mentioned above, reinsurance is also very important to the company. Reinsurance is an agreement between an insurer and a reinsurer under which claims are split between them in a agreed manner. This strategy can reduce the probability of suffering losses and diminish the impact of the huge amount of claims. Though it had been discussed in many papers, most of them are based on the one-dimensional risk model. With the development of insurance, multi-dimensional risk model has been paid more and more attention due to the fact that it can describe the effect bring by the dependent claims. So we also discussed the optimal proportional reinsurance in the two-dimensional classical risk model to minimize the ruin probability.Based on the analysis above, my dissertation is organized as follows.In the second chapter we will study the optimal stopping of the controlled process under the optimal dividend strategy in the classical risk model with capital injection. Through the optimal stopping time we illustrate when should the shareholder inject capital to rescue the company and when should they refuse to do so. From the HJB equation satisfied by the value function V(x), we find the optimal stopping timeτ*. Especially, when the claim is exponential distributed we derive the explicit expressions of V(x) andτ*. The optimal stopping strategy enhances the net profit of the shareholders when comparing with Kulenko & Schmidli (2008).In the former chapter, we find the restriction of the shareholder covering all deficit should be removed. So in the third chapter, we will research the optimal dividend and optimal capital injection in the classical risk model. According to whether the dividend was restricted, we establish the HJB equation satisfied by the value function V(x) respectively. Finally, we find the optimal dividend and optimal capital injection strategies. Especially, it is the first time that we propose the definition of optimal upper and lower capital injection barriers. For detail, we calculate V(x) and optimal strategies when the claim is exponential.In Chapter 4, we define the model and the objective function to be more realistic. Considering the company which is under supervising, so in order to operate it conveniently the surplus should not be less than a positive level m>0. Meantime, the transaction cost when dividend is payout will not be neglect, so we assume that dividend is with proportional transaction costs. The value function and the optimal strategy will be discussed.The last chapter is on the proportional reinsurance police in the two-dimensional classical risk model. Via HJB equation we find a candidate of value function V(x). Because the explicit expression of V(x) is very hard to be obtained, we study the Lundberg bounds and the Cramer-Lundberg approximation of V(x) and finally get the asymptotically optimal constant reinsurance coefficients. Adjustment coefficient plays an important role.