Research On The Expected Discounted Penalty Function And The Optimal Dividend In Some Risk Models

Risk theory is the important part in actuarial science. It is mainly about two prob-lems:risk and profit. The risk of the company can be described by some actuarial variables such as the ruin probability, the surplus immediately prior to ruin, the deficit at ruin and so on. The expected discounted penalty function includes the above mentioned risk variables and has been become an important actuarial variable. In addition to risk, the company is also concerned about the profit. The total dividends paid before the time of ruin are the most representative profit for the shareholders. How to maximize the total dividend payments is one of the most hot topics in finance. My dissertation focuses on the expected discounted penalty function in the renewal risk models and the optimal dividend in the dual risk models. It is organized as follows.Firstly, we discuss the expected discounted penalty function in the renewal risk mod-els under two cases:one is the compound Poisson income, the other is the investment and debit interest. In first case, we obtain the renewal equation satisfied by the expected discounted penalty function by the Lagrange interpolation formula and give the explicit result of the expected discounted penalty function when the distribution of claim be-longs to the rational distribution. In the second case, we derive the integro-differential equation satisfied by the expected discounted penalty function and give the absolute ruin probability in terms of hyper geometric functions.Secondly, we consider the optimal dividend with capital injection in the compound Poisson dual models. And the fixed and proportional transaction costs are involved in the optimal control problems. There are two cases in this part:one is the penalty for ruin; the other is the random time horizon. Under the two cases, we maximize the expected discounted dividends minus costs of capital injection before the ruin and find the associated optimal strategies.Thirdly, we discuss three optimal control problems in the spectrally positive Levy risk models. The optimal dividend with capital injection is considered in the first two problems, in which the dividend rate is respectively restricted and unrestricted. In ad-dition, the transaction costs are considered in the two problems. We give the maximum of the expected discounted dividends minus costs of capital injection before the ruin in terms of the scale function and obtain the associated optimal strategies. In the third problem, the dividends can only be paid at the arrival times of a Poisson process. The optimal dividend strategy is also founded to maximize the expected discounted dividends until the time of ruin.