In the research domain of portfolio and risk management, some optimal and control problems are involved. Due to the randomness of the environment, these control problems are characterized by stochastic optimal control model. Generally, in these control models, the price of equities,assets of firms and other market variables are taken as state variables. The portfolio ratio and stopping time of trade are considered as control variables. The objective of control is to maximize the total return or minimize the risk of investment. The main method to solve the optimal control problem is the dynamic programming principle(DPP). And a Hamilton-Jacobi-Bellman(HJB) can be derived by DPP. In this article, two kinds of control problems are studied. Under the background of insurance company, in the case that the investment portfolio contains European call options, and the objective of control is to maximize the total return. The other one, under the background of network lending, assuming the firm of P2 P invest to a bond of credit risk, and the objective of control is to maximize the risk of investment. These two problems, the closed-form solutions of the HJB Equation are both solved, and the verification theorem is also proved. Finally, the influence of parameters on the solution are analyzed to illustrate the results. This article is divided into five chapters.The first chapter illustrates the research background and the research methods and status in domestic and overseas. In chapter two, we simply reviewed the dynamic programming principle and the multidimensional It?o formula, which are the main research methods and theoretical basis throughout this article.In chapter three, optimal investment and proportional reinsurance strategy with options is studied. In the perspective of insurance company, we consider the optimal investment and proportional reinsurance strategy in the case that the investment portfolio contains European call options under the Black-Scholes model. By using stochastic control method, the utility maximization model and the corresponding HJB equation are obtained. What’s more, the closed-form solutions of the HJB Equation is solved, and the verification theorem is also proved. Finally, the influence of parameters on the solution is analyzed to illustrate the result.In chapter four, we measure the risk of P2 P firm. In the view of P2 P firm, we discuss the optimal investment strategy in the case that P2 P invest to risk equities and a bond of credit risk.The objective is to minimize the risk on the condition that the terminal wealth is a constant. To solve this problem, the mean-variance model is used. The mean- variance problem can be seen as an optimal control problem with equality constraints. However, by using the Lagrange duality theorem and Lagrange multiplier, the original problem will be converted into an optimal control problem without constraint, then the method of dynamic programming can be used to solve this problem.Finally, we make the numerical analysis.In chapter five, we summarize the main results of this article, the inadequate and the direction of future research. |