Study On Optimal Investment Problem With Default Risk In O - U Process

In financial mathematics, study the optimal investment problem by using stochastic control theory is an important area of research. With the development of global economy, investors and investment institutions are facing the investment decision-making problem almost every day. So it becomes especially important to research all kinds of optimal investment problem. In the in-ternational financial market today, all kinds of financial products is complicated, default risk in the financial market get more and more attentions from people. Under such a big background, the research about optimal investment problem whit default risk not only has important theoret-ical significance, but also has strong practical significance. The reason that this article chooses the process of O-U to describe the volatility of stock prices is that the process of O-U is more aligned with the actual financial markets than the geometric Brownian motion. By using the op-timal control theory and the theory of viscosity solution, this paper mainly discusses the optimal portfolio problem with the default risk under the process of O-U, the optimal investment problem with default contagion under the process of O-U and utility indifference price for a defaultable bond under the process of O-U.The first chapter introduces the background of the subject, research significance, the devel-opment situation at home and abroad and the main research content of this article.The second chapter introduces the basic preliminary knowledge which is used in the model of this paper. Include the process of O-U, default process, the theory of optimal control and viscosity solution and utility indifference price.In the third chapter, we assume that investment products chose by the investors is divided into deposit in bank, stock and defaultable bond under the process of O-U. We established the optimal investment model under the condition of defaultable. Through the stochastic optimal control theory, we combine the objective function of the investor and dynamic programming principle, get the HJB equation which meets the value function and then prove that the value function is a viscosity solution of HJB equation. At last, we solve the equation by using finite difference scheme which meets the positive coefficient conditions and the final result are analyzed.In the fourth chapter, we assume that the investors to buy the stocks of company A and the bond of company B, the two companies both have the risk of default and they both have a default contagion. We describe the influence which is brought by default contagion by observing the changes of default intensity, and establish the optimal investment problem model from the perspective of investors, then we get the approximate solution by using finite difference method and the numerical results and parameters are analyzed. In the fifth chapter, the utility indifference pricing method was used to study the pricing prob-lem of the defaultable bond under the process of O-U. We assume that the investor can optimize his own investment portfolio dynamically within the period of validity of defaultable bond, we take advantage the process of O-U to replace the traditional geometric Brownian motion to de-scribe the movement of stock price. And then we deduced the HJB equation and get the value of the utility indifference price by using the dynamic programming principle under on two occasions that investors to buy and not to buy the bond.In the sixth chapter, we summarize the content of the research in this paper and point out the problems can be further study.