Optimal Portfolio In Fractional Jumping Diffusion Process

In 1952, The paper "Porofolio Selection" which was published by H.Markowiz laid the theoretical foundation of financial mathematics.The average was repre-sented as the returns of the stock .And the covariance was represented as the risk of the stock.It quantify the "diversity" in the stock market.He wanted to struc-ture the optimal portfolio in financial assets so that to achieve the best balance of the average and the covariance.That is to say, if the average of asset returns is given, we should minimize the variance of the portfolio. If the covariance of the portfolio is given, we should maximize the average of asset returns.From then on, the research of optimal portfolio had aroused the interest of scholars,.They made a lot of further development and application, about the portfolio of classic Markowtiz’theroy.Not only portfolio investment, but also will be consuming wealth.Then Mer-ton to discuss the optimal portfolio of consumption, and created the stochastic optimal control method.Merton hypothesis time continuous financial model, the beginning of the continuous time asset portfolio theory.In this paper,we assume that asset price changes obey poisson jump fractional Brownian motion.Then,we discussed the utility functions for the power function of the optimal portfolio and the optimal portfolio of consumption.Firstly, in the introduction of the history of the portfolio strategy and the research significance, and research results.Secondly, in the second chapter, preliminary knowledge of introduction, the main is several types of the basic random process and random analysis of fractional Brownian motion.Then, in the third chapter discussed the financial market model, the assump-tion that changes in the underlying asset price are assumed to be with poisson jump of fractional Brownian motion, to study the optimal portfolio of asset alloca-tion.Specific means is: to establish a value function, even if the final total wealth maximum;Dynamic programming principle is used to derive the differential equa-tion of HJB;Finally get the optimal portfolio allocation policy. This solution can give individual investors in the investment decision-making to provide beneficial reference.In the fourth chapter, the investors in the investment in the process of con-sumption is took into account, the value function into the maximization of invest-ment accumulated wealth and consumption.The study of the optimal portfolio of consumption can give investors in investment decision-making advice in the process of consumption.Finally, in chapter 5,we will summarize the main research contents and progress,and the next research direction was prospected.