Research On Portfolio Optimization Problem Based On Stochastic Differential Games

Portfolio optimization problem not only has been a hot issue in the modern financial and investment market research, but also is one of the investment questions that concerned by investors. In modern financial market, investors want to get higher returns,but don’t bear excessive risk, which makes the portfolio problem thermalized. The core idea of portfolio theory is to allow investors to invest in different securities markets by the appropriate proportion of their all assets, so as to realize the expected benefits maximization under a special utility function in a period of time. That is the optimal strategy problem which is required in this paper. The real financial market is affected by major events, important factors levels in market(such as economic crisis, financial crisis,etc.), the stock price is not continuous jump. So, based on stochastic differential games, the problem of investment optimization is studied mainly in this paper when the shares subject to the three different kinds of processes with the competition.(1)The optimal portfolio decision problem is studied when the stock price follows the jump-diffusion process. Firstly, based on the idea of stochastic differential games, a portfolio optimization mathematical model is established when the stock price subjects to the jump-diffusion process. Secondly, followed by the logarithmic, exponential and power utility function, the Ito formula, functional variational method and stochastic control method are used to study the issues of two people competition optimal portfolio option respectively, and the explicit solutions are obtained. Lastly, the problem of the investment choice of two persons competition is studied under the general utility functions, and the formula of the optional strategy is obtained.(2)The optimal portfolio decision problem is studied when the stock price follows the Levy process. Based on the idea of the stochastic differential games, the mathematical model of investment portfolio optimization is established when the stock price subjects to the Levy process. By using the Ito formula of Ito-Levy process and functional variational method, the problem of the two people competition investment portfolio strategy is studied under the logarithm utility function, and the display expressions for optional policy are obtained.(3)The optimal portfolio decision problem is studied under the partial information when the stock price follows the jump-diffusion process. In the actual financial markets,investors can’t predict the future information flow changes, and only can observe the information flows generated by the stock price in the past. So, in this paper, firstly, the random yield of the risky assets is parameterized, and the specific representation of the drift is given in the stock price model, and then, the mathematical model of investment portfolio optimization is established under the partial information when the stock pricefollows the jump-diffusion process. Secondly, the portfolio problem under partial information is transformed into a portfolio problem with complete information by using the filtering techniques to estimate the drift filtering, the Girsanov theorem and the measure transformation. Lastly, the optimal portfolio decision problem is studied by using the Ito formula and functional variational method under the logarithm utility function, and then the display expression is achieved, which not only provides a more realistic strategy for securities investment in the modern financial market, also provides investors with an available investment decision.