Research Advance On Dynamic Portfolio Selection In Stochastic Environments

The research of portfolio selection has been accompanied by the development of the financial market. It is one of the most basic problems in the field of modern financial economics, especially in the field of asset pricing. The research of this problem plays an important role in promoting the development of financial economic theory. But the real investment environment is not an ideal environment and the same always. On the contrary, it is full of randomness and uncertainty, such as the uncertain investors’investment preferences, the uncertain income, the uncertain exchange rate volatility and the uncertain inflation risk, etc. These uncertain factors will have a greater impact on the returns of investors, and it also affects the final consumption and investment strategy. In order to reflect the actual situation of investors’portfolio choice as well as possible, it is very important to consider these uncertain factors when the researchers establish the optimal portfolio choice model. It has an important significance to study how to allocate the assets effectively so that we can achieve the best goal and obtain the optimal portfolio strategy in stochastic environments.The main content of this paper is to discuss the effect of different random factors on the optimal consumption and portfolio strategies in stochastic environments. Firstly, based on the complete financial market, we study the optimal consumption and portfolio with the stochastic labor income when the stochastic interest rate is supposed to be driven by CIR (Cox-Ingersoll-Ross)model. We get the corresponding HJB equation and the optimal consumption and portfolio strategies by using the Ito formula and the dynamic programming principle. Under the power utility, the explicit solution of the value function and the explicit solutions of the optimal consumption and portfolio strategies are derived. The effect of uncertainty of stochastic labor income and the risk aversion on optimal consumption and portfolio ratio are analyzed by numerical simulation at last, and the economic significance also is explained.Secondly, based on the exchange rate following a diffusion process and under time-varying investment opportunities, we study the effect of the random exchange rate on the foreign investor’s international portfolio strategies. We also give the equation of asset price denoted by the local currency with the Ito formula. Under the framework of maximizing the expected utility of investors’terminal wealth with the power utility, the optimal portfolio strategy is derived by using the above derivation method. The effect of exchange rate on optimal international portfolio strategy is analyzed by the numerical simulation.Finally, based on the random change of investment opportunity set, we assume that the state variables and asset price dynamics obey quadratic Markovian diffusion process, and we discuss the effect of random inflation on the optimal consumption and portfolio strategies. In terms of the stochastic dynamic programming principle and the Ito formula, we get the corresponding HJB equation and the optimal portfolio strategy. Under the constant relative risk aversion (CRRA) utility, we get the explicit solution of the value function and corresponding optimal strategy for the complete and incomplete financial markets. The effect of the inflation and risk aversion on an investor’s optimal portfolio strategy is analyzed by numerical simulation.