Application Of Martingale Analysis To Optimal Investment And Consumption Strategies

Martingale theory is a advancing theory of random process, the Martingale theory method is a strong research tool, and has became other new mathematics branch by combining other mathematics branches. In this paper, we apply the method of martingale analysis to optimal investment and consumption strategies, and some related consequences are acquired.From the microstructure of market, we conclude the concept and proprieties of martingale and discuss the optimal investment and consumption strategies of two financial assets in the financial market under the Black-Scholes frame. We use the classic methods-random dynamic programming and martingale analysis to discuss the optimal investment and consumption strategies of two financial assets in the financial market. Using the method of random dynamic programming, we get the implicit solutions; and using the method of martingale analysis, we get the explicit solutions. By comparing the two methods, we find that the method of martingale analysis has evident superiority in resolving the problem of investment and consumption to make the utility of consumption and terminal wealth maximize. Besides, using the equivalent martingale measure principle and the theory of random process, we research the optimal investment and consumption under the recursive utility function. The price of financial assets is influenced by some random factors such as the payoff index, inflation rate. So we generalize the market coefficient from the time determinacy continuous function to the random function in random systems, and built mathematical model, that is the new optimal investment and assumption strategies model of financial assets price is influenced by random factors. Then we use the theory of martingale analysis and random process to find the optimal investment and consumption strategies of the new model. Taking the utility function as the decisive basis, we give the concrete investment and consumption strategies to the HARA utility function investors. In conclusion, these results improve the investment and consumption model and have some practical application.