Optimal Investment And Consumption Model And Its Numerical Method

The optimal investment and consumption problem has a long history, each person or each economic entity almost every day faces a decision-making problem for investment and consumption. Facing the many uncertainty factors in life, and especially in recent years major financial emergencies and many questions of financial reform, people find that the classic Merton model has not completely adapted to the change of modern financial markets. In this paper, we consider there are many kinds of factors which impact the investment and consumption activities, and then use stochastic optimal control theory and duality theory to study several portfolio and consumption problems under the framework of continuous-time. we get the optimal strategies, and reveal the real meaning of its applications. This paper's main work and results are mainly in chapter 3, 4, 5. In chapter 3, assuming the investor's death event is stochastic, we study one kind of optimal investment and consumption problem with loan rate higher than deposit rate is studied for an investor who has stochastic income. First of all, the stochastic optimal control model of the optimal investment and consumption problem is established. The feedback forms of the optimal consumption and investment strategies are obtained by HJB equation which is derived from dynamic programming principle and duality theory; Second, for a special HARA case, we discuss specific forms of the optimal strategies; Finally, we also give the comparative analysis with the classic Merton problem. In chapter 4, we study the optimal investment and consumption problems with abnormal fluctuations in the financial market and two consumption goods, namely a perishable good and a durable good. First of all, the stochastic optimal control model of the optimal investment and consumption problem is established, and then by using dynamic programming principle, the optimal investment and consumption strategies are obtained explicitly for the power utility function case. Finally, we analyze the optimal strategies. The numerical method for the optimal consumption and investment strategy with transaction costs is discussed in chapter5. First, the problem's mathematical model and HJB equation are established. Generally, the exact solution of the HJB equation is hard to obtain. According to H.Liu's work (2004) and the available conclusions, a numerical method for computing the problem is established, and some numerical examples are given to illustrate the feasibility of the new methods. the results are match up to the original result.