| This dissertation consists of four chapters. Several topics from Financial Economics are studied in Chapter 1 to Chapter 3. In all cases, we formulate the problems as stochastic control problems. Dynamic programming principles are used to get the dynamic programming equations (DPEs). In Chapter 4, we investigate the numerical results for some models we considered.; In Chapter 1, some generalizations of the classical Merton's problem of investment portfolio optimization are studied. We consider an investor who can invest on two assets, one riskless asset, and one risky asset. The goal is to maximize the expectation of the total discounted consumption-based utility by choosing optimal investment-consumption strategy. Here we consider the HARA type utility, including the log utility. The interest rate for the riskless asset is randomly fluctuated, and it may be correlated with the price change of the risky asset. The method of subsolution-supersolution is developed to get a suitable classical solution of the DPE. We also investigate the application of viscosity solution methods to this problem in certain cases.; In Chapter 2, we consider an investment optimization problem on a finite time horizon. The goal is to choose an optimal investment strategy to maximize the final utility which is based on the final wealth. Under certain conditions, the value functions are obtained, and the optimal investment policies are given.; In Chapter 3, a national GDP model is studied. We investigate a country which can borrow foreign debt to invest or consume. One important requirement for the country is to keep the debt ratio (the ratio of debt to wealth) under a security level. Penalty is imposed if the level is reached. We formulate this as a stochastic control problem and investigate some properties of the value function.; In Chapter 4, the numerical results for several models we considered in Chapter 1 and Chapter 3 are given. The Markov chain approximation methods are used there. The numerical results turn out to be very consistent with our analytic results. |
