| In this thesis, we investigate a stock market and study various portfolio optimization problems in this market. To extend the Black-Sholes model, we consider a scheme-switching market model, in which both the drift and the volatility are driven by a discrete time, finite states Markov chain. We assume the Markov chain is independent of the underlying Brownian motions and only the price processes are observable. Using the Hidden Markov model filtering theory, all the parameters of the model can be estimated.;In this regime-switching model, we solve the problem of maximizing the utility from the terminal wealth. Approximate solutions are constructed for CRRA utility functions by applying dynamic programming in discrete time. In the case of random endowments involved, the problem of optimizing utility from terminal wealth with random endowments meets same difficulty as pricing non-traded assets by utility functions, that is, generally there is no explicit solution for a power law utility function. In this thesis, we derive an approximation of the optimal solution to the problem. Finally, using the same techniques, we solve the Markowitz' mean-variance portfolio selection problem and the Markowitz's mean-variance hedging problem in the regime-switching model. |
