Revision with unchanged content. This book deals with portfolio optimization under partial information. We consider an investor who can invest in a money market and a stock market. For our intended application a good model for the drift is of uttermost importance. We assume that the investor can only observe the stock prices but not the drift process; hence he has only partial information. The investor's objective is to maximize the expected utility of consumption and/or terminal wealth under partial information. We derive an explicit representation of the optimal consumption and trading strategies using Malliavin calculus. We show that the results apply to both classical models for the drift process, linear Gaussian dynamics and a continuous time Markov chain with finitely many states. We discuss several problems which might arise and how to overcome them, e.g. by dynamic risk constraints or non-constant volatility models. The results are applied to historical prices and yield promising results. The book is aimed at graduate students and researches interested in portfolio optimization under partial information.