On Portfolio Optimization In Finite Horizons

Portfolio optimization is a long studied problem in mathematical finance which seeks to identify the optimal trading strategy of a given investor. First studied in continuous time by Merton [11, 12] under a log-normal stock price, explicit solu- tions were given for various utility functions. The problem formulation has since been generalized to include more empirically valid stock price models, as well as more general utility functions. Under such general assumptions, the optimization problem becomes very difficult to solve, owing to the fully nonlinear HJB equation that results in these settings, for which the well-posedness has not been established. It thus often becomes necessary to obtain approximations to the optimal trading strategy. In this thesis, we present a strategy to approximate the optimal portfolio by first constructing classical sub- and super-solutions to the HJB equation from which the value function can be approximated. We then substitute our approxima- tion of the value function into the formula for the optimal portfolio obtained from the HJB equation, yielding a closed-form formula for a trading strategy which is close-to-optimal on small time horizons. Extensions of these results to larger finite horizons, as well as to other families of utility, are discussed in later chapters