Portfolio Management Under Transaction Costs: Models and Solution Schemes for Transaction Cost in Portfolio Choice Theory

The purpose of this project is to consider the portfolio optimization problem where the investor's objective is to maximize the long-term growth rate and quite possibly many other criteria like the expected utility of consumption. Accordingly , different transaction cost structures would be considered. This problem belongs to the class of stochastic control problems with singular controls( and maybe multiple controls in some situations), which are usually solved by computing solutions to the related partial differential equations called the free-boundary Hamilton-Jacobi-Bellman equations. The dimensionality of the HJB equals the number of stocks in the portfolio. The runtime of existing solution methods grow super-exponentially with dimension,making them unsuitable to compute optimal solutions to portfolio problems even with four stocks. In this project I will consider different transaction cost models and also consider the associated solution methods to solve them. Numerical schemes for solutions would only be presented for one or two dimensional case leaving higher dimensions an exercise for future discussion.