Expected utility maximization under transaction costs.

The expected utility maximization under transaction cost is studied. We consider a continuous-time model with one riskless asset, called the bond (or bank account) and one risky asset, called the stock. We consider an investor whom we suppose can decide to transfer money from bank account to the stock and conversely and these transfers involve transaction costs (both fixed and proportional). These costs are drawn from the bank account and the investor's net wealth is defined as the holdings in the bank account after closing the short position. The objective is to determine the amounts to invest in the risky and risk-free assets that maximize the expected value of the discounted utility of terminal wealth for an investor who is subjected to transaction costs. We formulated this problem as a non-singular stochastic optimal control problem to clearly obtain optimal trading strategies. Using Bellman's Dynamic programming Hamilton-Jacobi-Bellman equation that characterizes the value function is derived. From the value function we obtain a number of equalities and inequalities, and its derivatives as optimal conditions which we characterize as optimal strategies, transaction and no-transaction regions. We discretized our linearized HJB or PDE equation by using upwind finite difference methods which we solve using MattLab 7, and give optimal analysis.