A Contribution on Stochastic Optimal Control to Quantitative Finance

This work considers a consumption and investment decision problem for an~individual who has available a~riskless asset paying fixed interest rate and a~risky asset driven by Brownian motion price fluctuations. The individual is supposed to observe his or her current wealth only, when making transactions, that transactions incur costs, and that decisions to transact can be made at any time based on all current information. The transactions costs under consideration could be a fixed, linear or a nonlinear function of the amount transacted. In addition, the investor is charged a fixed fraction of total wealth as management fee. The investor's objective is to maximize the expected utility of consumption over a given horizon. On the basis of this model, the existence of an optimal solution is given. Optimal consumption and investment strategies are obtained in closed form for each type of transaction costs function. In addition, the optimal interval of time between transactions is also derived. Results show that, for each transaction cost, transaction interval satisfies a nonlinear equation, which depends on total wealth at the beginning of that intervals. If, at each tran-saction, there is no costs involved other than that of management fee which is a fixed fraction of current portfolio value, then the optimal interval of time between transactions is fixed, independent of time and current wealth.