A Closed-Form Solution For Optimal Dynamic Portfolio Problems

We address the optimal dynamic portfolio problems and focus on a Bayesian investor, who predicts the future with the past information and is given an initial wealth which is used for paying stocks. The stock’s drift is assumed as a linear mean reverting diffusion, which can’t be observed directly. Adopting the martingale approach and Cameron-martin theorem, the maximization of expected utility can be converted to a system of differential equations. For the case of a given utility function, a closed-form solution of the terminal wealth is provided.The organization and basic content of the paper is the following. Section1introduces background and the content, as well as some important tools used in the article. In section2, we describe a model of the maximization of expected utility about the Bayesian investor and make use of the martingale approach to obtain a converted form of the model. In section3, we obtain the expression of Zt by using the general Riccati equation, where Z, is a key parameter in the model. After Laplace transform of log (Zr/C) is carried out. In section4, under the given utility function, we obtain the closed-form solution of the terminal wealth. In section5, we conclude the paper and discuss possible extensions.