Portfolio Optimization Associated with a Weak Information

In this paper we consider an investor who trades in a complete financial market so as to maximize his expected utility of wealth at a prespecified time. We assume that he is in the following position : His portfolio decisions are based on a public information flow but he possesses extra information about the law of some functional of the future prices of a stock. Our basic question is then: How should he trade on the financial market to optimally exploit his extra information? We show that the optimal trading strategies corresponding to a such investor (called weakly informed) are derived from processes which are solutions of what we called in a previous paper a conditioned stochastic differential equation. Such processes can be realized as all the possible filters of the price process seen under the martingale probability measure but looked in a enlarged filtration. This filtering makes the link with the works of several authors who considered financial markets in which one insider possesses from the beginning extra information about the outcome ω by ω of some variable (this is a higher information level which we call strong).