Filtering problem for general modeling of the drift and application to portfolio optimization problems

We study the filtering problem and the maximization problem of expected utility from terminal wealth in a partial information context. The special features is that the only information available to the investor is the vector of sock prices. The mean rate of return processes are not directly observed and supposed to be driven by a process $\mu_{t}$ modeled by a stochastic differential equations. The main result in this paper is to show under which assumptions on the coefficients of the model, we can estimate the unobserved market price of risks. Using the innovation approach, we show that under globally Lipschitz conditions on the coefficients of $\mu_{t}$, the filters estimate of the risks satisfy a measure-valued Kushner-Stratonovich equations. On the other hand, using the pathwise density approach, we show that under a nondegenerate assumption and some regularity assumptions on the coefficients of $\mu_{t}$, the density of the conditional distribution of $\mu_{t}$ given the observation data, can be expressed in terms of the solution to a linear parabolic partial differential equation parameterized by the observation path. Also, we can obtain an explicit formulae for the optimal wealth, the optimal portfolio and the value function for the cases of logarithmic and power utility function.