Before revision:  We also show that we need to introduce an a priori models for the trend and the stochastic volatility in order to evaluate the filters processes. The dynamic programming or maximum principle are still applicable and the associated Bellman equationor Hamiltonian system are now in infinite dimension .
After revision:  Using the martingale duality approach in this partially observed incomplete model, we can characterize the value function and the optimal portfolio. The main result here is that the dual value function associated to the martingale approach can be expressed, via the dynamic programmingapproach, in terms of the solution to a semilinear partial differential equation. We illustrate our results with some examples of stochastic volatility models popular in the financial literature .