Non linear filtering and optimal investment under partial information for stochastic volatility models

This paper studies the question of filtering and maximizing terminal wealth from expected utility in a stochastic volatility models. The special feature is that the only information available to the investor is the one generated by the asset prices and, in particular, the return processes cannot be observed directly and assumed to be modelled by a stochastic differential equation. Using stochastic non-linear filtering and change of measure techniques, the partial observation context can be transformed into a full information context such that coefficients depend only on past history of observed prices (filters processes). The main difficulty is that these filters are valued in infinite-dimensional space: it satisfy a stochastic partial differential equations named "Kushner-Stratonovich equations". 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 equation or Hamiltonian system are now in infinite dimension.