PDE approach to utility maximization for mar- ket models with hidden Markov factors

We consider the problem of maximizing expected utility from terminal wealth for a power utility of the risk-averse type assuming that the dynamics of the risky assets are affected by hidden “economic factors” that evolve as a finite-state Markov process. For this partially observable stochastic control problem we determine a corresponding complete observation problem that turns out to be of the risk sensitive type and for which the Dynamic programming approach leads to a nonlinear PDE that, via a suitable transformation, can be made linear. By means of a probabilistic representation we obtain a unique viscosity solution to the latter PDE that induces a unique viscosity solution to the former. This probabilistic representation allows to obtain, on one hand regularity results, on the other a computational approach based on Monte Carlo simulation. Mathematics Subject Classification (2000). Primary 93E20; Secondary 91B28, 49L20, 49L25.