We consider a financial market model consisting of one bank account, N risky stocks and d economic factors. We assume that the mean returns of the risky stocks are affected by the economic factors which evolve as a continuous-time Markov chain with with finite states.
An investor who observes only asset prices tries to maximize the discounted expected CRRA utility of the consumption of his wealth (as well as that of terminal wealth in the case of finite time horizon).
Hence this is an optimal consumption/ investment problem under partial information. We treat the problem both on finite and infinite time horizons.