Optimal Portfolio-consumption with Habit Formation and Partial Observations: the Fully Explicit Solutions Approach

We consider a model of optimal investment and consumption with both habit-formation and partial observations in incomplete Itô processes markets. The individual investor develops addictive consumption habits gradually while he can only observe the market stock prices but not the instantaneous rates of return, which follow Ornstein-Uhlenbeck processes. Applying the Kalman-Bucy filtering theorem and Dynamic Programming arguments, we solve the associated HJB equation fully explicitly for this path dependent stochastic control problem in the case of power utility preferences. We will provide the optimal investment and consumption policies in explicit feedback forms using rigorous verification arguments.