Optimal Portfolio-Consumption with Habit Formation under Partial Observations

Abstract

The aim of my thesis consists of characterizing explicitly the optimal consumption and investment strategy for an investor, when her habit level process is incorporated in the utility formulation. For a continuous-time market model, I maximize the expected utility from terminal wealth and/or consumption. For this optimization problem, the thesis presents three novel contributions. Using the Kalman-Bucy filter theorem, I transform the optimization problem under the partial information into an equivalent optimization problem within a full information context. Using the stochastic control techniques, this latter problem is reduced to solve an associated Hamilton-Jacobi-Bellman equation (HJB hereafter). For the exponential utility, the solution to the HJB is explicitly described, while the optimal policies/controls as well as the optimal wealth process are described by a stochastic differential equation. Furthermore, I discuss qualitative analysis on the optimal policies for the exponential utility. These achievements constitute my first contribution in this thesis. The second contribution lies in considering a stochastic volatility model and addressing the same optimization problem using again the techniques of stochastic control. The third contribution of my thesis resides in combining the filtering techniques with the martingale approach to solve the optimization problem when the investor is endowed with the logarithm, power or exponential utility.