Optimizing Consumption and Investment: The Case of Partial Information

In Section 2 we present a stock market model where prices satisfy a stochastic differential equation with a stochastic drift process which is independent of the driving Brownian motion. The investor’s objective is to maximize the expected utility of consumption and terminal wealth under partial information, meaning that investment decisions are based on the knowledge of the stock prices only, cf. [2, 3]. Consumption and investment processes as well as the optimization problem are introduced in Section 3. Optimal consumption and optimal terminal wealth can be expressed in terms of the filter for the Radon Nikodym density of the risk neutral probability under which the price processes become martingales. The solution to this classical optimization problem is provided in Section 4 where consumption and investment strategies are computed based on Malliavin derivatives of the corresponding density process. The results apply to both classical models for the drift process, a linear Gaussian model (GD) and a continuous time Markov chain (HMM). In Section 5 and 6 we look at these two cases, and show that they satisfy all the conditions for the optimal strategies, see also [3, 5]. For proofs and further details we refer to [4] if not mentioned differently. In addition to [4] we compare in Section 7 the HMM with the GD model when applied to historical prices.