Optimal portfolio allocation by the martingale method in an incomplete and partially observable market

Abstract: In this thesis, we consider an agent who wants to maximize his expected utility of his terminal wealth with respect to the power utility by the martingale method. The assets that the agent can allocate his capital to are assumed to follow a stochastic differential equation and exhibits stochastic volatility. The stochastic volatility assumption will make the market incomplete and therefore, the martingale method will not have a unique solution. We resolve this issue by including fictitious assets that complete the market and solve the allocation problem in the completed market. From the optimal allocation in the completed market, we will adjust the drift parameter for the fictitious assets so that our allocation don't include the fictitious assets in the portfolio strategy. We consider also the case when the assets also has stochastic drift and the agent can only observe the price process, which makes the information in the market for the agent partially observable. Explicit results are presented for the full and partially observable case and a feedback solution is obtained in the full observable case when the asset and volatility are assumed to follow the Heston model.