Incorporating Estimation Error into Optimal Portfolio �llocation

In this report, we consider the problem concerned with incorporating estimation error into optimal consumption and portfolio selection in continuous time. The original optimal consumption and asset allocation problem in continuous time was solved by Merton in a series of papers [4, 5] and became widely known as “Merton’s Problem”. Merton made the assumption that the asset price processes {Si} N i=1 are given by Geometric Brownian Motion (GBM), where the parameter values are known. He was able to prove that the investment opportunity set can be generated by two portfolios or mutual funds of assets, which themselves obey Geometric Brownian Motion. This result is sometimes known as a two-fund separation theorem, and does not depend on the market being in equilibrium. Merton utilized the separation theorem in his development of the Inter-temporal Capital Asset Pricing Model in [6], but the mutual fund theorem is only dependent on the assumed properties of the asset price processes. Subsequent work on this problem has sought to generalize Merton’s work in numerous ways. For instance, some authors have considered more general asset price processes than GBM, e.g. Ito processes with deterministic (and even stochastic) time-dependent drift and diffusion parameters, and other general diffusion and Markov processes, or general semi-martingales. In the latter case, the additional assumption that the market is complete (or more generally, effectively complete) is required, and the method of solution uses the so-called Cox-Huang-Pliska method [9], which involves the use of the Martingale Represen-tation Theorem. Another generalization of the Merton problem involves the inclusion o