Portfolio management toward optimal consumption and terminal wealth
The problem we will consider is that of a single investor who wishes to optimize his consumption and/or terminal wealth over a specified finite time horizon. This investor is endowed with an initial sum which he may invest in n + 1 assets, one of which is risk-free. In a 1971 paper, Merton, who was interested in continuous time models for CAPM (Capital Asset Pricing Model), gave a closed-form solution via the Bellman equations when constant coefficients are assumed. Then in 1986 and 1987, Karatzas, Lehoczky, Sethi and Shreve were able to demonstrate for general utility functions explicit solutions could be given for both the consumption process and wealth processes.
We will show that for a class of utility functions from the HARA family, which is of significant interest to investors, that explicit information regarding the nature of the portfolio process can be derived. Moreover, we will show that such solutions can be reached by applying martingale methods.