Mean-Variance Portfolio Selection with a Fixed Flow of Investment in a Continuous-Time Framework

Abstract

We consider a mean-variance portfolio selection problem for a fixed flow of investment in a continuous time framework. We consider a market  structure that is characterized by a cash account, an indexed bond and a stock. We obtain the expected optimal terminal wealth for the investor. We also obtain a closed-form expressions for the value functions, the optimal investment and mean-variance strategies using the stochastic maximum principle. We assume that the indexed bond and stock are governed by geometric Brownian motions with constant drifts and volatilities. The aim of the investor is to maximize the expected terminal wealth while minimizing the variance. We find that the higher the value of the control parameter used in minimizing the variance, the lower the variance. Furthermore, we find that if the parameter used to minimize the variance tends to zero, both the expected wealth and the variance tend to infinity. This means that if a risky asset becomes more risky, the wealth of the investor is expected to be “unmeasureable” otherwise the portfolio should remain only in the cash account.

Keywords: mean-variance, portfolio selection, investment, stochastic  maximum principle.