Adaptive control of a continuous-time portfolio and consumption model

In this paper, an adaptive control problem is formulated and solved using Merton's stochastic differential equation for the wealth in a portfolio selection and consumption model. Since the asset prices are assumed to satisfy a log normal distribution, it suffices to consider two assets. It is assumed that the drift parameter for the price of the risky asset is unknown. A recursive family of estimators for this unknown parameter is defined and is shown to converge almost surely to the true value of the parameter. The controls in the equation for the wealth are obtained from the optimal controls where the estimates of the unknown parameter are substituted for the unknown parameter.This research was partially supported by NSF Grant No. ECS-84-03286-A01.The authors wish to thank P. Varaiya for some useful comments on this paper.