ON A CONTINUOUS-TIME STOCHASTIC-APPROXIMATION PROBLEM

This paper is concerned with a continuous time stochastic approximation/optimization problem. The algorithm is given by a pair of differential-integral equations. Our main effort is to derive the asymptotic properties of the algorithm. It is shown that as t --> infinity, a suitably normalized sequence of the estimation error, tau square-root t (x(tr)BAR - theta) is equivalent to a scaled sequence of the random noise process, namely, (1/square-root t) integral-ttau/0 xi(s) ds. Consequently, the asymptotic normality is obtained via a functional invariance theorem, and the asymptotic covariance matrix is shown to be the optimal one. As a result, the algorithm is asymptotically efficient.