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 ast , a suitably normalized sequence of the estimation error,t(¯x_tr–) is equivalent to a scaled sequence of the random noise process, namely, (1/t)₀^tr _sds. 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.Supported in part by the National Science Foundation, and in part by Wayne State University.Supported in part by Wayne State University through a research assistantship.