ON STOCHASTIC APPROXIMATION PROCEDURES IN CONTINUOUS TIME.

Abstract

Under appropriate conditions on a par tially known function : [0, co) x e RN we establish the existence of a wide class of stochastic dynamical systems with a closed loop relation so that the input X (. ) converges to the unknown root cc of the equation f (t == 0. The output Y() of the dynamical system is a nonlinear transformation of the input X (. ) involving f, and which is corrupted by additive noise terms modelled by the differentials. By martingale arguments we demonstrate convergence in mean and with probability one. The procedure may be considered as a continuous-time analog of the Robbins-Monroc scheme for discrete-time processes.