Continuous Procedures of Stochastic Approximation

It was shown in [R. Z. Khas'minskii, Stability of Systems of Differential Equations with Random Disturbances of Their Parameters, Nauka, Moscow, 1969] that the continuous variant of the Robbins–Monro stochastic approximation procedure with “white noise” inputs can be considered from the point of view of stability of the solution of a system of ordinary differential equations with damped random inputs. In the present work these as well as other procedures of stochastic approximation are studied in the case of continuous time. To this end the stability theorem in the case of damped random inputs proved in [Khas'minskii] is somewhat extended. Similarly as in [Khas'minskii], the conditions are given for convergence of the procedures under consideration in terms of the existence of the corresponding stochastic Lyapunov functions. The conditions for convergence of the stochastic-approximation procedures were proved in [J. R. Blum, Ann. Math. Stat., 1954, vol. 25, no. 2; T. A. Sakrison,Ann. Math. Stat., 1964, vol. 35, no. 2; T. Morozan, Stabilitiea Sistemelor Parametri Aleatori, Editura Acad. Rep. Socialiste România, Bucharest, 1969; E. M. Braverman and L. I. Rozonoer,Avt. Telemekh., 1969, no. 3, pp. 57–77, 87–03] in a similar manner.