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Abstract

The main asymptotic property of the continuous stochastic approximation procedure, namely the convergence to the Ornstein-Uhlenbeck process, is considered. The regression function of the procedure depends on a uniformly ergonomic Markov process, which describes the external inuence in the form of switches. To obtain sucient conditions for the convergence of the procedure, we use ergodic properties of the generating Markov process generator, the existence of a stationary distribution of this process, and the existence of the potential for a generating Markov process generator. The stochastic approximation procedure, as a random process, is constructed in the form of a dierential equation in the xed state of a Markov process with a corresponding generator of the equation. Asymptoticity over time is achieved by using a small parameter that normalizes time. This made it possible to obtain a normalized stochastic approximation procedure and its dierential representation. The generator of the obtained dierential equation is used to construct a generator of a two-component Markov process, which consists of a procedure and a switching process. The singular representation of the last generator by a small parameter makes it possible to solve the singular perturbation problem and determine the form of the limited generator. Such form denes the representation of a limited process as the Ornstein-Uhlenbeck diusion process. Note that the convergence to the limited process is weak, which follows from the Koroliuk's theorem. The conditions for the existence of the Lyapunov function for the dynamic procedure, which is averaged by the stationary distribution of the Markov process, are important. Additional conditions on the Lyapunov function make it possible to establish the boundedness of the residual terms of the solution of a singular perturbation. The one-dimensional case of the procedure can be expanded to multidimensional with the corresponding complication of calculations of the components of the limited generator. The work summarizes the studies of Nevelson and Khasminsky in the case of a direct inuence of the Markov process on the regression function.