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Short Communications

A criterion for convergence of continuous stochastic approximation procedures

A. P. Korostelev

Moscow

Abstract: For the a.s. convergence of the stochastic approximation procedure
$$ dX_s=\alpha(s)[\triangledown f(X_s)+\varphi(s,X_s)]\,ds+\beta(s)\sigma(s,X_s)\,dW_s $$
to a maximum point of $f$, the following condition is proved to be necessary and sufficient: for any $\lambda>0$
$$ \int_0^{\infty}\exp(-\lambda\gamma^{-2}(t))\,dt<\infty $$
where $dt=\alpha(s)\,ds$; $\gamma(t)=\beta(t)/\sqrt{\alpha(t)}$.

Received: 16.12.1975

 English version: Theory of Probability and its Applications, 1978, 22:3, 584–591

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