A criterion for convergence of continuous stochastic approximation procedures

A. P. Korostelev

Teoriâ veroâtnostej i ee primeneniâ, Tome 22 (1977) no. 3, pp. 595-602

Résumé

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 $dt=\alpha(s)\,ds$; $\gamma(t)=\beta(t)/\sqrt{\alpha(t)}$.