The Polyak weighted averaging procedure for Robbins-Monro type SDE
Given an adapted continuous increasing process \(K\) and a continuous local martingale \(m\), let \(Z\) be a strong solution of the Robbins-Monro type stochastic differential equation \(dZ_t= H_t(Z_t)dK_t+ \ell_t(Z_t) dm_t\), \(t\geq 0\), introduced by the authors and \textit{T. Sharia} [Stochastics Stochastics Rep. 61, No. 1/2, 67-87 (1997; Zbl 0885.60048)]. The authors consider the weighted average process NEWLINE\[NEWLINE\overline Z_t= \exp\Biggl\{-\int^t_0 g_s dK_s\Biggr\} \int^t_0 Z_s d\exp\Biggl\{\int^s_0 g_r dK_r\Biggr\},\qquad t\geq 0,NEWLINE\]NEWLINE where \(g\) is a predictable process with \(\int^{+\infty}_0 g_s dK_s= +\infty\). Using the asymptotic properties of the process \(Z= (Z_t)_{t\geq 0}\) as \(t\to+\infty\) investigated in former works by the authors and T. Sharia (loc. cit.) and the authors [in: Probability theory and mathematical statistics, 415-428 (1999)]. They study conditions under which one has asymptotic normality of \(\overline Z= (\overline Z_t)_{t\geq 0}\) as \(t\to+\infty\). In special cases the authors obtain results concerning averaging procedures for standard Robbins-Monro type procedures and those with slowly varying gains.
scientific article; zbMATH DE number 1713780