Semimartingale Stochastic Approximation Procedures and Recursive Estimation

TL;DR

This paper introduces a semimartingale stochastic approximation framework that unifies stochastic approximation algorithms and recursive estimation for semimartingale models, providing convergence and asymptotic analysis.

Contribution

It develops a general semimartingale stochastic approximation method with convergence, rate, and asymptotic properties, including Polyak averaging.

Findings

Conditions for convergence are established.

Asymptotic expansion results are provided.

Polyak averaging improves estimation stability.

Abstract

The semimartingale stochastic approximation procedure, namely, the Robbins-Monro type SDE is introduced which naturally includes both generalized stochastic approximation algorithms with martingale noises and recursive parameter estimation procedures for statistical models associated with semimartingales. General results concerning the asymptotic behaviour of the solution are presented. In particular, the conditions ensuring the convergence, rate of convergence and asymptotic expansion are established. The results concerning the Polyak weighted averaging procedure are also presented.