A Generalization of the Borkar-Meyn Theorem for Stochastic Recursive Inclusions

TL;DR

This paper extends the Borkar-Meyn stability theorem to stochastic recursive inclusions involving differential inclusions, providing new conditions for stability and convergence, and relaxing original assumptions.

Contribution

It generalizes the Borkar-Meyn theorem to differential inclusions and introduces new sufficient conditions for stability and convergence of stochastic recursive inclusions.

Findings

Extended stability theorem to differential inclusions.

Provided new sufficient conditions for convergence.

Relaxed assumptions of the original theorem.

Abstract

In this paper the stability theorem of Borkar and Meyn is extended to include the case when the mean field is a differential inclusion. Two different sets of sufficient conditions are presented that guarantee the stability and convergence of stochastic recursive inclusions. Our work builds on the works of Benaim, Hofbauer and Sorin as well as Borkar and Meyn. As a corollary to one of the main theorems, a natural generalization of the Borkar and Meyn Theorem follows. In addition, the original theorem of Borkar and Meyn is shown to hold under slightly relaxed assumptions. Finally, as an application to one of the main theorems we discuss a solution to the approximate drift problem.