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

In this paper, the stability theorem of Borkar and Meyn is extended to include the case when the mean field is a set-valued map. 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 Benaïm, 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. As an application to one of the main theorems, we discuss a solution to the “approximate drift problem.” Finally, we analyze the stochastic gradient algorithm with “constant-error gradient estimators” as yet another application of our main result.