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

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  • Arunselvan Ramaswamy

    (Department of Computer Science and Automation, Indian Institute of Science, Bangalore 560012, India)

  • Shalabh Bhatnagar

    (Department of Computer Science and Automation, Indian Institute of Science, Bangalore 560012, India)

Abstract

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.

Suggested Citation

  • Arunselvan Ramaswamy & Shalabh Bhatnagar, 2017. "A Generalization of the Borkar-Meyn Theorem for Stochastic Recursive Inclusions," Mathematics of Operations Research, INFORMS, vol. 42(3), pages 648-661, August.
    • Handle: RePEc:inm:ormoor:v:42:y:2017:i:3:p:648-661
    DOI: 10.1287/moor.2016.0821
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    References listed on IDEAS

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    1. Michel Benaim & Josef Hofbauer & Sylvain Sorin, 2005. "Stochastic Approximations and Differential Inclusions II: Applications," Levine's Bibliography 784828000000000098, UCLA Department of Economics.
    2. Michel Benaïm & Josef Hofbauer & Sylvain Sorin, 2005. "Stochastic Approximations and Differential Inclusions; Part II: Applications," Working Papers hal-00242974, HAL.
    3. Michel Benaïm & Josef Hofbauer & Sylvain Sorin, 2012. "Perturbations of Set-Valued Dynamical Systems, with Applications to Game Theory," Dynamic Games and Applications, Springer, vol. 2(2), pages 195-205, June.
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