Optimal control and zero-sum games for markov chains of mean-field type

Salah Eddine Choutri; Boualem Djehiche; Tembine Hamidou

doi:10.3934/mcrf.2019026

Optimal control and zero-sum games for markov chains of mean-field type

Salah Eddine Choutri, Boualem Djehiche, Tembine Hamidou

Research output: Contribution to journal › Article › peer-review

Abstract

We establish existence of Markov chains of mean-field type with unbounded jump intensities by means of a fixed point argument using the total variation distance. We further show existence of nearly-optimal controls and, using a Markov chain backward SDE approach, we suggest conditions for existence of an optimal control and a saddle-point for respectively a control problem and a zero-sum differential game associated with payoff functionals of mean-field type, under dynamics driven by such Markov chains of mean-field type.

Original language	English (US)
Pages (from-to)	571-605
Number of pages	35
Journal	Mathematical Control and Related Fields
Volume	9
Issue number	3
DOIs	https://doi.org/10.3934/mcrf.2019026
State	Published - Sep 2019

Keywords

Backward SDEs
Mean-field
Nonlinear Markov chain
Optimal control
Saddle point
Stochastic maximum principle
Thinning
Zero-sum game

ASJC Scopus subject areas

Control and Optimization
Applied Mathematics

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10.3934/mcrf.2019026

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title = "Optimal control and zero-sum games for markov chains of mean-field type",

abstract = "We establish existence of Markov chains of mean-field type with unbounded jump intensities by means of a fixed point argument using the total variation distance. We further show existence of nearly-optimal controls and, using a Markov chain backward SDE approach, we suggest conditions for existence of an optimal control and a saddle-point for respectively a control problem and a zero-sum differential game associated with payoff functionals of mean-field type, under dynamics driven by such Markov chains of mean-field type.",

keywords = "Backward SDEs, Mean-field, Nonlinear Markov chain, Optimal control, Saddle point, Stochastic maximum principle, Thinning, Zero-sum game",

author = "Choutri, {Salah Eddine} and Boualem Djehiche and Tembine Hamidou",

note = "Funding Information: Acknowledgments. The authors gratefully acknowledge support from the Swedish Research Council grant (2016-04086) and from the U.S. Air Force Office of Scientific Research under grant number FA9550-17-1-0259. We would like to thank Said Hamad{\`e}ne and the anonymous referees for their insightful remarks that helped improve the content of the paper. Publisher Copyright: {\textcopyright} 2019, American Institute of Mathematical Sciences. All rights reserved.",

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doi = "10.3934/mcrf.2019026",

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AU - Djehiche, Boualem

AU - Hamidou, Tembine

N1 - Funding Information: Acknowledgments. The authors gratefully acknowledge support from the Swedish Research Council grant (2016-04086) and from the U.S. Air Force Office of Scientific Research under grant number FA9550-17-1-0259. We would like to thank Said Hamadène and the anonymous referees for their insightful remarks that helped improve the content of the paper. Publisher Copyright: © 2019, American Institute of Mathematical Sciences. All rights reserved.

PY - 2019/9

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N2 - We establish existence of Markov chains of mean-field type with unbounded jump intensities by means of a fixed point argument using the total variation distance. We further show existence of nearly-optimal controls and, using a Markov chain backward SDE approach, we suggest conditions for existence of an optimal control and a saddle-point for respectively a control problem and a zero-sum differential game associated with payoff functionals of mean-field type, under dynamics driven by such Markov chains of mean-field type.

AB - We establish existence of Markov chains of mean-field type with unbounded jump intensities by means of a fixed point argument using the total variation distance. We further show existence of nearly-optimal controls and, using a Markov chain backward SDE approach, we suggest conditions for existence of an optimal control and a saddle-point for respectively a control problem and a zero-sum differential game associated with payoff functionals of mean-field type, under dynamics driven by such Markov chains of mean-field type.

KW - Backward SDEs

KW - Mean-field

KW - Nonlinear Markov chain

KW - Optimal control

KW - Saddle point

KW - Stochastic maximum principle

KW - Thinning

KW - Zero-sum game

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Optimal control and zero-sum games for markov chains of mean-field type

Abstract

Keywords

ASJC Scopus subject areas

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