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Pareto Optimal Cooperative Control of Mean-Field Backward Stochastic Differential System in Finite Horizon

Authors:
Saranya Ganesan at Gandhigram Rural Institute
Saranya Ganesan
R. Deepa
R. Deepa
R. Deepa
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Muthukumar Palanisamy at Gandhigram Rural Institute
Muthukumar Palanisamy
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Abstract

This research article aims to investigate a new type of Pareto cooperative differential game governed by backward stochastic differential equations of mean-field type. By the characterization of Pareto optimal solutions, the proposed Pareto game problem is converted into a set of optimal control problems with single weighted objective function which is constrained by mean-field backward stochastic differential equations. First, we derive the necessary conditions for Pareto optimality of the proposed system in finite time horizon. Next, the sufficient conditions are established with the conclusion that the necessary conditions are sufficient under some convexity assumptions. Finally, for the better understanding of theoretical results, we discuss the linear quadratic optimal control problem and a mathematical transportation problem.
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Dynamic Games and Applications
https://doi.org/10.1007/s13235-024-00566-7
Pareto Optimal Cooperative Control of Mean-Field Backward
Stochastic Differential System in Finite Horizon
G. Saranya1·R. Deepa2·P. Muthukumar1
Accepted: 10 May 2024
© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2024
Abstract
This research article aims to investigate a new type of Pareto cooperative differential game
governed by backward stochastic differential equations of mean-field type. By the character-
ization of Pareto optimal solutions, the proposed Pareto game problem is converted into a set
of optimal control problems with single weighted objective function which is constrained by
mean-field backward stochastic differential equations. First, we derive the necessary condi-
tions for Pareto optimality of the proposed system in finite time horizon. Next, the sufficient
conditions are established with the conclusion that the necessary conditions are sufficient
under some convexity assumptions. Finally, for the better understanding of theoretical results,
we discuss the linear quadratic optimal control problem and a mathematical transportation
problem.
Keywords Cooperative differential game ·Mean-field backward stochastic differential
equations ·Linear quadratic optimal control ·Pareto optimal control ·Weighted sum
optimization
Mathematics Subject Classification 91A15 ·49K15 ·60H10 ·58E17
1 Introduction and Model Formulation
Game theory [1,22] is a branch of mathematics and economics that analyzes strategic
interactions among rational decision-makers. It provides a framework for understanding the
outcomes of situations where the success of an individual’s decision depends on the choices
made by others. Game theory is widely applied in various fields, including economics, politi-
BP. Muthukumar
pmuthukumargri@gmail.com
G. Saranya
saranyaganesangri@gmail.com
R. Deepa
deepa.maths1729@gmail.com
1Department of Mathematics, The Gandhigram Rural Institute (Deemed to be University),
Gandhigram 624 302, Tamil Nadu, India
2Department of Mathematics, Panimalar Engineering College, Chennai 600123, Tamil Nadu, India
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
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