Explicit Characterization of Feedback Nash Equilibria for Indefinite, Linear-Quadratic, Mean-Field-Type Stochastic Zero-Sum Differential Games with Jump-Diffusion Models

Jun Moon and Wonhee Kim
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Jun Moon: Department of Electrical Engineering, Hanyang University, Seoul 04763, Korea
Wonhee Kim: School of Energy Systems Engineering, Chung-Ang University, Seoul 06974, Korea

Mathematics, 2020, vol. 8, issue 10, 1-23

Abstract: We consider the indefinite, linear-quadratic, mean-field-type stochastic zero-sum differential game for jump-diffusion models (I-LQ-MF-SZSDG-JD). Specifically, there are two players in the I-LQ-MF-SZSDG-JD, where Player 1 minimizes the objective functional, while Player 2 maximizes the same objective functional. In the I-LQ-MF-SZSDG-JD, the jump-diffusion-type state dynamics controlled by the two players and the objective functional include the mean-field variables, i.e., the expected values of state and control variables, and the parameters of the objective functional do not need to be (positive) definite matrices. These general settings of the I-LQ-MF-SZSDG-JD make the problem challenging, compared with the existing literature. By considering the interaction between two players and using the completion of the squares approach, we obtain the explicit feedback Nash equilibrium, which is linear in state and its expected value, and expressed as the coupled integro-Riccati differential equations (CIRDEs). Note that the interaction between the players is analyzed via a class of nonanticipative strategies and the “ordered interchangeability” property of multiple Nash equilibria in zero-sum games. We obtain explicit conditions to obtain the Nash equilibrium in terms of the CIRDEs. We also discuss the different solvability conditions of the CIRDEs, which lead to characterization of the Nash equilibrium for the I-LQ-MF-SZSDG-JD. Finally, our results are applied to the mean-field-type stochastic mean-variance differential game, for which the explicit Nash equilibrium is obtained and the simulation results are provided.

Keywords: mean-field stochastic differential equations with jump diffusions; stochastic zero-sum differential games; Nash equilibrium; coupled integro-Riccati differential equations (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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