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.
Highlights
The mean-field-type stochastic differential equation (MF-SDE) is known as a class of stochastic differential equations (SDEs), in which the expected values of state and control variables are included.the theory of MF-SDEs can be traced back to studying of McKean–Vlasov SDEs for analyzing interacting particle systems at the macroscopic level [1,2]
We discuss the different solvability conditions of the coupled integro-Riccati differential equations (CIRDEs), which lead to characterization of the Nash equilibrium for the I-LQ-MF-SZSDG-JD
We have considered the indefinite, linear-quadratic, mean-field-type stochastic zero-sum differential game for jump-diffusion models (I-LQ-MF-SZSDG-JD), in which the cost parameters do not need to be definite matrices
Summary
The mean-field-type stochastic differential equation (MF-SDE) is known as a class of stochastic differential equations (SDEs), in which the expected values of state and control variables are included. We note that the references mentioned above considered only the mean-field-type stochastic control problem and nonzero-sum game for jump-diffusion models. It is necessary to extend the earlier results mentioned above to the framework of stochastic zero-sum games in order to analyze a more complex interaction between the players through the dynamics and the objective functional in the mean-field sense using the concept of strategies. This is addressed in our paper; the precise problem formulation and a detailed comparison with the existing literature are given below. We consider the indefinite, linear-quadratic, mean-field-type stochastic zero-sum differential game for jump-diffusion models (I-LQ-MF-SZSDG-JD).
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