This paper establishes Nash equilibrium for mean field games using stochastic differential equations, suggesting optimal strategies with discounting effects.
In this paper, we study the infinite-time mean field games with discounting, establishing an equilibrium where individual optimal strategies collectively regenerate the mean-field distribution. To solve this problem, we partition all agents into a representative player and the social equilibrium. When the optimal strategy of the representative player shares the same feedback form with the strategy of the social equilibrium, we say the system achieves a Nash equilibrium. We construct a Nash equilibrium using the stochastic maximum principle and infinite-time forward-backward stochastic differential equations(FBSDEs). By employing the elliptic master equations, a class of distribution-dependent elliptic PDEs , we provide a representation for the Nash equilibrium. We prove the Yamada-Watanabe theorem and show the weak uniqueness for infinite-time FBSDEs. And we prove that the solutions to a system of infinite-time FBSDEs can be employed to construct viscosity solutions for a class of distribution-dependent elliptic PDEs.
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Yang et al. (2025) studied this question.
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