Zero-sum stochastic differential games of generalized McKean–Vlasov type

By Huyen Pham, Andrea Cosso. Journal des Mathématiques Pures et Appliquées, 2018

66 citations recorded by the source corpus.

We study zero-sum stochastic differential games where the state dynamics of the two players is governed by a generalized McKean-Vlasov (or mean-field) stochastic differential equation in which the distribution of both state and controls of each player appears in the drift and diffusion coefficients, as well as in the running and terminal payoff functions. We prove the dynamic programming principle (DPP) in this general setting, which also includes the control case with only one player, where it is the first time that DPP is proved for open-loop controls. We also show that the upper and lower value functions are viscosity solutions to a corresponding upper and lower Master Bellman-Isaacs equation. Our results extend the seminal work of Fleming and Souganidis [15] to the McKean-Vlasov setting.

Subfields: Demography, Finance, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/6b6110349376a0d8a9c8da1032562e78ab54a1d0

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Citations in the corpus

Lagrangian, Eulerian and Kantorovich formulations of multi-agent optimal control problems: Equivalence and Gamma-convergence (2020, background): “in [14, 23] for the case of random differential games and in [39] for multi-agent systems.”
Master Bellman equation in the Wasserstein space: Uniqueness of viscosity solutions (2021, background): “Concerning second-order equations, papers [41, 42, 4, 17, 16] focus on the existence of viscosity solutions, proving that the value function solves in the viscosity sense the Master Bellman equation.”
Optimal control of path-dependent McKean–Vlasov SDEs in infinite-dimension (2020, background): “The optimal control problem of McKean-Vlasov SDEs is sometimes called extended or generalized (see [1, 27]) when the coefficients also depend on the law of the control process Pαs (as in the present framework, see equation (2.”