2023/02/10 by Charles Bertucci, Bertucci, Charles, Jean‐Michel Lasry +3 · 1 citation
Decision Sciences · Economics, Econometrics and Finance · #Analysis of PDEs (math.AP) #Economic theories and models #FOS: Mathematics #Game Theory and Applications #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2302.05218
openalex publication_date 2023/02/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We develop a theory of existence and uniqueness of solutions of MFG master equations when the initial condition is Lipschitz continuous. Namely, we show that as long as the solution of the master equation is Lipschitz continuous in space, it is uniquely defined. Because we do not impose any structural assumptions, such as monotonicity for instance, there is a maximal time of existence for the notion of solution we provide. We analyze three cases: the case of a finite state space, the case of master equation set on a Hilbert space, and finally on the set of probability measures, all in cases involving common noises. In the last case, the Lipschitz continuity we refer to is on the gradient of the value function with respect to the state variable of the player.