Abstract
In this paper, we study Mean Field games with common noise based on nonlinear stable-like processes. The MFG limit is specified by a single quasi-linear deterministic infinite-dimensional partial differential second order backward equation. The main result is that any its solution provides an 1/N-Nash equilibrium for the initial game of N agents. Our approach is based on interpreting the common noise as a kind of binary interaction of agents and our previous results on regularity and sensitivity with respect to the initial conditions of the solution to the nonlinear stochastic differential equations of McKean-Vlasov type generated by stable-like processes.
Highlights
Mean field games with common noise present a quickly developing part of the mean field game theory
In [12], existence of weak solutions for mean field games with common noise is shown to hold under very general assumptions, existence and uniqueness of a strong solution are proved under additional assumptions
Mean field games with common noise based on nonlinear stable-like processes are studied in this paper
Summary
Mean field games with common noise present a quickly developing part of the mean field game theory. Under appropriate regularity assumptions on the coefficients b, σcom and a in (1), the corresponding Markov evolution of the empirical measures μNt converges, as N → ∞ to the unique solution μt of the following nonlinear stochastic partial differential equation of the McKean-Vlasov type generated by the stable-like process, d(f, μt) This equation is written in the weak form meaning that it should hold for all f ∈ C2(Rd). Let Ck(M≤λ(Rd)) denote the space of functionals such that the kth order variational derivatives are well defined and represent continuous functions of all variables with measures considered in their weak topology.
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