Kernel-Based Heterogeneous Mean Field Games on a Continuum Type Space

Abstract

This paper studies a class of heterogeneous mean field games on a continuum type space, in which interactions are generated by a Hilbert-Schmidt kernel operator acting on the type-indexed mean state profile. The control set is allowed to be a general closed convex subset of a Euclidean space, and the optimal feedback is characterized through the unique maximizer of the Hamiltonian. For a prescribed mean-field path, the stochastic maximum principle leads to a family of fully coupled forward-backward stochastic differential equations (FBSDEs) indexed by the agent type. Under Lipschitz conditions and a monotonicity assumption on the reduced coefficients, the frozen FBSDE is globally well posed. We then formulate the consistency problem as a fixed-point problem for the kernel mean-field path. When the operator norm of the interaction kernel is sufficiently small, the resulting map has an invariant ball and is continuous and compact, so that Schauder's theorem yields existence of a mean field Nash equilibrium. Uniqueness follows under an aggregate monotonicity condition adapted to the kernel coupling. A linear-quadratic example illustrates how finite-rank interactions and heterogeneous risk preferences produce coupled collective modes, and yields a spectral smallness condition for existence and an explicit sufficient condition for uniqueness.