Abstract McKean-Vlasov and HJB equations, their fractional versions and related forward-backward systems on Riemannian manifolds

TL;DR

This paper develops a unified theoretical framework for analyzing nonlinear fractional pseudo-differential equations, including McKean-Vlasov and HJB equations, on Banach spaces and Riemannian manifolds, relevant to stochastic control and mean-field games.

Contribution

It introduces a class of abstract fractional equations encompassing key models in stochastic processes and control, enabling unified analysis of forward-backward systems on manifolds.

Findings

Unified approach to McKean-Vlasov and HJB equations

Framework applicable to fractional and manifold settings

Facilitates analysis of mean-field game systems

Abstract

We introduce a class of abstract nonlinear fractional pseudo-differential equations in Banach spaces that includes both the Mc-Kean-Vlasov-type equations describing nonlinear Markov processes and the Hamilton-Jacobi-Bellman(HJB)-Isaacs equation of stochastic control and games thus allowing for a unified analysis of these equations. This leads to an effective theory of coupled forward-backward systems (forward McKean-Vlasov evolution and backward HJB-Isaacs evolution) that are central to the modern theory of mean-field games.