Bellman systems with mean field dependent dynamics

TL;DR

This paper studies nonlinear Bellman systems linked to stochastic differential games with highly mean field dependent dynamics, introducing new methods to prove existence of solutions and maximum principles for systems with critical growth.

Contribution

It presents a novel approach to establish existence and regularity of solutions for complex PDE systems with critical growth due to mean field dependence in stochastic games.

Findings

Existence of weak solutions under new structural assumptions

Development of a new maximum/minimum principle method for critical growth systems

Applicable to a broad class of mean field dependent stochastic differential games

Abstract

We deal with nonlinear elliptic and parabolic systems that are the Bellman like systems associated to stochastic differential games with mean field dependent dynamics. The key novelty of the paper is that we allow heavily mean field dependent dynamics. This in particular leads to a system of PDE's with critical growth, for which it is rare to have an existence and/or regularity result. In the paper, we introduce a structural assumptions that cover many cases in stochastic differential games with mean filed dependent dynamics for which we are able to establish the existence of a weak solution. In addition, we present here a completely new method for obtaining the maximum/minimum principles for systems with critical growths, which is a starting point for further existence and also qualitative analysis.