Reflected Mean-Field Backward Stochastic Differential Equations. Approximation and Associated Nonlinear PDEs

TL;DR

This paper studies reflected mean-field backward stochastic differential equations (BSDEs), their approximation via penalization, and links them to nonlinear nonlocal PDEs with obstacles, expanding probabilistic methods in mean-field theory.

Contribution

It introduces a new class of reflected mean-field BSDEs, establishes their approximation through penalization, and connects them to nonlinear PDEs with obstacles.

Findings

Reflected mean-field BSDEs characterized as limits of penalized equations.

Existence and uniqueness of solutions for the new class of BSDEs.

Probabilistic interpretation of obstacle PDEs via these BSDEs.

Abstract

Mathematical mean-field approaches have been used in many fields, not only in Physics and Chemistry, but also recently in Finance, Economics, and Game Theory. In this paper we will study a new special mean-field problem in a purely probabilistic method, to characterize its limit which is the solution of mean-field backward stochastic differential equations (BSDEs) with reflections. On the other hand, we will prove that this type of reflected mean-field BSDEs can also be obtained as the limit equation of the mean-field BSDEs by penalization method. Finally, we give the probabilistic interpretation of the nonlinear and nonlocal partial differential equations with the obstacles by the solutions of reflected mean-field BSDEs.