Forward and Backward Stochastic Differential Equations with normal constraint in law

TL;DR

This paper studies the well-posedness of stochastic differential equations with constraints on their probability law, exploring their connection to PDEs on Wasserstein space and analyzing associated particle systems.

Contribution

It introduces a framework for SDEs with law constraints reflected along normal vectors, linking them to PDEs on Wasserstein space and analyzing particle systems with common noise.

Findings

Established well-posedness of constrained SDEs

Connected stochastic equations to PDEs with Neumann and obstacle conditions

Analyzed particle systems with mean field interactions

Abstract

In this paper we investigate the well-posedness of backward or forward stochastic differential equations whose law is constrained to live in an a priori given (smooth enough) set and which is reflected along the corresponding ''normal'' vector. We also study the associated interacting particle system reflected in mean field and asymptotically described by such equations. The case of particles submitted to a common noise as well as the asymptotic system is studied in the forward case. Eventually, we connect the forward and backward stochastic differential equations with normal constraints in law with partial differential equations stated on the Wasserstein space and involving a Neumann condition in the forward case and an obstacle in the backward one.