Forward and Backward Stochastic Differential Equations with normal constraint in law

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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.