Forward and Backward Stochastic Differential Equations with normal constraint in law

Philippe Briand (1) , Pierre Cardaliaguet (2) , Paul-Eric Chaudru de Raynal (1) , Ying Hu (3)

1 LAMA - Laboratoire de Mathématiques
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
3 IRMAR - Institut de Recherche Mathématique de Rennes

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.

Domaines: Probabilités [math.PR]; Equations aux dérivées partielles [math.AP]

Philippe Briand, Pierre Cardaliaguet, Paul-Eric Chaudru de Raynal, Ying Hu. Forward and Backward Stochastic Differential Equations with normal constraint in law. Stochastic Processes and their Applications, 2020, 130 (12), pp.7021-7097. ⟨10.1016/j.spa.2020.07.007⟩. ⟨hal-02053777⟩