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.

中文翻译：

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具有法向正态约束的前向和后向随机微分方程

摘要 在本文中，我们研究了后向或前向随机微分方程的适定性，其定律被约束在先验给定（足够平滑）的集合中，并沿相应的“法向”向量反映。我们还研究了反映在平均场中并由此类方程渐近描述的相关相互作用粒子系统。在前向情况下研究了粒子受到共同噪声以及渐近系统的影响。最后，我们将具有法向约束的前向和后向随机微分方程与在 Wasserstein 空间上陈述的偏微分方程联系起来，并且在前向情况下涉及诺依曼条件，在后向情况下涉及障碍。