Reflected backward stochastic differential equations with rough drivers

Abstract: In this paper, we investigate reflected backward stochastic differential equations driven by rough paths (rough RBSDEs), which can be viewed as probabilistic representations of nonlinear rough partial differential equations (rough PDEs) or stochastic partial differential equations (SPDEs) with obstacles. Furthermore, we demonstrate that solutions to rough RBSDEs solve the corresponding optimal stopping problems within a rough framework. This development provides effective and practical tools for pricing American options in the context of the rough volatility model, thus playing a crucial role in advancing the understanding and application of option pricing in complex market regimes. The well-posedness of rough RBSDEs is established using a variant of the Doss-Sussman transformation. Moreover, we show that rough RBSDEs can be approximated by a sequence of penalized BSDEs with rough drivers. For applications, we first develop the viscosity solution theory for rough PDEs with obstacles via rough RBSDEs. Second, we solve the corresponding optimal stopping problem and establish its connection with an American option pricing problem in the rough path setting.