Reflected backward stochastic differential equations with rough drivers

TL;DR

This paper develops a theory for reflected backward stochastic differential equations driven by rough paths, linking them to rough PDEs and optimal stopping problems, with applications in American option pricing under rough volatility.

Contribution

It introduces the well-posedness of rough RBSDEs using a Doss-Sussman transformation and connects solutions to rough PDEs and optimal stopping problems, advancing mathematical tools for complex financial models.

Findings

Established well-posedness of rough RBSDEs.

Connected rough RBSDE solutions to viscosity solutions of rough PDEs.

Applied the theory to American option pricing in rough volatility models.

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.