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Reflected backward stochastic differential equations with rough drivers

Hanwu Li, Huilin Zhang, Kuan Zhang

TL;DR

The paper develops a theory for reflected backward SDEs driven by rough paths, establishing well-posedness via a Doss–Sussman transformation and convergence of penalized approximations. It connects rough RBSDEs to obstacle rough PDEs through a nonlinear Feynman–Kac representation and extends viscosity solution methods to rough PDEs with obstacles. A central contribution is the characterization of an optimal stopping problem in the rough-path setting, with direct implications for American option pricing under rough volatility. Together, these results provide a robust framework for pricing and control problems in markets with rough, non-Markovian dynamics and obstacles, by unifying rough path theory, RBSDEs, and obstacle PDEs.

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.

Reflected backward stochastic differential equations with rough drivers

TL;DR

The paper develops a theory for reflected backward SDEs driven by rough paths, establishing well-posedness via a Doss–Sussman transformation and convergence of penalized approximations. It connects rough RBSDEs to obstacle rough PDEs through a nonlinear Feynman–Kac representation and extends viscosity solution methods to rough PDEs with obstacles. A central contribution is the characterization of an optimal stopping problem in the rough-path setting, with direct implications for American option pricing under rough volatility. Together, these results provide a robust framework for pricing and control problems in markets with rough, non-Markovian dynamics and obstacles, by unifying rough path theory, RBSDEs, and obstacle PDEs.

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.
Paper Structure (15 sections, 30 theorems, 292 equations)

This paper contains 15 sections, 30 theorems, 292 equations.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Structure of the paper
  4. Preliminaries
  5. Notations and basic definitions
  6. Property of rough flows
  7. Rough reflected BSDEs
  8. Well-posedness of rough RBSDEs
  9. Approximation via penalization
  10. Applications
  11. Obstacle problem for rough PDEs
  12. Optimal stopping problem
  13. Results on RBSDEs and rough BSDEs
  14. Some facts about PDEs with obstacles
  15. Some elementary facts about increasing processes

Key Result

Lemma 2.2

Suppose (Hpr) holds and $\mathbf{X}\in C^{0,p\text{-var}}([0,T],G^{[p]}(\mathbb R^{l}))$ with Then, the solution to e:flow$\phi^{T'}$ belongs to $\mathcal{D}_{3}([0,T']\times\mathbb R^d;\mathbb R).$ Moreover, there exists a positive constant $C = C(C_H,C_{\mathbf{X}},T)$ independent of $T'$ such that, uniformly over $t\in[0,T'],$$x\in\mathbb R^d,$ and $y\in\mathbb{R}$ where the derivatives of $\p

Theorems & Definitions (69)

  • Remark 2.1
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.2
  • Remark 2.3
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Remark 3.1
  • Definition 3.1
  • ...and 59 more