Rough backward stochastic differential equations
Abstract
We develop an intrinsic well-posedness theory for nonlinear backward stochastic differential equations (BSDEs) driven simultaneously by Brownian motion and a (level-$2$) rough path of finite p-variation. Unlike earlier approaches based on smooth approximation or transformation methods, we formulate the equation directly by viewing its solution as a rough semimartingale (in the sense of \cite{friz2023rough}). This framework is particularly suited to BSDEs, whose Brownian martingale component is only implicitly defined and lacks the \emph{a priori} time regularity required by stochastic-sewing-based controlled rough path methods \cite{fhl21,allan2024rough}. We establish comparison, existence, uniqueness, and stability under the natural regularity condition H∈ Cbγ, γ>p. The main analytical difficulty, namely, the loss of integrability arising from nonlinear composition, is overcome through conditional p-variation norms with BMO-type properties. Finally, by randomizing the rough driver as a Brownian rough path, we establish a direct correspondence between rough BSDEs and backward doubly stochastic differential equations (BDSDEs).
Related
- On the Pontryagin Maximum Principle for Rough Stochastic Differential Equations
- Backward and reflected backward stochastic differential equations under initial enlargement of Brownian filtrations
- Backward doubly stochastic differential equations with or without reflection under weak conditions
- Backward stochastic partial differential equations with unbounded random coefficients
- Conditional Reflected Backward Stochastic Differential Equations
BibTeX
@misc{friz2026rough,
title = {{Rough backward stochastic differential equations}},
author = {Peter K. Friz and Jian Song and Huilin Zhang and Kuan Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06606},
eprint = {2609.06606},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06606},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Rough backward stochastic differential equations
AU - Peter K. Friz
AU - Jian Song
AU - Huilin Zhang
AU - Kuan Zhang
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.06606
UR - https://arxiv.org/abs/2609.06606
AB - We develop an intrinsic well-posedness theory for nonlinear backward stochastic differential equations (BSDEs) driven simultaneously by Brownian motion and a (level-$2$) rough path of finite p-variation. Unlike earlier approaches based on smooth approximation or transformation methods, we formulate the equation directly by viewing its solution as a rough semimartingale (in the sense of \cite{friz2023rough}). This framework is particularly suited to BSDEs, whose Brownian martingale component is only implicitly defined and lacks the \emph{a priori} time regularity required by stochastic-sewing-based controlled rough path methods \cite{fhl21,allan2024rough}. We establish comparison, existence, uniqueness, and stability under the natural regularity condition H∈ C_b^γ, γ>p. The main analytical difficulty, namely, the loss of integrability arising from nonlinear composition, is overcome through conditional p-variation norms with BMO-type properties. Finally, by randomizing the rough driver as a Brownian rough path, we establish a direct correspondence between rough BSDEs and backward doubly stochastic differential equations (BDSDEs).
ER -