粗糙倒向随机微分方程
Rough backward stochastic differential equations
- Technische Universität Berlin（柏林工业大学）
- Weierstraß Institut, Berlin（柏林韦伊拉斯特拉瑟研究所）
- Research Center for Mathematics and Interdisciplinary Sciences（山东大学数学与交叉科学研究中心；教育部非线性期望前沿科学中心）
- Frontiers Science Center for Nonlinear Expectations, Ministry of Education, Shandong University（西湖大学先进理论科学研究所）
- Institute for Theoretical Sciences, Westlake Institute for Advanced Study, Westlake University
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出一种直接基于粗糙半鞅的粗糙倒向随机微分方程理论，在正则性条件下建立适定性，并建立与倒向双随机微分方程的对应关系。
AI中文摘要：
我们为同时由布朗运动和一个有限$p$变差的（$2$级）粗糙路径驱动的非线性倒向随机微分方程（BSDEs）发展了一套内在的适定性理论。与早期基于光滑逼近或变换方法的方法不同，我们直接通过将其解视为粗糙半鞅（在\cite{friz2023rough}的意义下）来表述方程。该框架特别适用于BSDEs，因为其布朗鞅分量仅被隐式定义，缺乏基于随机缝合的控制粗糙路径方法\cite{fhl21,allan2024rough}所需的先验时间正则性。我们在自然正则性条件$H\in C_b^\gamma$，$\gamma>p$下建立了比较、存在性、唯一性和稳定性。主要的分析困难，即由非线性复合引起的可积性损失，通过具有BMO型性质的条件$p$变差范数得以克服。最后，通过将粗糙驱动随机化为布朗粗糙路径，我们建立了粗糙BSDEs与倒向双随机微分方程（BDSDEs）之间的直接对应关系。
英文摘要：
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 $H\in C_b^γ$, $γ>p$. 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).
