Rough backward SDEs with discontinuous Young drivers

TL;DR

The work develops a comprehensive theory for rough backward SDEs driven by a deterministic rough path $W$ with finite $q$-variation and Brownian noise, distinguishing forward-type and Marcus-type integrals. It proves global well-posedness via local fixed-point arguments, wins robust apriori bounds, and constructs a stable solution map by lifting equations to decorated paths with a Skorokhod-type metric, enabling Wong–Zakai-like stability results with respect to $W$. The authors extend the framework to backward doubly SDEs (BDSDEs) driven by an independent process $L$ and show RBSDEs arise as conditional solutions to such BDSDEs upon freezing $L$, with a measurable-selection approach enabling a full probabilistic representation. Collectively, the paper provides intrinsic, constructive existence-uniqueness results, a stability theory under rough noise perturbations, and a bridge to BDSDEs for a broad class of jumpy rough drivers, with potential implications for nonlinear SPDEs and rough PDEs driven by irregular noise.

Abstract

We study solutions to backward differential equations that are driven hybridly by a deterministic discontinuous rough path $W$ of finite $q$-variation for $q \in [1, 2)$ and by Brownian motion $B$. To distinguish between integration of jumps in a forward- or Marcus-sense, we refer to these equations as forward- respectively Marcus-type rough backward stochastic differential equations (RBSDEs). We establish global well-posedness by proving global apriori bounds for solutions and employing fixed-point arguments locally. Furthermore, we lift the RBSDE solution and the driving rough noise to the space of decorated paths endowed with a Skorokhod-type metric and show stability of solutions with respect to perturbations of the rough noise. Finally, we prove well-posedness for a new class of backward doubly stochastic differential equations (BDSDEs), which are jointly driven by a Brownian martingale $B$ and an independent discontinuous stochastic process $L$ of finite $q$-variation. We explain, how our RBSDEs can be understood as conditional solutions to such BDSDEs, conditioned on the information generated by the path of $L$.

Theorems & Definitions (41)

• Definition 2.1
• Lemma 2.2
• Lemma 2.3
• Remark 2.4
• Theorem 3.1
• Theorem 3.2: Local existence and uniqueness
• Remark 3.3
• Theorem 3.4: Global existence and uniqueness
• Theorem 3.5
• Definition 4.1