Multidimensional Backward Stochastic Differential Equations with Rough Drifts

TL;DR

This paper investigates multidimensional backward stochastic differential equations with rough, potentially random drifts, establishing existence and uniqueness results, introducing a new rough stochastic integral, and linking to rough PDE systems.

Contribution

It introduces a new $p$-rough stochastic integral, extends existence and uniqueness results to multidimensional rough BSDEs with general terminal conditions, and connects these to rough PDEs.

Findings

Established existence and uniqueness of solutions under various conditions.

Introduced a new $p$-rough stochastic integral for $p ange [2,3)$.

Connected rough BSDEs to systems of rough partial differential equations.

Abstract

In this paper, we study a multidimensional backward stochastic differential equation (BSDE) with an additional rough drift (rough BSDE), and give the existence and uniqueness of the adapted solution, either when the terminal value and the geometric rough path are small, or when each component of the rough drift only depends on the corresponding component of the first unknown variable (but dropped is the one-dimensional assumption of Diehl and Friz [Ann. Probab. 40 (2012), 1715-1758]). We also introduce a new notion of the $p$-rough stochastic integral for $p \in \left[2, 3\right)$, and then succeed in giving -- through a fixed-point argument -- a general existence and uniqueness result on a multidimensional rough BSDE with a general square-integrable terminal value, allowing the rough drift to be random and time-varying but having to be linear; furthermore, we connect it to a system of rough partial differential equations.