Multidimensional backward stochastic differential equations with rough drifts

Liang, Jiahao; Tang, Shanjian

doi:10.1090/tran/9237

Multidimensional backward stochastic differential equations with rough drifts
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by Jiahao Liang and Shanjian Tang;

Trans. Amer. Math. Soc. 378 (2025), 201-257

DOI: https://doi.org/10.1090/tran/9237

Published electronically: October 31, 2024

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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 we drop the one-dimensional assumption of Diehl and Friz [Ann. Probab. 40 (2012), pp. 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.

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