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October 3, 2025Open Access

Backward stochastic differential equations with nonlinear Young drivers II

Key Points

The study finds new methods to solve backward stochastic differential equations with unbounded drivers, improving theoretical foundations.
A key result is the derivation of a nonlinear Feynman-Kac formula linked to partial differential equations driven by Young signals.
The proposed localization method reveals significant error estimates connecting bounded domain Cauchy-Dirichlet problems to Cauchy counterparts.
These results highlight the potential for advancements in solving non-Lipschitz PDEs through novel mathematical approaches.

Abstract

This paper continues our previous work (Part I, arXiv: 2504. 18632v3) on the well-posedness of backward stochastic differential equations (BSDEs) involving a nonlinear Young integral of the form ₓ^Tg (Yₑ) η (dr, Xₑ), with particular focus on the case where the driver η (t, x) is unbounded. To address this setting, we develop a new localization method that extends solvability from BSDEs with bounded drivers to those with unbounded ones. As a direct application, we derive a nonlinear Feynman-Kac formula for a class of partial differential equations driven by Young signals (Young PDEs). Moreover, employing the proposed localization method, we obtain error estimates that compare Cauchy-Dirichlet problems on bounded domains with their whole-space Cauchy counterparts, with special attention to non-Lipschitz PDEs.

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