Backward stochastic differential equations with nonlinear Young drivers II

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 $\int_{t}^{T}g(Y_{r})\eta(dr,X_{r})$, with particular focus on the case where the driver $\eta(t,x)$ 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.