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.
Song et al. (Fri,) studied this question.