Backward stochastic differential equations with nonlinear Young drivers I

This paper (alongside its companion, Part II \cite{BSDEYoung-II}) investigates backward stochastic differential equations (BSDEs) involving a nonlinear Young integral of the form $\int_{t}^{T}g(Y_{r})\eta(dr,X_{r})$, where the driver $\eta(t,x)$ is a space-time H\"older continuous function and $X$ is a diffusion process. Solutions to such equations provide a probabilistic interpretation of the solutions to stochastic partial differential equations (SPDEs) driven by space-time noise. Assuming the driver $\eta(t,x)$ is bounded, we establish the existence and uniqueness of the solutions to these BSDEs via a modified Picard iteration method. We then derive a comparison principle by analyzing the associated linear BSDEs and establish regularity properties of the solutions. As an application, we obtain Feynman-Kac formulae for a class of linear stochastic heat equations subject to Neumann boundary conditions.