We consider a class of normally reflected stochastic differential equations in non-smooth time-dependent domain whose boundary is W^1, {p} in time and with convex time sections. We show existence and uniqueness of the solution by providing a strong approximation for this type of equations using a sequence of standard diffusions, where the provided approximation is controlled through additional regularization of the domain by smooth time-dependent domains converging to the original one. Then in this Markovian framework, we study the corresponding generalized backward stochastic differential equation and derive a probabilistic (Feynman–Kac) representation of the solution to a partial differential equation with Cauchy-Neumann boundary conditions on non-smooth time-dependent domains. As a by-product, we obtain an approximation of this PDE by a sequence of partial differential equations without Neumann boundary condition.
Jakani et al. (Wed,) studied this question.