  We study the large deviations of current-type observables defined for Markov
diffusion processes evolving in smooth bounded regions of $\mathbb{R}^d$ with
reflections at the boundaries. We derive for these the correct boundary
conditions that must be imposed on the spectral problem associated with the
scaled cumulant generating function, which gives, by Legendre transform, the
rate function characterizing the likelihood of current fluctuations. Two
methods for obtaining the boundary conditions are presented, based on the
diffusive limit of random walks and on the Feynman--Kac equation underlying the
evolution of generating functions. Our results generalize recent works on
density-type observables, and are illustrated for an $N$-particle single-file
diffusion on a ring, which can be mapped to a reflected $N$-dimensional
diffusion.
