  Reflected diffusions in convex polyhedral domains arise in a variety of
applications, including interacting particle systems, queueing networks,
biochemical reaction networks and mathematical finance. Under suitable
conditions on the data, we establish pathwise differentiability of such a
reflected diffusion with respect to its defining parameters --- namely, its
initial condition, drift and diffusion coefficients, and (oblique) directions
of reflection along the boundary of the domain. We characterize the
right-continuous regularization of a pathwise derivative of the reflected
diffusion as the pathwise unique solution to a constrained linear stochastic
differential equation with jumps whose drift and diffusion coefficients, domain
and directions of reflection depend on the state of the reflected diffusion.
The proof of this result relies on properties of directional derivatives of the
associated (extended) Skorokhod reflection map and their characterization in
terms of a so-called derivative problem, and also involves establishing certain
path properties of the reflected diffusion at nonsmooth parts of the boundary
of the polyhedral domain, which may be of independent interest. As a corollary,
we obtain a probabilistic representation for derivatives of expectations of
functionals of reflected diffusions, which is useful for sensitivity analysis
of reflected diffusions.
