One idea is to approach the problem through stochastic differential equations with reflections off the boundary of domains. For example, [this paper by Tanaka (1979)](http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.hmj/1206135203) considers a stochastic differential equation with reflections off the boundary of a convex set. They prove existence and uniqueness for coefficients defined on the domain only, and as far as I can see, in the proof they don't extend the coefficients to all of $\mathbb{R}^d$. If unique solutions exist with reflecting boundary conditions for all time, then presumably the original equation has a solution up to hitting the boundary. A more recent reference is [Lions and Sznitman (1984)](http://onlinelibrary.wiley.com/doi/10.1002/cpa.3160370408/abstract).

This is a bit inelegant in that an SDE stopped at the boundary should be simpler than an SDE reflected off a boundary. But a least you don't need to extend the domain of the coefficients.