  We prove existence and uniqueness for semimartingale reflecting diffusions in
2-dimensional piecewise smooth domains with varying, oblique directions of
reflection on each "side", under geometric, easily verifiable conditions. Our
conditions are optimal in the sense that, in the case of a convex polygon with
constant direction of reflection on each side, they reduce to the conditions of
Dai and Williams (1996), which are necessary for existence of Reflecting
Brownian Motion. Moreover our conditions allow for cusps. Our argument is based
on a new localization result for constrained martingale problems which holds
quite generally: as an additional example, we show that it holds for diffusions
with jump boundary conditions.
