Reflected brownian motion in a wedge: Semimartingale property

In this paper, the object of study is reflected Brownian motion in a two-dimensional wedge with constant direction of reflection on each side of the wedge. The basic question considered here is “When is this process a semimartingale?”. It is first shown that a related process, defined by specifying the corner of the wedge to be an absorbing state, rather than an instantaneous one, is a semimartingale. Conditions for the existence and uniqueness of the process for which the corner is an instantaneous state were given by Vardhan and Williams (“Brownian motion in a wedge with oblique reflection”, Comm. Pure Appl. Math., to appear). Under these conditions, it is shown that starting away from the corner, the process is a semimartingale if and only if there is a convex combination of the directions of reflection that points into the wedge. This equivalence is also shown to hold starting from the corner, except in one unresolved case for which the wedge angle exceeds π and the directions of reflection are exactly opposed.