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Reflected Brownian motion in a wedge: Semimartingale property. (English) Zbl 0535.60042
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 S. R. S. Varadhan and the author, Brownian motion in a wedge with oblique reflection, 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 \(\pi\) and the directions of reflection are exactly opposed.

MSC:
60G60 Random fields
60J65 Brownian motion
60G44 Martingales with continuous parameter
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