On the roughness of the paths of RBM in a wedge

Reflected Brownian motion (RBM) in a wedge is a 2-dimensional stochastic process Z whose state space in ℝ2 is given in polar coordinates by S = {(r, θ) : r ≥ 0, 0 ≤ θ ≤ ξ } for some 0 < ξ < 2π. Let α = (θ1 + θ2)/ξ, where -π/2 < θ1, θ2 < π/2 are the directions of reflection of Z off each of the two edges of the wedge as measured from the corresponding inward facing normal. We prove that in the case of 1<α <2, RBM in a wedge is a Dirichlet process. Specifically, its unique Doob-Meyer type decomposition is given by Z = X + Y, where X is a two-dimensional Brownian motion and Y is a continuous process of zero energy. Furthermore, we show that for p >α, the strong p-variation of the sample paths of Y is finite on compact intervals, and, for 0 < p ≤ α, the strong p-variation of Y is infinite on [0,T ] whenever Z has been started from the origin. We also show that on excursion intervals of Z away from the origin, (Z,Y ) satisfies the standard Skorokhod problem for X. However, on the entire time horizon (Z,Y ) does not satisfy the standard Skorokhod problem for X, but nevertheless we show that it satisfies the extended Skorkohod problem.

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