2019
On the roughness of the paths of RBM in a wedge
Publication
Publication
Annales de l'Institut Henri Poincaré  Probability and Statistics , Volume 55  Issue 3 p. 1566 1598
Reflected Brownian motion (RBM) in a wedge is a 2dimensional 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 DoobMeyer type decomposition is given by Z = X + Y, where X is a twodimensional Brownian motion and Y is a continuous process of zero energy. Furthermore, we show that for p >α, the strong pvariation of the sample paths of Y is finite on compact intervals, and, for 0 < p ≤ α, the strong pvariation 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.
Additional Metadata  

Dirichlet process, Extended Skorokhod problem, Pvariation, Reflected Brownian motion  
dx.doi.org/10.1214/18AIHP928  
Annales de l'Institut Henri Poincaré  Probability and Statistics  
Organisation  Centrum Wiskunde & Informatica, Amsterdam, The Netherlands 
Lakner, P, Reed, J, & Zwart, A.P. (2019). On the roughness of the paths of RBM in a wedge. Annales de l'Institut Henri Poincaré  Probability and Statistics, 55(3), 1566–1598. doi:10.1214/18AIHP928
