2016/05/06 by Peter Lakner, Lakner, Peter, Josh Reed +3
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #math.PR
paper · pdf · doi:10.48550/arxiv.1605.02020
arxiv created 2016/05/06 · openalex publication_date 2016/05/06 · arxiv updated 2016/05/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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,theta): r >= 0, 0 <= theta <= xi for some 0 < xi < 2 pi. Let alpha= (theta1+theta2)/xi, where -pi/2 < theta1,theta2 < pi/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 < alpha < 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 > alpha , the strong p-variation of the sample paths of Y is finite on compact intervals, and, for 0 < p <= alpha, 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.