Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

On the roughness of the paths of RBM in a wedge

Peter Lakner, Josh Reed, and Bert Zwart

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Abstract

Reflected Brownian motion (RBM) in a wedge is a 2-dimensional stochastic process $Z$ whose state space in $\mathbb{R}^{2}$ is given in polar coordinates by $S=\{(r,\theta):r\geq0,0\leq\theta\leq\xi\}$ for some $0<\xi<2\pi$. Let $\alpha=(\theta_{1}+\theta_{2})/\xi$, where $-\pi/2<\theta_{1},\theta_{2}<\pi/2$ 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 $0<p\leq\alpha$, 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.

Résumé

Le mouvement Brownien réfléchi (RBM) dans un coin est un processus stochastique 2-dimensionnel $Z$ dont l’espace d’états dans $\mathbb{R}^{2}$ est donné en coordonnées polaires par $S=\{(r,\theta):r\geq0,0\leq\theta\leq\xi\}$ pour un $0<\xi<2\pi$. Soit $\alpha=(\theta_{1}+\theta_{2})/\xi$, où $-\pi/2<\theta_{1},\theta_{2}<\pi/2$ sont les angles de réflexion de $Z$ sur chacun des côtés du cône, mesurés à partir des normales rentrantes correspondantes. Nous montrons que dans le cas $1<\alpha<2$, le RBM dans un coin est un processus de Dirichlet. Plus précisément, son unique décomposition de Doob-Meyer est donnée par $Z=X+Y$, où $X$ est un mouvement brownien 2-dimensionnel et $Y$ est un processus continu d’énergie zéro. De plus, nous montrons que pour $p>\alpha$, la $p$-variation forte des trajectoires de $Y$ est finie sur les intervalles compacts, et, pour $0<p\leq\alpha$, la $p$-variation forte de $Y$ est infinie sur $[0,T]$ dès que $Z$ est issu de l’origine. Nous montrons aussi que sur les intervalles d’excursions de $Z$ en dehors de l’origine, $(Z,Y)$ satisfait le problème de Skorokhod standard pour $X$. Sur l’intervalle de temps infini, $(Z,Y)$ ne satisfait pas le problème de Skorokhod standard pour $X$, mais satisfait néanmoins le problème de Skorokhod étendu.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 3 (2019), 1566-1598.

Dates
Received: 15 June 2016
Revised: 30 July 2018
Accepted: 23 August 2018
First available in Project Euclid: 25 September 2019

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1569398879

Digital Object Identifier
doi:10.1214/18-AIHP928

Mathematical Reviews number (MathSciNet)
MR4010945

Subjects
Primary: 60J65: Brownian motion [See also 58J65] 60J27: Continuous-time Markov processes on discrete state spaces 60G17: Sample path properties 60G52: Stable processes 60J55: Local time and additive functionals

Keywords
Reflected Brownian motion Extended Skorokhod problem Dirichlet process $p$-variation

Citation

Lakner, Peter; Reed, Josh; Zwart, Bert. On the roughness of the paths of RBM in a wedge. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 3, 1566--1598. doi:10.1214/18-AIHP928. https://projecteuclid.org/euclid.aihp/1569398879


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