Reflecting Brownian motion

In probability theory, reflected Brownian motion (or regulated Brownian motion, both with the acronym RBM) is a Wiener process in a space with reflecting boundaries.

RBMs have been shown to describe queueing models experiencing heavy traffic as first proposed by Kingman and proven by Iglehart and Whitt.

A d–dimensional reflected Brownian motion Z is a on ${\displaystyle \mathbb {R} _{+}^{d}}$ uniquely defined by

where X(t) is an unconstrained Brownian motion and

with Y(t) a d–dimensional vector where

The reflection matrix describes boundary behaviour. In the interior of ${\displaystyle \scriptstyle \mathbb {R} _{+}^{d}}$ the process behaves like a Wiener process, on the boundary "roughly speaking, Z is pushed in direction R^j whenever the boundary surface ${\displaystyle \scriptstyle \{z\in \mathbb {R} _{+}^{d}:z_{j}=0\}}$ is hit, where R^j is the jth column of the matrix R."

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