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Reflected Brownian motion

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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.

Contents

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

Definition

A d–dimensional reflected Brownian motion Z is a stochastic process on R + d uniquely defined by

  • a d–dimensional drift vector μ
  • a d×d non-singular covariance matrix Σ and
  • a d×d reflection matrix R.

    • where X(t) is an unconstrained Brownian motion and

    with Y(t) a d–dimensional vector where

  • Y is continuous and non–decreasing with Y(0) = 0
  • Yj only increases at times for which Zj = 0 for j = 1,2,...,d
  • Z(t) ∈  R + d , t ≥ 0.

    • The reflection matrix describes boundary behaviour. In the interior of R + d the process behaves like a Wiener process, on the boundary "roughly speaking, Z is pushed in direction Rj whenever the boundary surface { z ∈ R + d : z j = 0 } is hit, where Rj is the jth column of the matrix R."

    Stability conditions

    Stability conditions are known for RBMs in 1, 2, and 3 dimensions. "The problem of recurrence classification for SRBMs in four and higher dimensions remains open." In the special case where R is an M-matrix then necessary and sufficient conditions for stability are

    1. R is a non-singular matrix and
    2. R−1μ < 0.

    One dimension

    The marginal distribution (transient distribution) of a one-dimensional Brownian motion starting at 0 restricted to positive values (a single reflecting barrier at 0) with drift μ and variance σ2 is

    for all t ≥ 0, (with Φ the cumulative distribution function of the normal distribution) which yields (for μ < 0) when taking t → ∞ an exponential distribution

    For fixed t, the distribution of Z(t) coincides with the distribution of the running maximum M(t) of the Brownian motion,

    But be aware that the distributions of the processes as a whole are very different. In particular, M(t) is increasing in t, which is not the case for Z(t).

    The heat kernel for reflected Brownian motion at p b :

    f ( x , p b ) = e − ( ( x − u ) / a ) 2 / 2 + e − ( ( x + u − 2 p b ) / a ) 2 / 2 a ( 2 π ) 1 / 2

    For the plane above x ≥ p b

    Multiple dimensions

    The stationary distribution of a reflected Brownian motion in multiple dimensions is tractable analytically when there is a product form stationary distribution, which occurs when the process is stable and

    where D = diag(Σ). In this case the probability density function is

    where ηk = 2μkγk/Σkk and γ = R−1μ. Closed-form expressions for situations where the product form condition does not hold can be computed numerically as described below in the simulation section.

    One dimension

    In one dimension the simulated process is the absolute value of a Wiener process. The following MATLAB program creates a sample path.

    The error involved in discrete simulations has been quantified.

    Multiple dimensions

    QNET allows simulation of steady state RBMs.

    Other boundary conditions

    Feller described possible boundary condition for the process

  • absorption or killed Brownian motion, a Dirichlet boundary condition
  • instantaneous reflection, as described above a Neumann boundary condition
  • elastic reflection, a Robin boundary condition
  • delayed reflection (the time spent on the boundary is positive with probability one)
  • partial reflection where the process is either immediately reflected or is absorbed
  • sticky Brownian motion.

    • References

    Reflected Brownian motion Wikipedia
    (Text) CC BY-SA


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