Reflected Brownian motion

In probability theory, reflected Brownian motion (or regulated Brownian motion,[1][2] both with the acronym RBM) is a Wiener process in a space with reflecting boundaries.[3] In the physical literature, this process describes diffusion in a confined space and it is often called confined Brownian motion. For example it can describe the motion of hard spheres in water confined between two walls.[4]

RBMs have been shown to describe queueing models experiencing heavy traffic[2] as first proposed by Kingman[5] and proven by Iglehart and Whitt.[6][7]

Definition

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

• a d–dimensional drift vector μ
• a d×d non-singular covariance matrix Σ and
• a d×d reflection matrix R.[8]

where X(t) is an unconstrained Brownian motion and[9]

${\displaystyle Z(t)=X(t)+RY(t)}$

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) ∈ ${\displaystyle \mathbb {R} _{+}^{d}}$, t ≥ 0.

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 Rj whenever the boundary surface ${\displaystyle \scriptstyle \{z\in \mathbb {R} _{+}^{d}:z_{j}=0\}}$ is hit, where Rj is the jth column of the matrix R."[9]

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."[9] In the special case where R is an M-matrix then necessary and sufficient conditions for stability are[9]

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

Marginal and stationary distribution

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

${\displaystyle \mathbb {P} (Z(t)\leq z)=\Phi \left({\frac {z-\mu t}{\sigma t^{1/2}}}\right)-e^{2\mu z/\sigma ^{2}}\Phi \left({\frac {-z-\mu t}{\sigma t^{1/2}}}\right)}$

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[2]

${\displaystyle \mathbb {P} (Z

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

${\displaystyle Z(t)\sim M(t)=\sup _{s\in [0,t]}X(s).}$

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 ${\displaystyle p_{b}}$:

${\displaystyle f(x,p_{b})={\frac {e^{-((x-u)/a)^{2}/2}+e^{-((x+u-2p_{b})/a)^{2}/2}}{a(2\pi )^{1/2}}}}$

For the plane above ${\displaystyle x\geq p_{b}}$

Multiple dimensions

The stationary distribution of a reflected Brownian motion in multiple dimensions is tractable analytically when there is a product form stationary distribution,[10] which occurs when the process is stable and[11]

${\displaystyle 2\Sigma =RD+DR'}$

where D = diag(Σ). In this case the probability density function is[8]

${\displaystyle p(z_{1},z_{2},\ldots ,z_{d})=\prod _{k=1}^{d}\eta _{k}e^{-\eta _{k}z_{k}}}$

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.

Simulation

One dimension

In one dimension the simulated process is the absolute value of a Wiener process. The following MATLAB program creates a sample path.[12]

% rbm.m
n = 10^4; h=10^(-3); t=h.*(0:n); mu=-1;
X = zeros(1, n+1); M=X; B=X;
B(1)=3; X(1)=3;
for k=2:n+1
Y = sqrt(h) * randn; U = rand(1);
B(k) = B(k-1) + mu * h - Y;
M = (Y + sqrt(Y ^ 2 - 2 * h * log(U))) / 2;
X(k) = max(M-Y, X(k-1) + h * mu - Y);
end
subplot(2, 1, 1)
plot(t, X, 'k-');
subplot(2, 1, 2)
plot(t, X-B, 'k-');


The error involved in discrete simulations has been quantified.[13]

Multiple dimensions

QNET allows simulation of steady state RBMs.[14][15][16]

Other boundary conditions

Feller described possible boundary condition for the process[17][18][19]

References

1. ^ Dieker, A. B. (2011). "Reflected Brownian Motion". Wiley Encyclopedia of Operations Research and Management Science. doi:10.1002/9780470400531.eorms0711. ISBN 9780470400531.
2. ^ a b c Harrison, J. Michael (1985). Brownian Motion and Stochastic Flow Systems (PDF). John Wiley & Sons. ISBN 978-0471819394.
3. ^ Veestraeten, D. (2004). "The Conditional Probability Density Function for a Reflected Brownian Motion". Computational Economics. 24 (2): 185–207. doi:10.1023/B:CSEM.0000049491.13935.af. S2CID 121673717.
4. ^ Faucheux, Luc P.; Libchaber, Albert J. (1994-06-01). "Confined Brownian motion". Physical Review E. 49 (6): 5158–5163. doi:10.1103/PhysRevE.49.5158. ISSN 1063-651X.
5. ^ Kingman, J. F. C. (1962). "On Queues in Heavy Traffic". Journal of the Royal Statistical Society. Series B (Methodological). 24 (2): 383–392. doi:10.1111/j.2517-6161.1962.tb00465.x. JSTOR 2984229.
6. ^ Iglehart, Donald L.; Whitt, Ward (1970). "Multiple Channel Queues in Heavy Traffic. I". Advances in Applied Probability. 2 (1): 150–177. doi:10.2307/3518347. JSTOR 3518347. S2CID 202104090.
7. ^ Iglehart, Donald L.; Ward, Whitt (1970). "Multiple Channel Queues in Heavy Traffic. II: Sequences, Networks, and Batches" (PDF). Advances in Applied Probability. 2 (2): 355–369. doi:10.2307/1426324. JSTOR 1426324. S2CID 120281300. Retrieved 30 Nov 2012.
8. ^ a b Harrison, J. M.; Williams, R. J. (1987). "Brownian models of open queueing networks with homogeneous customer populations" (PDF). Stochastics. 22 (2): 77. doi:10.1080/17442508708833469.
9. ^ a b c d Bramson, M.; Dai, J. G.; Harrison, J. M. (2010). "Positive recurrence of reflecting Brownian motion in three dimensions" (PDF). The Annals of Applied Probability. 20 (2): 753. arXiv:1009.5746. doi:10.1214/09-AAP631. S2CID 2251853.
10. ^ Harrison, J. M.; Williams, R. J. (1992). "Brownian Models of Feedforward Queueing Networks: Quasireversibility and Product Form Solutions". The Annals of Applied Probability. 2 (2): 263. doi:10.1214/aoap/1177005704. JSTOR 2959751.
11. ^ Harrison, J. M.; Reiman, M. I. (1981). "On the Distribution of Multidimensional Reflected Brownian Motion". SIAM Journal on Applied Mathematics. 41 (2): 345–361. doi:10.1137/0141030.
12. ^ Kroese, Dirk P.; Taimre, Thomas; Botev, Zdravko I. (2011). Handbook of Monte Carlo Methods. John Wiley & Sons. p. 202. ISBN 978-1118014950.
13. ^ Asmussen, S.; Glynn, P.; Pitman, J. (1995). "Discretization Error in Simulation of One-Dimensional Reflecting Brownian Motion". The Annals of Applied Probability. 5 (4): 875. doi:10.1214/aoap/1177004597. JSTOR 2245096.
14. ^ Dai, Jim G.; Harrison, J. Michael (1991). "Steady-State Analysis of RBM in a Rectangle: Numerical Methods and A Queueing Application". The Annals of Applied Probability. 1 (1): 16–35. CiteSeerX 10.1.1.44.5520. doi:10.1214/aoap/1177005979. JSTOR 2959623.
15. ^ Dai, Jiangang "Jim" (1990). "Section A.5 (code for BNET)" (PDF). Steady-state analysis of reflected Brownian motions: characterization, numerical methods and queueing applications (Ph. D. thesis) (Thesis). Stanford University. Dept. of Mathematics. Retrieved 5 December 2012.
16. ^ Dai, J. G.; Harrison, J. M. (1992). "Reflected Brownian Motion in an Orthant: Numerical Methods for Steady-State Analysis" (PDF). The Annals of Applied Probability. 2 (1): 65–86. doi:10.1214/aoap/1177005771. JSTOR 2959654.
17. Skorokhod, A. V. (1962). "Stochastic Equations for Diffusion Processes in a Bounded Region. II". Theory of Probability and Its Applications. 7: 3–23. doi:10.1137/1107002.
18. ^ Feller, W. (1954). "Diffusion processes in one dimension". Transactions of the American Mathematical Society. 77: 1–31. doi:10.1090/S0002-9947-1954-0063607-6. MR 0063607.
19. ^ Engelbert, H. J.; Peskir, G. (2012). "Stochastic Differential Equations for Sticky Brownian Motion" (PDF). Probab. Statist. Group Manchester Research Report (5).
20. ^ Chung, K. L.; Zhao, Z. (1995). "Killed Brownian Motion". From Brownian Motion to Schrödinger's Equation. Grundlehren der mathematischen Wissenschaften. Vol. 312. p. 31. doi:10.1007/978-3-642-57856-4_2. ISBN 978-3-642-63381-2.
21. ^ Itō, K.; McKean, H. P. (1996). "Time changes and killing". Diffusion Processes and their Sample Paths. pp. 164. doi:10.1007/978-3-642-62025-6_6. ISBN 978-3-540-60629-1.