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]
RBMs have been shown to describe queueing models experiencing heavy traffic[2] as first proposed by Kingman[4] and proven by Iglehart and Whitt.[5][6]
Definition[]
A d–dimensional reflected Brownian motion Z is a stochastic process on uniquely defined by
- a d–dimensional drift vector μ
- a d×d non-singular covariance matrix Σ and
- a d×d reflection matrix R.[7]
where X(t) is an unconstrained Brownian motion and[8]
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) ∈ , t ≥ 0.
The reflection matrix describes boundary behaviour. In the interior of the process behaves like a Wiener process; on the boundary "roughly speaking, Z is pushed in direction Rj whenever the boundary surface is hit, where Rj is the jth column of the matrix R."[8]
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."[8] In the special case where R is an M-matrix then necessary and sufficient conditions for stability are[8]
- R is a non-singular matrix and
- 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
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]
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 :
For the plane above
Multiple dimensions[]
The stationary distribution of a reflected Brownian motion in multiple dimensions is tractable analytically when there is a product form stationary distribution,[9] which occurs when the process is stable and[10]
where D = diag(Σ). In this case the probability density function is[7]
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.[11]
% 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.[12]
Multiple dimensions[]
QNET allows simulation of steady state RBMs.[13][14][15]
Other boundary conditions[]
Feller described possible boundary condition for the process[16][17][18]
- absorption[16] or killed Brownian motion,[19] a Dirichlet boundary condition
- instantaneous reflection,[16] as described above a Neumann boundary condition
- elastic reflection, a Robin boundary condition
- delayed reflection[16] (the time spent on the boundary is positive with probability one)
- partial reflection[16] where the process is either immediately reflected or is absorbed
- sticky Brownian motion.[20]
See also[]
References[]
- ^ Dieker, A. B. (2011). "Reflected Brownian Motion". Wiley Encyclopedia of Operations Research and Management Science. doi:10.1002/9780470400531.eorms0711. ISBN 9780470400531.
- ^ a b c Harrison, J. Michael (1985). Brownian Motion and Stochastic Flow Systems (PDF). John Wiley & Sons. ISBN 978-0471819394.
- ^ 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.
- ^ 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.
- ^ 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.
- ^ 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. Retrieved 30 Nov 2012.
- ^ 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.
- ^ 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.
- ^ 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.
- ^ 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.
- ^ Kroese, Dirk P.; Taimre, Thomas; Botev, Zdravko I. (2011). Handbook of Monte Carlo Methods. John Wiley & Sons. p. 202. ISBN 978-1118014950.
- ^ 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.
- ^ 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.
- ^ Dai, Jiangang "Jim" (1990). "Section A.5 (code for BNET)". Steady-state analysis of reflected Brownian motions: characterization, numerical methods and queueing applications (Ph. D. thesis) (PDF) (Thesis). Stanford University. Dept. of Mathematics. Retrieved 5 December 2012.
- ^ 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.
- ^ a b c d e 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.
- ^ 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.
- ^ Engelbert, H. J.; Peskir, G. (2012). "Stochastic Differential Equations for Sticky Brownian Motion" (PDF). Probab. Statist. Group Manchester Research Report (5).
- ^ Chung, K. L.; Zhao, Z. (1995). "Killed Brownian Motion". From Brownian Motion to Schrödinger's Equation. Grundlehren der mathematischen Wissenschaften. 312. p. 31. doi:10.1007/978-3-642-57856-4_2. ISBN 978-3-642-63381-2.
- ^ 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.
- Wiener process