Reflected Brownian motion

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

• a d–dimensional drift vector μ
• a d×d non-singular covariance matrix Σ and
• a d×d reflection matrix R.{{Cite journal | last1 = Harrison | first1 = J. M. | author-link1 = J. Michael Harrison| last2 = Williams | first2 = R. J. | doi = 10.1080/17442508708833469 | title = Brownian models of open queueing networks with homogeneous customer populations| journal = Stochastics| volume = 22 | issue = 2 | pages = 77 | year = 1987 | url = https://www.ima.umn.edu/preprints/Jan87Dec87/321.pdf}}

where X(t) is an unconstrained Brownian motion with drift μ and variance Σ, and

::$Z(t)\; =\; X(t)\; +\; R\; Y(t)$

with Y(t) a d–dimensional vector where

• Y is continuous and non–decreasing with Y(0) = 0
• Y_j only increases at times for which Z_j = 0 for j = 1,2,...,d
• Z(t) ∈ $\backslash mathbb\; R^d\_+$, t ≥ 0.

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

The process Y_j is the local time of the process on the corresponding section of the boundary.

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."{{Cite journal | last1 = Bramson | first1 = M. | last2 = Dai | first2 = J. G. | last3 = Harrison | first3 = J. M. | author-link3 = J. Michael Harrison| doi = 10.1214/09-AAP631 | title = Positive recurrence of reflecting Brownian motion in three dimensions | journal = The Annals of Applied Probability | volume = 20 | issue = 2 | pages = 753 | year = 2010 | url = http://www2.isye.gatech.edu/people/faculty/dai/publications/bramsonDaiHarrison10.pdf| arxiv = 1009.5746 | s2cid = 2251853 }} 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.

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

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

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{{cite book | title = Brownian Motion and Stochastic Flow Systems | first = J. Michael | last = Harrison | author-link = J. Michael Harrison | year = 1985 | publisher = John Wiley & Sons | isbn = 978-0471819394 | url = http://faculty-gsb.stanford.edu/harrison/Documents/BrownianMotion-Stochasticms.pdf}}

::

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

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

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 $p\_b$:

$f(x,p\_b)=\backslash frac\{e^\{-((x-u)/a)^2/2\}+e^\{-((x+u-2p\_b)/a)^2/2\}\}\{a(2\backslash pi)^\{1/2\}\}$

For the plane above $x\; \backslash ge\; p\_b$

The stationary distribution of a reflected Brownian motion in multiple dimensions is tractable analytically when there is a product form stationary distribution,{{Cite journal | last1 = Harrison | first1 = J. M. | author-link1 = J. Michael Harrison| last2 = Williams | first2 = R. J. | doi = 10.1214/aoap/1177005704 | title = Brownian Models of Feedforward Queueing Networks: Quasireversibility and Product Form Solutions | journal = The Annals of Applied Probability | volume = 2 | issue = 2 | pages = 263 | year = 1992 | jstor = 2959751| doi-access = free }} which occurs when the process is stable and{{Cite journal | last1 = Harrison | first1 = J. M. | author-link1 = J. Michael Harrison | last2 = Reiman | first2 = M. I. | doi = 10.1137/0141030 | title = On the Distribution of Multidimensional Reflected Brownian Motion | journal = SIAM Journal on Applied Mathematics | volume = 41 | issue = 2 | pages = 345–361 | year = 1981 }}

::$2\; \backslash Sigma\; =\; RD\; +\; DR\text{'}$

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

::$p(z\_1,z\_2,\backslash ldots,z\_d)\; =\; \backslash prod\_\{k=1\}^d\; \backslash eta\_k\; e^\{-\backslash eta\_k\; z\_k\}$

where η_k = 2μ_kγ_k/Σ_kk 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.

In one dimension the simulated process is the absolute value of a Wiener process. The following MATLAB program creates a sample path.{{cite book| page = [https://archive.org/details/handbookmontecar00kroe/page/n224 202] | title = Handbook of Monte Carlo Methods | url = https://archive.org/details/handbookmontecar00kroe | url-access = limited | first1=Dirk P. | last1= Kroese|author-link1=Dirk Kroese | first2= Thomas |last2=Taimre|first3= Zdravko I.|last3= Botev | publisher = John Wiley & Sons |year = 2011 | isbn = 978-1118014950}}

% 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.{{Cite journal | last1 = Asmussen | first1 = S. | last2 = Glynn | first2 = P. | last3 = Pitman | first3 = J. | doi = 10.1214/aoap/1177004597 | jstor = 2245096| title = Discretization Error in Simulation of One-Dimensional Reflecting Brownian Motion | journal = The Annals of Applied Probability | volume = 5 | issue = 4 | pages = 875 | year = 1995 | doi-access = free }}

[http://www2.isye.gatech.edu/~dai/QNET/ QNET] allows simulation of steady state RBMs.{{cite journal | last1 = Dai | first1 = Jim G. | last2 = Harrison | first2 = J. Michael | author-link2 = J. Michael Harrison | year = 1991 | title = Steady-State Analysis of RBM in a Rectangle: Numerical Methods and A Queueing Application | journal = The Annals of Applied Probability | volume = 1 | issue = 1 | pages = 16–35 | jstor=2959623 | doi=10.1214/aoap/1177005979| citeseerx = 10.1.1.44.5520 }}{{cite thesis| title = Steady-state analysis of reflected Brownian motions: characterization, numerical methods and queueing applications (Ph. D. thesis) | publisher = Stanford University. Dept. of Mathematics | year = 1990| first = Jiangang "Jim" | last = Dai | chapter-url = http://www2.isye.gatech.edu/~dai/publications/dai90Dissertation.pdf | access-date = 5 December 2012 | chapter = Section A.5 (code for BNET)}}{{cite journal | last1 = Dai | first1 = J. G. | last2 = Harrison | first2 = J. M. | author-link2 = J. Michael Harrison | year = 1992 | title = Reflected Brownian Motion in an Orthant: Numerical Methods for Steady-State Analysis | journal = The Annals of Applied Probability | volume = 2 | issue = 1 | pages = 65–86 | jstor = 2959654 | url = http://www2.isye.gatech.edu/people/faculty/dai/publications/daiHarrison92.pdf | doi=10.1214/aoap/1177005771| doi-access = free }}

Feller described possible boundary condition for the process{{Cite journal | last1 = Skorokhod | first1 = A. V. | author-link1 = Anatoliy Skorokhod| doi = 10.1137/1107002 | title = Stochastic Equations for Diffusion Processes in a Bounded Region. II | journal = Theory of Probability and Its Applications | volume = 7 | pages = 3–23| year = 1962 }}{{Cite journal | last1 = Feller | first1 = W. | author-link1 = William Feller| doi = 10.1090/S0002-9947-1954-0063607-6 | title = Diffusion processes in one dimension | journal = Transactions of the American Mathematical Society | volume = 77 | pages = 1–31 | year = 1954 | mr = 0063607 | doi-access = free }}{{cite journal | url = http://www.maths.manchester.ac.uk/~goran/skorokhod.pdf | title = Stochastic Differential Equations for Sticky Brownian Motion | first1 = H. J. | last1 = Engelbert | first2 = G. | last2 = Peskir | journal = Probab. Statist. Group Manchester Research Report | issue = 5 | year = 2012}}

• absorption or killed Brownian motion,{{Cite book | last1 = Chung | first1 = K. L. | last2 = Zhao | first2 = Z. | chapter = Killed Brownian Motion | doi = 10.1007/978-3-642-57856-4_2 | title = From Brownian Motion to Schrödinger's Equation | series = Grundlehren der mathematischen Wissenschaften | volume = 312 | pages = 31 | year = 1995 | isbn = 978-3-642-63381-2 }} 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.{{Cite book | last1 = Itō | first1 = K. | author-link1 = Kiyoshi Itō| last2 = McKean | first2 = H. P. | author-link2 = Henry McKean| doi = 10.1007/978-3-642-62025-6_6 | chapter = Time changes and killing | title = Diffusion Processes and their Sample Paths | url = https://archive.org/details/diffusionprocess00kito | url-access = limited | pages = [https://archive.org/details/diffusionprocess00kito/page/n182 164] | year = 1996 | isbn = 978-3-540-60629-1 }}

• Skorokhod problem

{{Reflist}}

{{Queueing theory}}

Category:Wiener process

Category:Articles with example MATLAB/Octave code