Reflected Brownian motion
{{Use American English|date = January 2019}}
{{Short description|Wiener process with reflecting spatial boundaries}}
In probability theory, reflected Brownian motion (or regulated Brownian motion,{{Cite book | last1 = Dieker | first1 = A. B. | chapter = Reflected Brownian Motion | doi = 10.1002/9780470400531.eorms0711 | title = Wiley Encyclopedia of Operations Research and Management Science | year = 2011 | isbn = 9780470400531 }} both with the acronym RBM) is a Wiener process in a space with reflecting boundaries.{{Cite journal | last1 = Veestraeten | first1 = D. | title = The Conditional Probability Density Function for a Reflected Brownian Motion | doi = 10.1023/B:CSEM.0000049491.13935.af | journal = Computational Economics | volume = 24 | issue = 2 | pages = 185–207 | year = 2004 | s2cid = 121673717 }} 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.{{Cite journal|last=Faucheux|first=Luc P.|last2=Libchaber|first2=Albert J.|date=1994-06-01|title=Confined Brownian motion|url=https://link.aps.org/doi/10.1103/PhysRevE.49.5158|journal=Physical Review E|language=en|volume=49|issue=6|pages=5158–5163|doi=10.1103/PhysRevE.49.5158|issn=1063-651X}}
RBMs have been shown to describe queueing models experiencing heavy traffic as first proposed by Kingman{{cite journal | last1 = Kingman | first1 = J. F. C. | author-link1 = John Kingman | year = 1962 | title = On Queues in Heavy Traffic | journal = Journal of the Royal Statistical Society. Series B (Methodological) | volume = 24 | issue = 2 | pages = 383–392 |jstor=2984229| doi = 10.1111/j.2517-6161.1962.tb00465.x }} and proven by Iglehart and Whitt.{{cite journal | last1 = Iglehart | first1 = Donald L. | last2 = Whitt | first2 = Ward | author-link2 = Ward Whitt | year = 1970 | title = Multiple Channel Queues in Heavy Traffic. I | journal = Advances in Applied Probability | volume = 2 | issue = 1 | pages = 150–177 | jstor = 3518347 | doi=10.2307/3518347| s2cid = 202104090 }}{{cite journal | last1 = Iglehart | first1 = Donald L. | last2 = Ward | first2 = Whitt | author-link2 = Ward Whitt | year = 1970 | title = Multiple Channel Queues in Heavy Traffic. II: Sequences, Networks, and Batches | journal = Advances in Applied Probability | volume = 2 | issue = 2 | pages = 355–369 | jstor = 1426324 | access-date = 30 Nov 2012 | url = http://www.columbia.edu/~ww2040/MultipleChannel1970II.pdf | doi=10.2307/1426324| s2cid = 120281300 }}
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.{{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
::
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."
The process Yj is the local time of the process on the corresponding section of the boundary.
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."{{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
- 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{{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,
::
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,{{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 }}
::
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.
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.{{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 }}
=Multiple dimensions=
[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 }}
Other boundary conditions
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 }}
See also
References
{{Reflist}}
{{Queueing theory}}