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词条 Reflected Brownian motion
释义

  1. Definition

  2. Stability conditions

  3. Marginal and stationary distribution

     One dimension  Multiple dimensions 

  4. Simulation

     One dimension  Multiple dimensions 

  5. Other boundary conditions

  6. See also

  7. References

{{Use American English|date = January 2019}}{{Short description|Wiener process with reflecting spatial boundaries}}

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.[2]

RBMs have been shown to describe queueing models experiencing heavy traffic[2] as first proposed by Kingman[3] and proven by Iglehart and Whitt.[4][5]

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.[6]

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

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."[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."[7] In the special case where R is an M-matrix then necessary and sufficient conditions for stability are[7]

  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

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[8]

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[6]

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

  • Skorokhod problem

References

1. ^{{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 | pmid = | pmc = }}
2. ^{{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 | pmid = | pmc = }}
3. ^{{cite journal | last1 = Kingman | first1 = J. F. C. | authorlink1 = 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 }}
4. ^{{cite journal | last1 = Iglehart | first1 = Donald L. | last2 = Whitt | first2 = Ward | authorlink2 = 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}}
5. ^{{cite journal | last1 = Iglehart | first1 = Donald L. | last2 = Ward | first2 = Whitt | authorlink2 = 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 | accessdate = 30 Nov 2012 | url = http://www.columbia.edu/~ww2040/MultipleChannel1970II.pdf | doi=10.2307/1426324}}
6. ^1 {{Cite journal | last1 = Harrison | first1 = J. M. | authorlink1 = 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| pmid = | pmc = }}
7. ^1 2 3 {{Cite journal | last1 = Bramson | first1 = M. | last2 = Dai | first2 = J. G. | last3 = Harrison | first3 = J. M. | authorlink3 = 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| pmid = | pmc = | arxiv = 1009.5746 }}
8. ^1 2 {{cite book | title = Brownian Motion and Stochastic Flow Systems | first = J. Michael | last = Harrison | authorlink = J. Michael Harrison | year = 1985 | publisher = John Wiley & Sons | isbn = 978-0471819394 | url = http://faculty-gsb.stanford.edu/harrison/Documents/BrownianMotion-Stochasticms.pdf}}
9. ^{{Cite journal | last1 = Harrison | first1 = J. M. | authorlink1 = 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 | pmid = | pmc = | jstor = 2959751}}
10. ^{{Cite journal | last1 = Harrison | first1 = J. M. | authorlink1 = 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 | pmid = | pmc = }}
11. ^{{cite book| page = 202 | title = Handbook of Monte Carlo Methods | first1=Dirk P. | last1= Kroese | first2= Thomas |last2=Taimre|first3= Zdravko I.|last3= Botev | publisher = John Wiley & Sons |year = 2011 | isbn = 978-1118014950}}
12. ^{{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 | pmid = | pmc = }}
13. ^{{cite journal | last1 = Dai | first1 = Jim G. | last2 = Harrison | first2 = J. Michael | authorlink2 = 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}}
14. ^{{cite dissertation| 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 | url = http://www2.isye.gatech.edu/~dai/publications/dai90Dissertation.pdf | accessdate = 5 December 2012 | chapter = Section A.5 (code for BNET)}}
15. ^{{cite journal | last1 = Dai | first1 = J. G. | last2 = Harrison | first2 = J. M. | authorlink2 = 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 | format = | accessdate = | doi=10.1214/aoap/1177005771}}
16. ^1 2 3 4 {{Cite journal | last1 = Skorokhod | first1 = A. V. | authorlink1 = Anatoliy Skorokhod| doi = 10.1137/1107002 | title = Stochastic Equations for Diffusion Processes in a Bounded Region. II | journal = Theory of Probability & its Applications | volume = 7 | pages = 3–23| year = 1962 | pmid = | pmc = }}
17. ^{{Cite journal | last1 = Feller | first1 = W. | authorlink1 = 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 | pmid = | mr = 0063607 }}
18. ^{{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}}
19. ^{{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 | pmid = | pmc = }}
20. ^{{Cite book | last1 = Itō | first1 = K. | authorlink1 = Kiyoshi Itō| last2 = McKean | first2 = H. P. | authorlink2 = Henry McKean| doi = 10.1007/978-3-642-62025-6_6 | chapter = Time changes and killing | title = Diffusion Processes and their Sample Paths | pages = 164 | year = 1996 | isbn = 978-3-540-60629-1 | pmid = | pmc = }}
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1 : Wiener process
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