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| In [[probability theory]], '''reflected Brownian motion''' (or '''regulated Brownian motion''',<ref>{{cite doi|10.1002/9780470400531.eorms0711}}</ref><ref name="harrison-book" /> both with the acronym '''RBM''') is a [[Wiener process]] in a space with reflecting boundaries.<ref>{{cite doi|10.1023/B:CSEM.0000049491.13935.af}}</ref>
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| RBMs have been shown to describe [[queueing model]]s experiencing [[heavy traffic approximation|heavy traffic]]<ref name="harrison-book" /> as first proposed by [[John Kingman|Kingman]]<ref>{{cite jstor|2984229}}</ref> and proven by Iglehart and [[Ward Whitt|Whitt]].<ref>{{cite jstor|3518347}}</ref><ref>{{cite jstor|1426324}}</ref>
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| ==Definition==
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| A ''d''–dimensional reflected Brownian motion ''Z'' is a [[stochastic process]] on <math>\scriptstyle \mathbb R^d_+</math> uniquely defined by
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| * a ''d''–dimensional drift vector ''μ''
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| * a ''d''×''d'' non-singular covariance matrix ''Σ'' and
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| * a ''d''×''d'' reflection matrix ''R''.<ref name="open">{{cite doi|10.1080/17442508708833469}}</ref>
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| where ''X''(''t'') is an unconstrained [[Brownian motion]] and<ref name="Bramson" />
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| ::<math>Z(t) = X(t) + R Y(t)</math>
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| with ''Y''(''t'') a ''d''–dimensional vector where
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| * ''Y'' is continuous and non–decreasing with ''Y''(0) = 0
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| * ''Y''<sub>''j''</sub> only increases at times for which ''Z''<sub>''j''</sub> = 0 for ''j'' = 1,2,...,''d''
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| * ''Z''(''t'') ∈ S, t ≥ 0.
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| The reflection matrix describes boundary behaviour. In the interior of <math>\scriptstyle \mathbb R^d_+</math> the process behaves like a [[Wiener process]], on the boundary "roughly speaking, ''Z'' is pushed in direction ''R''<sup>''j''</sup> whenever the boundary surface <math>\scriptstyle \{ z \in \mathbb R^d_+ : z_j=0\}</math> is hit, where ''R''<sup>''j''</sup> is the ''j''th column of the matrix ''R''."<ref name="Bramson" />
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| ==Stability conditions==
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| 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."<ref name="Bramson">{{cite doi|10.1214/09-AAP631}}</ref> In the special case where ''R'' is an [[M-matrix]] then necessary and sufficient conditions for stability are<ref name="Bramson" />
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| # ''R'' is a [[non-singular matrix]] and
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| # ''R''<sup>−1</sup>''μ'' < 0.
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| ==Stationary distribution==
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| ===One dimension===
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| The 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 ''σ''<sup>2</sup> is
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| ::<math>\mathbb P(Z \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)</math>
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| 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]]<ref name="harrison-book">{{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 = 0471819395 | url = http://faculty-gsb.stanford.edu/harrison/Documents/BrownianMotion-Stochasticms.pdf}}</ref>
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| ::<math>\mathbb P(Z<z) = 1-e^{2\mu z/\sigma^2}.</math>
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| ===Multiple dimensions===
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| The stationary distribution of a reflected Brownian motion in multiple dimensions is tractable analytically when there is a [[product form stationary distribution]],<ref>{{cite doi|10.1214/aoap/1177005704}}</ref> which occurs when the process is stable and<ref>{{cite doi|10.1137/0141030}}</ref>
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| ::<math>2 \Sigma = RD + DR'</math>
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| where ''D'' = [[diagonal matrix|diag]](''Σ''). In this case the [[probability density function]] is<ref name="open" />
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| ::<math>p(z_1,z_2,\ldots,z_d) = \prod_{k=1}^d \eta_k e^{-\eta_k z_k}</math>
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| where ''η''<sub>''k''</sub> = 2''μ''<sub>''k''</sub>''γ''<sub>''k''</sub>/''Σ''<sub>''kk''</sub> and ''γ'' = ''R''<sup>−1</sup>''μ''. [[Closed-form expression]]s for situations where the product form condition does not hold can be computed numerically as described below in the simulation section.
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| ==Hitting times==
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| ===One dimension===
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| Write ''T''(''y'') for the first time a one dimensional RBM starting at 0 reaches the level ''y''. Then<ref name="harrison-book" />
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| ::<math>\mathbb P(T(y)>t) = \Phi\left(\frac{y-\mu t}{\sigma t^{1/2}}\right) -e^{2\mu y/\sigma^2}\Phi\left(\frac{-y-\mu t}{\sigma t^{1/2}}\right).</math>
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| ==Simulation== | |
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| ===One dimension===
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| In one dimension the simulated process is the [[absolute value]] of a [[Wiener process]]. The following [[MATLAB]] program creates a sample path.<ref>{{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 = 1118014952}}</ref>
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| <source lang="matlab"> | |
| %rbm.m
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| n=10^4; h=10^(-3); t=h.*(0:n); mu=-1;
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| X=zeros(1,n+1); M=X; B=X;
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| B(1)=3; X(1)=3;
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| for k=2:n+1
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| Y=sqrt(h)*randn; U=rand(1);
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| B(k)=B(k-1)+mu*h-Y;
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| M=(Y + sqrt(Y^2-2*h*log(U)))/2;
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| X(k)=max(M-Y,X(k-1)+h*mu-Y);
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| end
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| subplot(2,1,1)
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| plot(t,X,'k-');
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| subplot(2,1,2)
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| plot(t,X-B,'k-');
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| </source>
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| The error involved in discrete simulations has been quantified.<ref>{{cite doi|10.1214/aoap/1177004597}}</ref> An exact simulation method for time-dependent RBM is also proposed.<ref>{{cite journal|last1=Mousavi|first1=Mohammad|first2=Peter W.|last2= Glynn|title=Exact Simulation of Non-stationary Reflected Brownian Motion|journal=arXiv preprint|arxiv=1312.6456|year=2013|month=Dec}}</ref>
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| ===Multiple dimensions===
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| [http://www2.isye.gatech.edu/~dai/QNET/ QNET] allows simulation of steady state RBMs.<ref>{{cite jstor|2959623}}</ref><ref>{{cite journal| 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)}}</ref><ref>{{cite jstor| 2959654}}</ref> | |
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| ==Other boundary conditions==
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| Feller described possible boundary condition for the process<ref name="skorokhod">{{cite doi|10.1137/1107002}}</ref><ref>{{cite doi|10.1090/S0002-9947-1954-0063607-6}}</ref><ref>{{cite journal | url = http://www.maths.manchester.ac.uk/~goran/skorokhod.pdf | title = Stochastic Differential Equations
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| for Sticky Brownian Motion | first1 = H. J. | last1 = Engelbert | first2 = G. | last2 = Peskir | journal = Probab. Statist. Group Manchester Research Report | issue = 5 | year = 2012}}</ref>
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| * absorption<ref name="skorokhod" /> or killed Brownian motion,<ref>{{cite doi|10.1007/978-3-642-57856-4_2}}</ref> a [[Dirichlet boundary condition]]
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| * instantaneous reflection,<ref name="skorokhod" /> as described above a [[Neumann boundary condition]]
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| * elastic reflection, a [[Robin boundary condition]]
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| * delayed reflection<ref name="skorokhod" /> (the time spent on the boundary is positive with probability one)
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| * partial reflection<ref name="skorokhod" /> where the process is either immediately reflected or is absorbed
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| * sticky Brownian motion.<ref>{{cite doi|10.1007/978-3-642-62025-6_6}}</ref>
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| ==See also==
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| * [[Skorokhod problem]]
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| ==References==
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| {{Reflist}}
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| {{Queueing theory}}
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| [[Category:Stochastic processes]]
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| [[Category:Variants of random walks]]
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