Reflected Brownian motion

A d–dimensional reflected Brownian motion Z is a stochastic process on \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. 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 • Yj only increases at times for which Zj = 0 for j = 1,2,...,d • Z(t) ∈ \mathbb R^d_+, t ≥ 0. The reflection matrix describes boundary behaviour. In the interior of \scriptstyle \mathbb R^d_+ the process behaves like a Wiener process; on the boundary "roughly speaking, Z is pushed in direction Rj whenever the boundary surface \scriptstyle \{ z \in \mathbb R^d_+ : z_j=0\} 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." 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 ::\mathbb P(Z(t) \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) for all t ≥ 0, (with Φ the cumulative distribution function of the normal distribution) which yields (for μ \mathbb P(Z For fixed t, the distribution of Z(t) coincides with the distribution of the running maximum M(t) of the Brownian motion, ::Z(t) \sim M(t)=\sup_{s\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)=\frac{e^{-((x-u)/a)^2/2}+e^{-((x+u-2p_b)/a)^2/2}}{a(2\pi)^{1/2}} For the plane above x \ge p_b Multiple dimensions The stationary distribution of a reflected Brownian motion in multiple dimensions is tractable analytically when there is a product form stationary distribution, which occurs when the process is stable and ::2 \Sigma = RD + DR' where D = diag(Σ). In this case the probability density function is ::p(z_1,z_2,\ldots,z_d) = \prod_{k=1}^d \eta_k e^{-\eta_k z_k} where ηk = 2''μ'k'γ''k/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. % 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. Multiple dimensions QNET allows simulation of steady state RBMs. ==Other boundary conditions==

Feller described possible boundary condition for the process • absorption a Dirichlet boundary condition • instantaneous reflection, ==See also==