Reflected Brownian Motion

Core Idea

Reflected Brownian motion (RBM) is a process constrained to stay non-negative by an instantaneous reflection at the origin. It is constructed as |W(t)| (for a standard Brownian motion W) or equivalently as the solution to the Skorokhod reflection problem: X(t) = W(t) + L(t) ≥ 0, where L(t) is the local time at zero — a continuous, non-decreasing process that increases only when X(t) = 0. The local time measures the "time spent at the boundary" in a generalized sense and is central to the study of stochastic processes with reflecting barriers, queueing theory, and free-boundary problems.

Explainer

Reflected Brownian motion (RBM) is Brownian motion constrained to a domain — most commonly the half-line [0,∞) — by instantaneous elastic reflection at the boundary. The simplest construction takes a standard Brownian motion W(t) and defines X(t) = |W(t)|. By Lévy's theorem, this is equivalent to X(t) = W(t) + L_t^0(W), where L_t^0(W) is the local time of W at zero — a continuous, non-decreasing process that increases only on the (Lebesgue-null) set of times when W(t) = 0. The local time "pushes" the process away from zero just enough to maintain non-negativity.

The Skorokhod reflection problem provides the general framework. Given a continuous function w(t) with w(0) ≥ 0, find a pair (x, l) such that x(t) = w(t) + l(t) ≥ 0, where l is non-decreasing with l(0) = 0 and l increases only when x(t) = 0 (formally, ∫₀^∞ x(t) dl(t) = 0). The unique solution is x(t) = w(t) - min(0, inf_{s ≤ t} w(s)) and l(t) = -min(0, inf_{s ≤ t} w(s)) = max_{s ≤ t}(-w(s))⁺. When w is a Brownian path, x is reflected Brownian motion and l is the local time at zero. The Skorokhod map w ↦ x is continuous in the sup-norm topology, which makes it a powerful tool for proving weak convergence of reflected processes (if w_n → w, then the reflected processes converge too).

Local time is one of the most subtle objects in stochastic calculus. The local time L_t^a(W) at level a measures how much time Brownian motion spends near a up to time t — but "time spent at a point" has Lebesgue measure zero for a continuous process, so local time captures something finer. It is defined through the occupation times formula: ∫₀ᵗ g(W(s))ds = ∫_{-∞}^∞ g(a) L_t^a da. The local time is the Radon-Nikodym derivative of the occupation measure (time spent in each region) with respect to Lebesgue measure. As a function of a, L_t^a(W) is a.s. continuous (jointly in t and a for Brownian motion). As a function of t, L_t^a is non-decreasing and grows on the (topologically large but measure-zero) set of times when W visits a.

Tanaka's formula connects local time to stochastic calculus: |W(t)| = ∫₀ᵗ sgn(W(s)) dW(s) + L_t^0(W). This is the Itô formula applied to the non-smooth function f(x) = |x|. The local time term replaces the ½f''(x)dt correction that would appear for smooth f — since |x|'' = 2δ₀(x) in the distributional sense, the correction is ½ · 2 · "δ₀(W(t))dt" = dL_t^0. Tanaka's formula generalizes to convex functions (the Itô-Tanaka formula) and to semimartingales, making local time a bridge between non-smooth analysis and stochastic calculus.

Reflected Brownian motion has deep applications in queueing theory and mathematical finance. In heavy-traffic queueing, the workload in a GI/GI/1 queue converges (after diffusion scaling) to RBM with drift — the non-negativity of the queue length forces the reflection, and the local time represents cumulative server idle time. Harrison's program extends this to queueing networks, where multidimensional RBM in the positive orthant models interacting queues with reflection directions determined by the routing structure. In finance, RBM appears in models with regulated prices (currency bands, interest rate floors) and in the study of barriers and knockouts in exotic option pricing, where the boundary behavior of diffusions at barriers determines payoff structures.