Brownian bridge

Definition and basic formulae

On the canonical interval [0,1], if W_t is standard Brownian motion, the Brownian bridge B_t (pinned at 0 at time 1) can be written as B_t = W_t - t W_1. It is a mean-zero Gaussian process with covariance

• Cov(B_s,B_t) = min(s,t) - s t, for s,t in [0,1],
• Var(B_t) = t(1-t), which vanishes at the endpoints 0 and 1.

Properties and representations

The Brownian bridge is continuous, Gaussian, and time-inhomogeneous. It is non-stationary because variance depends on time, but it is a (time-inhomogeneous) Markov process. There are several useful constructions: conditioning a Brownian motion on its terminal value; representing the bridge by subtracting a linear correction from the motion; and series expansions (Karhunen–Loève) that express the bridge as an infinite sum of orthogonal sine functions with independent normal coefficients.

History and theoretical role

Brownian bridges arise naturally in limit theorems. In empirical process theory, suitably scaled cumulative sum processes converge to a Brownian bridge (Donsker's theorem), which underpins the distribution theory of goodness-of-fit statistics such as the Kolmogorov–Smirnov test. In diffusion theory the term "pinned" or "conditioned" diffusion refers to the same idea of fixing endpoints.

Applications and examples

Practical uses include: (1) sampling paths constrained to meet boundary conditions in Monte Carlo algorithms, (2) modeling deviations of cumulative empirical distributions in statistics, and (3) constructing bridge-type processes in financial and physical models where an endpoint is enforced. Bridges are also used to compute conditional hitting probabilities and to reduce variance in path-dependent simulations.

Distinctions and notable facts

Unlike Brownian motion, the bridge returns to a prescribed endpoint and has variance zero at that time. Generalizations include bridges between arbitrary endpoints and bridges of other Markov processes. For further technical background and proofs of the identities above consult standard texts on Gaussian processes and stochastic calculus.

See also basic introductions to stochastic processes and Brownian motion for broader context.