Brownian bridge

A Brownian bridge is a continuous-time gaussian process B(t) whose probability distribution is the conditional probability distribution of a standard Wiener process W(t) (a mathematical model of Brownian motion) subject to the condition (when standardized) that W(T) = 0, so that the process is pinned to the same value at both t = 0 and t = T. More precisely: The expected value of the bridge at any t in the interval [0,T] is zero, with variance

, implying that the most uncertainty is in the middle of the bridge, with zero uncertainty at the nodes.

, or s(T − t)/T if s < t. The increments in a Brownian bridge are not independent.

is a standard Wiener process (i.e., for

is normally distributed with expected value

, and the increments are stationary and independent), then is a Brownian bridge for

is a Brownian bridge for

is a standard normal random variable independent of

, then the process is a Wiener process for

More generally, a Wiener process

can be decomposed into Another representation of the Brownian bridge based on the Brownian motion is, for

The Brownian bridge may also be represented as a Fourier series with stochastic coefficients, as where

are independent identically distributed standard normal random variables (see the Karhunen–Loève theorem).

A Brownian bridge is the result of Donsker's theorem in the area of empirical processes.

It is also used in the Kolmogorov–Smirnov test in the area of statistical inference.

, then the cumulative distribution function of

A standard Wiener process satisfies W(0) = 0 and is therefore "tied down" to the origin, but other points are not restricted.

In a Brownian bridge process on the other hand, not only is B(0) = 0 but we also require that B(T) = 0, that is the process is "tied down" at t = T as well.

Just as a literal bridge is supported by pylons at both ends, a Brownian Bridge is required to satisfy conditions at both ends of the interval [0,T].

(In a slight generalization, one sometimes requires B(t1) = a and B(t2) = b where t1, t2, a and b are known constants.)

Suppose we have generated a number of points W(0), W(1), W(2), W(3), etc.

of a Wiener process path by computer simulation.

It is now desired to fill in additional points in the interval [0,T], that is to interpolate between the already generated points W(0) and W(T).

The solution is to use a Brownian bridge that is required to go through the values W(0) and W(T).

For the general case when W(t1) = a and W(t2) = b, the distribution of B at time t ∈ (t1, t2) is normal, with mean and variance and the covariance between B(s) and B(t), with s < t is