Brownian bridge

From Wikipedia, the free encyclopedia

This article or section does not adequately cite its references or sources.
Please help improve this article by adding citations to reliable sources. (help, get involved!)
This article has been tagged since August 2006.

Brownian motion, pinned at both ends. This uses a Brownian bridge.

A Brownian bridge is a continuous-time stochastic process whose probability distribution is the conditional probability distribution of a Wiener process B(t) (a mathematical model of Brownian motion) given the condition that B(0) = B(1) = 0. Equivalently, if W(t) is a standard Wiener process (i.e., for t ≥ 0, W(t) is normally distributed with expected value 0 and variance t, and the increments are stationary and independent), then W(t) − t W(1) is a Brownian bridge. The increments in a Brownian bridge are clearly not independent.

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(1) = 0, that is the process is "tied down" at t = 1 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 [0,1]. (In a slight generalization, one sometimes requires B(T₁) = a and B(T₂) = b where T₁, T₂, a and b are known constants).

A Brownian bridge is useful in several situations. First, 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 [0,1], that is to interpolate between the already generated points W(0) and W(1). The solution is to use a Brownian bridge that is required to go through the values W(0) and W(1). Furthermore, a Brownian bridge is the result of Donsker's theorem in the area of empirical processes.

There is also a natural decomposition of a Brownian motion B(t) for t ∈ [0, n] into

$B(t) = W_n(t) + \frac{t}{\sqrt{n}} Z$

where W_n(t) is a Brownian bridge over time interval [0, n] and Z is a standard Gaussian random variable.

The expected value of the bridge is zero, with variance t(1 − t), implying that the most uncertainty is in the middle of the bridge, with zero uncertainty at the nodes. The covariance of B(s) and B(t) is s(1 − t) if s < t.

For the general case when W(T₁) = a and W(T₂) = b, the distribution of W at time T ∈ (T₁, T₂) is normal, with mean