# Brownian bridge

A **Brownian bridge** is a continuous-time stochastic process *B*(*t*) whose probability distribution is the conditional probability distribution of a 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 at the origin at both *t=0* and *t=T*. More precisely:

The expected value of the bridge is zero, with variance , 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*(T − *t*)/T if *s* < *t*.
The increments in a Brownian bridge are not independent.

## Relation to other stochastic processes[edit]

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

is a Brownian bridge for *t* ∈ [0, T]. It is independent of *W*(T)^{[1]}

Conversely, if *B*(*t*) is a Brownian bridge and *Z* is a standard normal random variable independent of *B*, then the process

is a Wiener process for *t* ∈ [0, 1]. More generally, a Wiener process *W*(*t*) for *t* ∈ [0, *T*] can be decomposed into

Another representation of the Brownian bridge based on the Brownian motion is, for *t* ∈ [0, T]

Conversely, for *t* ∈ [0, ∞]

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.

## Intuitive remarks[edit]

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*(*t*_{1}) = *a* and *B*(*t*_{2}) = *b* where *t*_{1}, *t*_{2}, *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).

## General case[edit]

For the general case when *B*(*t*_{1}) = *a* and *B*(*t*_{2}) = *b*, the distribution of *B* at time *t* ∈ (*t*_{1}, *t*_{2}) is normal, with mean

and the covariance between *B*(*s*) and *B*(*t*), with *s* < *t* is

## References[edit]

**^**Aspects of Brownian motion, Springer, 2008, R. Mansuy, M. Yor page 2

- Glasserman, Paul (2004).
*Monte Carlo Methods in Financial Engineering*. New York: Springer-Verlag. ISBN 0-387-00451-3. - Revuz, Daniel; Yor, Marc (1999).
*Continuous Martingales and Brownian Motion*(2nd ed.). New York: Springer-Verlag. ISBN 3-540-57622-3.