# 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:

$B_{t}:=(W_{t}\mid W_{T}=0),\;t\in [0,T]$ The expected value of the bridge is zero, with variance $\textstyle {\frac {t(T-t)}{T}}$ , 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(T − t)/T if s < t. The increments in a Brownian bridge are not independent.

## Relation to other stochastic processes

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

$B(t)=W(t)-{\frac {t}{T}}W(T)\,$ is a Brownian bridge for t ∈ [0, T]. It is independent of W(T)

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

$W(t)=B(t)+tZ\,$ is a Wiener process for t ∈ [0, 1]. More generally, a Wiener process W(t) for t ∈ [0, T] can be decomposed into

$W(t)=B\left({\frac {t}{T}}\right)+{\frac {t}{\sqrt {T}}}Z.$ Another representation of the Brownian bridge based on the Brownian motion is, for t ∈ [0, T]

$B(t)={\frac {(T-t)}{\sqrt {T}}}W\left({\frac {t}{T-t}}\right).$ Conversely, for t ∈ [0, ∞]

$W(t)={\frac {T+t}{T}}B\left({\frac {Tt}{T+t}}\right).$ The Brownian bridge may also be represented as a Fourier series with stochastic coefficients, as

$B_{t}=\sum _{k=1}^{\infty }Z_{k}{\frac {{\sqrt {2T}}\sin(k\pi t/T)}{k\pi }}$ where $Z_{1},Z_{2},\ldots$ 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

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).

## General case

For the general case when B(t1) = a and B(t2) = b, the distribution of B at time t ∈ (t1t2) is normal, with mean

$a+{\frac {t-t_{1}}{t_{2}-t_{1}}}(b-a)$ and the covariance between B(s) and B(t), with s < t is

${\frac {(t_{2}-t)(s-t_{1})}{t_{2}-t_{1}}}.$ 