# 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:
    $B_{t}:=(W_{t}\mid W_{T}=0),\;t\in [0,T]$
The expected value of the bridge at any t in the interval [0,T] 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 $\min(s,t)-{\frac {s\,t}{T}}$, or s(T − t)/T if s < t. The increments in a Brownian bridge are not independent.

## Relation to other stochastic processes

If W(t){\textstyle W(t)} is a standard Wiener process (i.e., for t≥0{\textstyle t\geq 0}, W(t){\textstyle W(t)} is normally distributed with expected value 0{\textstyle 0} and variance t{\textstyle 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]{\textstyle t\in [0, T]}. It is independent of W(T){\textstyle W(T)}
Conversely, if B(t){\textstyle B(t)} is a Brownian bridge for t∈[0, 1]{\textstyle t\in [0, 1]} and Z{\textstyle Z} is a standard normal random variable independent of B{\textstyle B}, then the process
    $W(t)=B(t)+tZ\,$
is a Wiener process for t∈[0, 1]{\textstyle t\in [0, 1]}. More generally, a Wiener process W(t){\textstyle W(t)} for t∈[0, T]{\textstyle t\in [0, T]} can be decomposed into
    $W(t)={\sqrt {T}}B\left({\frac {t}{T}}\right)+{\frac {t}{\sqrt {T}}}Z.$
Another representation of the Brownian bridge based on the Brownian motion is, for t∈[0, T]{\textstyle t\in [0, T]}
    $B(t)={\frac {T-t}{\sqrt {T}}}W\left({\frac {t}{T-t}}\right).$
Conversely, for t∈[0, ∞]{\textstyle t\in [0, \infty ]}
    $W(t)={\frac {T+t}{T}}B\left({\frac {Tt}{T+t}}\right).$
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.
Let $K=\sup _{t\in [0,1]}|B(t)|$, then the cumulative distribution function of K{\textstyle K} is given by$\operatorname {Pr} (K\leq x)=1-2\sum _{k=1}^{\infty }(-1)^{k-1}e^{-2k^{2}x^{2}}={\frac {\sqrt {2\pi }}{x}}\sum _{k=1}^{\infty }e^{-(2k-1)^{2}\pi ^{2}/(8x^{2})}.$

## 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(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

For the general case when W(t$_1$) = a and W(t$_2$) = b, the distribution of B at time t ∈ (t$_1$, t$_2$) is normal, with mean
    $a+{\frac {t-t_{1}}{t_{2}-t_{1}}}(b-a)$
and variance
    ${\frac {(t_{2}-t)(t-t_{1})}{t_{2}-t_{1}}},$
and the covariance between B(s) and B(t), with s < t is
    ${\frac {(t_{2}-t)(s-t_{1})}{t_{2}-t_{1}}}.$