Brownian bridge

${\displaystyle B_t:=(W_t\mid W_T=0),t\in [0,T]}$

The expected value of the bridge at any ${\displaystyle t}$ in the interval ${\displaystyle [0,T]}$ is zero, with variance ${\displaystyle \frac{t(T-t)}{T}}$, 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 ${\displaystyle min(s,t)-\frac{st}{T}}$, or ${\displaystyle \frac{s(T-t)}{T}}$ if ${\displaystyle s<t}$. 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 $t\ge 0$, $W(t)$ is normally distributed with expected value $0$ and variance $t$, and the increments are stationary and independent), then

${\displaystyle B(t)=W(t)-\frac{t}{T}W(T)}$

is a Brownian bridge for $t\in [0,T]$. It is independent of $W(T)$

Conversely, if $B(t)$ is a Brownian bridge for $t\in [0,1]$ and $Z$ is a standard normal random variable independent of $B$, then the process

${\displaystyle W(t)=B(t)+tZ}$

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

${\displaystyle W(t)=\sqrt{T}B\left(\frac{t}{T}\right)+\frac{t}{\sqrt{T}}Z.}$

Another representation of the Brownian bridge based on the Brownian motion is, for $t\in [0,T]$

${\displaystyle B(t)=\frac{T-t}{\sqrt{T}}W\left(\frac{t}{T-t}\right).}$

Conversely, for $t\in [0,\mathrm{\infty }]$

${\displaystyle W(t)=\frac{T+t}{T}B\left(\frac{Tt}{T+t}\right).}$

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

${\displaystyle B_t=\sum _{k=1}^{\mathrm{\infty }}{Z}_k\frac{\sqrt{2T}\sin (k\pi t/T)}{k\pi }}$

where ${\displaystyle Z_1,Z_2,\dots }$ 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 ${\displaystyle K=\underset{t\in [0,1]}{sup}|B(t)|}$, for a Brownian bridge with ${\displaystyle T=1}$; then the cumulative distribution function of $K$ is given by $${\displaystyle \mathrm{Pr}(K\le x)=1-2\sum _{k=1}^{\mathrm{\infty }}(-1{)}^{k-1}{e}^{-2k^2{x}^2}=\frac{\sqrt{2\pi }}{x}\sum _{k=1}^{\mathrm{\infty }}{e}^{-(2k-1{)}^2{\pi }^2/(8x^2)}.}$$

Decomposition by zero-crossings

The Brownian bridge can be "split" by finding the last zero ${\displaystyle {\tau }_{-}}$ before the midpoint, and the first zero ${\displaystyle {\tau }_{+}}$ after, forming a (scaled) bridge over ${\displaystyle [0,{\tau }_{-}]}$, an excursion over ${\displaystyle [{\tau }_{-},{\tau }_{+}]}$, and another bridge over ${\displaystyle [{\tau }_{+},1]}$. The joint pdf of ${\displaystyle {\tau }_{-},{\tau }_{+}}$ is given by

${\displaystyle \rho \left({\tau }_{-},{\tau }_{+}\right)=\frac{1}{2\pi \sqrt{{\tau }_{-}(1-{\tau }_{+})({\tau }_{+}-{\tau }_{-}{)}^3}}}$

which can be conditionally sampled as

${\displaystyle {\tau }_{+}=\frac{1}{1+{\sin }^2\left(\frac{\pi }{2}{U}_1\right)}\in \left(\frac{1}{2},1\right)}$

${\displaystyle {\tau }_{-}=\frac{U_2^2{\tau }_{+}}{2{\tau }_{+}+U_2^2-1}\in \left(0,\frac{1}{2}\right)}$

where ${\displaystyle U_1,U_2}$ are uniformly distributed random variables over (0,1).