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:

${\displaystyle B_{t}:=(W_{t}\mid W_{T}=0),\;t\in [0,T]}$

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 ${\displaystyle \min(s,t)-{\frac {s\,t}{T}}}$, or ${\displaystyle {\frac {s(T-t)}{T}}}$ if ${\displaystyle s<t}$. The increments in a Brownian bridge are not independent.