Brownian bridge

A standard Brownian bridge is the centered Gaussian process $B(t)=W(t)-tW(1)$ on $[0,1]$. Its covariance is $\min (s,t)-st$ and it satisfies $B(0)=B(1)=0$.

For standard Brownian motion, the Brownian bridge $U_t=B_t-t{B}_1$, $0\le t\le 1$, is independent of $B_1$. Every finite vector of bridge coordinates is jointly multivariate normal with $B_1$, and its cross-covariances with $B_1$ vanish. The independence of uncorrelated jointly normal variables and independence extended from generating pi-systems prove independence of the whole stochastic process.

For a one-dimensional diffusion of variance rate $2D$ conditioned to start at $x>0$ and end at $y>0$ after time $\Delta t$, the bridge crosses zero with probability $\exp [-xy/(D\Delta t)]$. This corrects endpoint-only absorption tests in time-stepping schemes.