5.3 The Brownian Bridge
Definition 5.3.1 (Brownian Bridge).label Let $(\Omega, \cf, \bp)$ be a probability space, $a \in \real^{d}$, and $X: \Omega \to C([0, 1]; \real^{d})$ be a Gaussian process, then $X$ is a Brownian bridge from $0$ to $a$ such that:
- (1)
For each $t \in [0, 1]$, $\ev(X_{t}) = at$.
- (2)
For each $s, t \in [0, 1]$, $\text{Cov}(X_{s}, X_{t}) = (s \wedge t - st)I$.
Theorem 5.3.2 ([Theorem 40.3, RW89]).label Let $a \in \real^{d}$, then the following are equivalent definitions of the distribution of the Brownian bridge from $0$ to $a$.
- (1)
There exists a Brownian motion $B$ such that for each $t \in [0, 1]$,
\[X_{t} = B_{t} + t(a - B_{1})\] - (2)
There exists a Brownian motion $B$ such that for each $t \in [0, 1]$,
\[X_{t} = at + (1 - t)B_{t/(1-t)}\] - (3)
$X$ is a continuous process such that
\[Y_{t} = X_{t} - at + \int_{0}^{1} \frac{X_{s} - as}{1 - s}ds\]is a Brownian motion.
- (4)
The distribution of $X$ is the $h$-transform of the classical Wiener measure on $C([0, 1]; \real^{d})$, where
\[h(t, x) = \frac{1}{[2\pi(1 - t)]^{d/2}}\exp\braks{-\frac{\norm{x - a}_{\real^d}^{2}}{2(1 - t)}}\]