Brownian bridge
A Brownian bridge is a continuous-time B(t) whose probability distribution is the conditional probability distribution of a Wiener process W(t) (a mathematical model of Brownian motion) subject to the condition that W(T) = 0, so that the process is pinned at the origin at both t=0 and t=T. More precisely:
The expected value of the bridge is zero, with variance , 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 s(T − t)/T if s < t. The increments in a Brownian bridge are not independent.
If W(t) is a standard Wiener process (i.e., for t ≥ 0, W(t) is normally distributed with expected value 0 and variance t, and the increments are stationary and independent), then
is a Brownian bridge for t ∈ [0, T]. It is independent of W(T)
Conversely, if B(t) is a Brownian bridge and Z is a standard normal random variable independent of B, then the process
is a Wiener process for t ∈ [0, T]. More generally, a Wiener process W(t) for t ∈ [0, T] can be decomposed into
Another representation of the Brownian bridge based on the Brownian motion is, for t ∈ [0, T]
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