A Brownian bridge can be defined as standard Brownian motion conditioned on hitting zero at a fixed future time T, or as any continuous process with the same distribution as this. Rather than conditioning, a slightly easier approach is to subtract a linear term from the Brownian motion, chosen such that the resulting process hits zero at the time T. This is equivalent, but has the added benefit of being independent of the original Brownian motion at all later times.
Lemma 1 Let X be a standard Brownian motion and be a fixed time. Then, the process
over is independent from .
Proof: As the processes are joint normal, it is sufficient that there is zero covariance between them. So, for times , we just need to show that is zero. Using the covariance structure we obtain,
as required. ⬜
This leads us to the definition of a Brownian bridge.
Definition 2 A continuous process is a Brownian bridge on the interval if and only it has the same distribution as for a standard Brownian motion X.
In case that , then B is called a standard Brownian bridge.
There are actually many different ways in which Brownian bridges can be defined, which all lead to the same result.
- As a Brownian motion minus a linear term so that it hits zero at T. This is definition 2.
- As a Brownian motion X scaled as . See lemma 9 below.
- As a joint normal process with prescribed covariances. See lemma 7 below.
- As a Brownian motion conditioned on hitting zero at T. See lemma 14 below.
- As a Brownian motion restricted to the times before it last hits zero before a fixed positive time T, and rescaled to fit a fixed time interval. See lemma 15 below.
- As a Markov process. See lemma 13 below.
- As a solution to a stochastic differential equation with drift term forcing it to hit zero at T. See lemma 18 below.
There are other constructions beyond these, such as in terms of limits of random walks, although I will not cover those in this post. Continue reading “Brownian Bridges”