Having previously looked at Brownian bridges and excursions, I now turn to a third kind of process which can be constructed either as a conditioned Brownian motion or by extracting a segment from Brownian motion sample paths. Specifically, the *Brownian meander*, which is a Brownian motion conditioned to be positive over a unit time interval. Since this requires conditioning on a zero probability event, care must be taken. Instead, it is cleaner to start with an alternative definition by appropriately scaling a segment of a Brownian motion.

For a fixed positive times *T*, consider the last time *σ* before *T* at which a Brownian motion *X* is equal to zero,

(1) |

On interval [*σ*, *T*], the path of *X* will start from 0 and then be either strictly positive or strictly negative, and we may as well restrict to the positive case by taking absolute values. Scaling invariance says that *c*^{-1/2}*X*_{ct} is itself a standard Brownian motion for any positive constant *c*. So, scaling the path of *X* on [*σ*, 1] to the unit interval defines a process

(2) |

over 0 ≤ *t* ≤ 1; This starts from zero and is strictly positive at all other times.

Scaling invariance shows that the law of the process *B* does not depend on the choice of fixed time *T* The only remaining ambiguity is in the choice of the fixed time *T*.

Lemma 1The distribution ofBdefined by (2) does not depend on the choice of the timeT> 0.

*Proof:* Consider any other fixed positive time *T̃*, and use the construction above with *T̃*, *σ̃*, *B̃* in place of *T*, *σ*, *B* respectively. We need to show that *B̃* and *B* have the same distribution. Using the scaling factor *S* = *T̃*/*T*, then *X*′_{t} = *S*^{-1/2}*X*_{tS} is a standard Brownian motion. Also, *σ*′= *σ̃*/*S* is the last time before *T* at which *X*′ is zero. So,

has the same distribution as *B*. ⬜

This leads to the definition used here for Brownian meanders.

Definition 2A continuous process{B_{t}}_{t ∈ [0, 1]}is aBrownian meanderif and only it has the same distribution as (2) for a standard Brownian motionXand fixed timeT> 0.

In fact, there are various alternative — but equivalent — ways in which Brownian excursions can be defined and constructed.

- As a scaled segment of a Brownian motion before a time
*T*and after it last hits 0. This is definition 2. - As a Brownian motion conditioned on being positive. See theorem 4 below.
- As a segment of a Brownian excursion. See lemma 5.
- As the path of a standard Brownian motion starting from its minimum, in either the forwards or backwards direction. See theorem 6.
- As a Markov process with specified transition probabilities. See theorem 9 below.
- As a solution to an SDE. See theorem 12 below.