Simulating a Brownian excursion from a Brownian bridge

Let's consider \(B_t \) a standard Brownian motion in \([0,1]\). We can think of the process \[B_t-tB_1 \text{ for } t\in[0,1],\] which is called a Brownian bridge, as a Brownian motion conditioned to be \(1\) at time \(t=1\). As one can imagine, if we are able to simulate Brownian motion (which we are), then a Brownian bridge is simple to simulate.

What happens if we further want to condition a Brownian motion to stay positive in \([0,1]\)?

This new process, called a Brownian excursion is not so simple to simulate. Here we show one possible way, sometimes called the Vervaat Transform, to transform a Brownian bridge into a Brownian excursion.

To get a Brownian excursion from a given Brownian bridge,

1. Find its (almost surely unique) minimum.
2. Cyclically shift to the minimum

Equivalently, and this is what we simulate here,

1. Find its (almost surely unique) minimum.
2. Swap the parts before and after the minimum.
3. Translate vertically.

The details of why this transformation is indeed a Brownian excursion can be found in Vervaat's paper from 1979: A relation between Brownian bridge and Brownian excursion.

This transformation is useful, for example, to easily see that the range (max-min) of a Brownian bridge has the same distribution as the maximum of a Brownian excursion.