Standard Brownian motion, , is defined to be a real-valued process satisfying the following properties.

- .
- is normally distributed with mean 0 and variance
*t*–*s*independently of , for any . -
*B*has continuous sample paths.

As always, it only really matters is that these properties hold *almost surely*. Now, to apply the techniques of stochastic calculus, it is assumed that there is an underlying filtered probability space , which necessitates a further definition; a process *B* is a Brownian motion on a filtered probability space if in addition to the above properties it is also adapted, so that is -measurable, and is independent of for each . Note that the above condition that is independent of is not explicitly required, as it also follows from the independence from . According to these definitions, a process is a Brownian motion if and only if it is a Brownian motion with respect to its natural filtration.

The property that has zero mean independently of means that Brownian motion is a martingale. Furthermore, we previously calculated its quadratic variation as . An incredibly useful result is that the converse statement holds. That is, Brownian motion is the *only* local martingale with this quadratic variation. This is known as *Lévy’s characterization*, and shows that Brownian motion is a particularly general stochastic process, justifying its ubiquitous influence on the study of continuous-time stochastic processes.

Theorem 1 (Lévy’s Characterization of Brownian Motion)LetXbe a local martingale with . Then, the following are equivalent.

Xis standard Brownian motion on the underlying filtered probability space.Xis continuous and is a local martingale.Xhas quadratic variation .

Continue reading “Lévy’s Characterization of Brownian Motion”