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.
- X is standard Brownian motion on the underlying filtered probability space.
- X is continuous and is a local martingale.
- X has quadratic variation .