Brownian motion is by far the most important stochastic process. It is the archetype of Gaussian processes, of continuous time martingales, and of Markov processes. It is basic to the study of stochastic differential equations, financial mathematics, and filtering, to name only a few of its applications.

In this chapter we define Brownian motion and consider some of its elementary aspects. Later chapters will take up the construction of Brownian motion and properties of Brownian motion paths.

Definition and basic properties

Let (Ω, ℱ, ℙ) be a probability space and let {ℱt} be a filtration, not necessarily satisfying the usual conditions.

Definition 2.1Wt = Wt(ω) is a one-dimensional Brownian motion with respect to {ℱt} and the probability measure ℙ, started at 0, if

1. (1)Wt is ℱt measurable for each t ≥ 0.

2. (2)W0 = 0, a.s.

3. (3)Wt − Ws is a normal random variable with mean 0 and variance t − s whenever s < t.

4. (4)Wt − Ws is independent of ℱs whenever s < t.

5. (5)Wt has continuous paths.

If instead of (2) we have W0 = x, we say we have a Brownian motion started at x. Definition 2.1(4) is referred to as the independent increments property of Brownian motion. The fact that Wt1 – Ws has the same law as Wt–s, which follows from Definition 2.1(3), is called the stationary increments property. When no filtration is specified, we assume the filtration is the filtration generated by W, i.e., ℱt = σ (Ws; s ≤ t). Sometimes a one-dimensional Brownian motion started at 0 is called a standard Brownian motion.