Brownian motion

Definition.

One-dimensional Brownian motion is a stochastic process $W(t)$ , defined for $t\\in[0,\\infty)$ such that

1.
$W(0)=0$ almost surely
2.
The sample paths $t\\mapsto W(t)$ are almost surely continuous.
3.
For any finite sequence of times $t_{0}<t_{1}<\\cdots<t_{n}$ ,the increments
$W(t_{1})-W(t_{0}),W(t_{2})-W(t_{1}),\\ldots,W(t_{n})-W(t_{n-1})$
are independent.
4.
For any times $s<t$ , $W(t)-W(s)$ is normally distributed with mean zeroand variance $t-s$ .

Definition.

A $d$ -dimensional Brownian motion is a stochastic process $W(t)=(W_{1}(t),\\dots,W_{d}(t))$ in $\\mathbb{R}^{d}$ whose coordinate processes $W_{i}(t)$ areindependent one-dimensional Brownian motions.

Figure 1: Sample paths of a standard Brownian motion

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