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 zero and 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 $W_{i}(t)$ are independent one-dimensional Brownian motions.

Figure 1: Sample paths of a standard Brownian motion

Title	Brownian motion
Canonical name	BrownianMotion
Date of creation	2013-03-22 15:12:46
Last modified on	2013-03-22 15:12:46
Owner	skubeedooo (5401)
Last modified by	skubeedooo (5401)
Numerical id	16
Author	skubeedooo (5401)
Entry type	Definition
Classification	msc 60J65
Synonym	Wiener process
Related topic	WienerMeasure
Related topic	StochasticCalculusAndSDE