# Brownian motion

###### Definition.

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

1. 1.

$W(0)=0$ almost surely

2. 2.

The sample paths $t\mapsto W(t)$ are almost surely continuous.

3. 3.

For any finite sequence of times $t_{0}, 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. 4.

For any times $s, $W(t)-W(s)$ 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.

Title Brownian motion BrownianMotion 2013-03-22 15:12:46 2013-03-22 15:12:46 skubeedooo (5401) skubeedooo (5401) 16 skubeedooo (5401) Definition msc 60J65 Wiener process WienerMeasure StochasticCalculusAndSDE