Stocks & Commodities V. 25:2 (44-48): Sidebar: Brownian motion and fractional Brownian motion by Radha Panini

Stocks & Commodities V. 25:2 (44-48): Sidebar: Brownian motion and fractional Brownian motion by Radha Panini

BROWNIAN MOTION AND FRACTIONAL BROWNIAN MOTION

Brownian motion: A random process {Wt: t > 0} is a standard Brownian motion if:

1. For each t> 0 and s>0, Wt+s – Wt is normally distributed with mean 0 and variance s

2. Wt+s – Wt is independent of Wt

3. Wt is a continuous function of time and W0 = 0.

An important property of a Brownian motion is that the covariance of Wt is given by:

E(WtWs) = min{s,t}