quadratic variation of Brownian motion


Let (Wt)tR+ be a standard Brownian motionMathworldPlanetmath. Then, its quadratic variation exists and is given by


As Brownian motion is a martingaleMathworldPlanetmath and, in particular, is a semimartingale then its quadratic variation must exist (http://planetmath.org/QuadraticVariationOfASemimartingale). We just need to compute its value along a sequence of partitionsMathworldPlanetmath.

If P={0=t0t1tm=t} is a partition (http://planetmath.org/SubintervalPartition) of the intervalMathworldPlanetmath [0,t], then the quadratic variation on P is


Using the property that the increments Wtk-Wtk-1 are independentPlanetmathPlanetmath normal random variables with mean zero and varianceMathworldPlanetmath tk-tk-1, the mean and variance of [W]P are

𝔼[[W]P] =k=1m𝔼[(Wtk-Wtk-1)2]=k=1m(tk-tk-1)=t,
Var[[W]P] =k=1mVar[(Wtk-Wtk-1)2]=k=1m2(tk-tk-1)2

Here, |P|=maxk(tk-tk-1) is the mesh of the partition. If (Pn)n=1,2, is a sequence of partitions of [0,t] with mesh going to zero as n then,


as n. This shows that [W]Pnt in the L2 norm and, in particular, converges in probability. So, [W]t=t.

Title quadratic variation of Brownian motion
Canonical name QuadraticVariationOfBrownianMotion
Date of creation 2013-03-22 18:41:25
Last modified on 2013-03-22 18:41:25
Owner gel (22282)
Last modified by gel (22282)
Numerical id 6
Author gel (22282)
Entry type Theorem
Classification msc 60H10
Classification msc 60J65
Related topic QuadraticVariation