quadratic variation of Brownian motion
As Brownian motion is a martingale 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 partitions.
If is a partition (http://planetmath.org/SubintervalPartition) of the interval , then the quadratic variation on is
Here, is the mesh of the partition. If is a sequence of partitions of with mesh going to zero as then,
as . This shows that in the norm and, in particular, converges in probability. So, .
|Title||quadratic variation of Brownian motion|
|Date of creation||2013-03-22 18:41:25|
|Last modified on||2013-03-22 18:41:25|
|Last modified by||gel (22282)|