quadratic variation of Brownian motion

Theorem.

Let ${\mathrm{(}{W}_t\mathrm{)}}_{t\mathrm{\in }{\mathrm{R}}_{\mathrm{+}}}$ be a standard Brownian motion. Then, its quadratic variation exists and is given by

$${[W]}_t=t.$$

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 $P=\{0=t_0\le t_1\le \mathrm{\cdots }\le t_m=t\}$ is a partition (http://planetmath.org/SubintervalPartition) of the interval $[0,t]$, then the quadratic variation on $P$ is

$${[W]}^P=\sum _{k=1}^m{(W_{t_k}-W_{t_{k-1}})}^2.$$

Using the property that the increments $W_{t_k}-W_{t_{k-1}}$ are independent normal random variables with mean zero and variance $t_k-t_{k-1}$, the mean and variance of ${[W]}^P$ are

$𝔼\left[{[W]}^P\right]$$={\displaystyle \sum _{k=1}^m}𝔼\left[{(W_{t_k}-W_{t_{k-1}})}^2\right]={\displaystyle \sum _{k=1}^m}(t_k-t_{k-1})=t,$

$\mathrm{Var}\left[{[W]}^P\right]$$={\displaystyle \sum _{k=1}^m}\mathrm{Var}\left[{(W_{t_k}-W_{t_{k-1}})}^2\right]={\displaystyle \sum _{k=1}^m}2{(t_k-t_{k-1})}^2$

$\le 2|P|{\displaystyle \sum _{k=1}^m}(t_k-t_{k-1})=2|P|t.$

Here, $|P|={\max }_k(t_k-t_{k-1})$ is the mesh of the partition.If ${(P_n)}_{n=1,2,\mathrm{\dots }}$ is a sequence of partitions of $[0,t]$ with mesh going to zero as $n\to \mathrm{\infty }$ then,

$$𝔼\left[{({[W]}^{P_n}-t)}^2\right]\le 2|P_n|t\to 0$$

as $n\to \mathrm{\infty }$. This shows that ${[W]}^{P_n}\to t$ in the $L^2$ norm and, in particular, converges in probability. So, ${[W]}_t=t$.