Quadratic variation
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In mathematics, quadratic variation that is particularly useful for the analysis of Brownian motion and martingales. Quadratic variation is just one kind of variation of a function (see function variation).
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[edit] Definition
The quadratic variation of a function f on the interval [0, T] is defined as
where P ranges over partitions of the interval [0,T] and the norm of the partition is the mesh. More generally, the quadratic covariation of two functions f and g on the interval [0,T] is
![[f,g]_T = \lim_{||P|| \to 0}\sum_{k=0}^{n-1}\left(f(t_{k+1})-f(t_k)\right)\left(g(t_{k+1})-g(t_k)\right).](../../../math/2/c/1/2c1a0d790d3a4e92e3e72426d528250a.png)
Many authors denote the quadratic variation of f by [f,f] instead of 
. The quadratic covariation may be written in terms of the quadratic variation by the polarization identity:
![[f,g]_t=\frac{1}{4}([f+g,f+g]+[f-g,f-g]).](../../../math/d/b/5/db56cd78be9d6057ac195cdc7dd99106.png)
[edit] Quadratic differentiability
[edit] Theorem
If f is differentiable, then 
[edit] Proof
Let P be the partition 
where ||P|| denotes the norm of the partition. Notice that |f'(t)| is continuous on a compact set [0, T] and therefore attains a maximum M. Then
where 
by the mean value theorem.
