Quadratic variation
From Wikipedia, the free encyclopedia
In mathematics, quadratic variation is a concept in real analysis that gives an alternative to testing for differentiability. It is particularly useful for the analysis of Brownian motion and martingales. Quadratic variation is just one kind of variation of a function, please see Function variation for more information.
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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
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where 
by the Mean Value Theorem.
