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.^{[dubious — see talk page]} 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.

1 Definition
2 Quadratic differentiability
- 2.1 Theorem
- 2.2 Proof

[edit] Definition

The quadratic variation of a function f on the interval [0, T] is defined as

$\langle f\rangle_T = \lim_{||P|| \to 0}\sum_{k=0}^{n-1}\left(f(t_{k+1})-f(t_k)\right) ^ 2.$

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

$[f,g]_T = \lim_{||P|| \to 0}\sum_{k=0}^{n-1}\left(f(t_{k+1})-f(t_k)\right)\left(g(t_{k+1})-g(t_k)\right).$

Many authors denote the quadratic variation of f by [f,f] instead of $\langle f\rangle$ . The quadratic covariation may be written in terms of the quadratic variation by the polarization identity:

$[f,g]_t=\frac{1}{4}([f+g,f+g]+[f-g,f-g]).$

[edit] Quadratic differentiability

[edit] Theorem

If f is differentiable, then $\langle f\rangle (T) = 0.$ ^{[dubious — see talk page]}

[edit] Proof

Let P be the partition $0 = t_0 < t_1 < \cdots < t_n = T$ where ||P|| denotes the norm of the partition. Notice that |f'(t)| is continuous^{[dubious — see talk page]} on a compact set [0, T] and therefore attains a maximum M. Then

$\begin{align} \lim_{||P|| \to 0}\sum_{k=0}^{n-1}|f(t_{k+1})-f(t_k)|^2 & {}=\lim_{||P|| \to 0}\sum_{k=0}^{n-1}(f'(t_k^*))^2|t_{k+1} - t_k|^2 \\ & {} \le\lim_{||P|| \to 0}\sum_{k=0}^{n-1}M^2|t_{k+1} - t_k| ||P|| \\ &{} \le M^2 T \lim_{||P|| \to 0}||P||\,=\,0 \end{align}$