Quadratic variation

In mathematics, quadratic variation is used in the analysis of such as Brownian motion and other martingales. Quadratic variation is just one kind of variation of a process.

Suppose that X_t is a real-valued stochastic process defined on a probability space ${\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )}$ and with time index t ranging over the non-negative real numbers. Its quadratic variation is the process, written as [X]_t, defined as

where P ranges over partitions of the interval [0,t] and the norm of the partition P is the mesh. This limit, if it exists, is defined using convergence in probability. Note that a process may be of finite quadratic variation in the sense of the definition given here and its paths be nonetheless almost surely of infinite quadratic variation for every t>0 in the classical sense of taking the supremum of the sum over all partitions; this is in particular the case for Brownian Motion.

More generally, the covariation (or cross-variance) of two processes X and Y is

The covariation may be written in terms of the quadratic variation by the polarization identity:

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