Definition
Suppose that Xt is a real-valued stochastic process defined on a probability space and with time index t ranging over the non-negative real numbers. Its quadratic variation is the process, written as t, defined as
where P ranges over partitions of the interval 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 a.s. 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 quadratic covariation (or quadratic cross-variance) of two processes X and Y is
The quadratic covariation may be written in terms of the quadratic variation by the polarization identity:
Read more about this topic: Quadratic Variation
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