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
Famous quotes containing the word definition:
“The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.”
—Ralph Waldo Emerson (1803–1882)
“It is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possess—after many mysteries—what one loves.”
—François, Duc De La Rochefoucauld (1613–1680)
“No man, not even a doctor, ever gives any other definition of what a nurse should be than this—”devoted and obedient.” This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.”
—Florence Nightingale (1820–1910)