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:
“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)
“It’s a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was “mine.””
—Jane Adams (20th century)
“... if, as women, we accept a philosophy of history that asserts that women are by definition assimilated into the male universal, that we can understand our past through a male lens—if we are unaware that women even have a history—we live our lives similarly unanchored, drifting in response to a veering wind of myth and bias.”
—Adrienne Rich (b. 1929)