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 definition of good prose is proper words in their proper places; of good verse, the most proper words in their proper places. The propriety is in either case relative. The words in prose ought to express the intended meaning, and no more; if they attract attention to themselves, it is, in general, a fault.”
—Samuel Taylor Coleridge (1772–1834)
“Beauty, like all other qualities presented to human experience, is relative; and the definition of it becomes unmeaning and useless in proportion to its abstractness. To define beauty not in the most abstract, but in the most concrete terms possible, not to find a universal formula for it, but the formula which expresses most adequately this or that special manifestation of it, is the aim of the true student of aesthetics.”
—Walter Pater (1839–1894)
“Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
—The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)