Wikipedia / Quadratic variation
Wikipedia / Quadratic variation
In mathematics, quadratic variation is used in the analysis of stochastic processes such as Brownian motion and other martingaless. Quadratic variation is just one kind of variation of a process.
Definition
is a real-valued stochastic process defined on a probability space, defined as
ranges over partitions of the interval 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 1-variation 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 is
The covariation may be written in terms of the quadratic variation by the polarization identity:
.
Finite variation processes
is said to have finite variation if it has bounded variation over every finite time interval (with probability 1). Such processes are very common including, in particular, all continuously differentiable functions. The quadratic variation exists for all continuous finite variation processes, and is zero.
This statement can be generalized to non-continuous processes. Any càdlàg. Then, the quadratic variation is given by
.
goes to zero.
Itô processes
The quadratic variation of a standard Brownian motion sense, but not pathwise. This generalizes to Itô processes that, by definition, can be expressed in terms of Itô integrals
is a Brownian motion. Any such process has quadratic variation given by
Semimartingales
Quadratic variations and covariations of all semimartingales can be shown to exist. They form an important part of the theory of stochastic calculus, appearing in Itô's lemma, which is the generalization of the chain rule to the Itô integral. The quadratic covariation also appears in the integration by parts formula
.
Alternatively this can be written as a stochastic differential equation:
Martingales
All càdlàg martingales, and local martingales is the unique right-continuous and increasing process (without using stochastic calculus) is given in Karandikar–Rao (2014).
A useful result for square integrable martingales is the Itô isometry, which can be used to calculate the variance of Itô integrals,
is a bounded predictable process, and is often used in the construction of the Itô integral.
Another important result is the Burkholder–Davis–Gundy inequality, named for Donald Burkholder, Burgess Davis, and Richard Floyd Gundy, the inequality is
.
An alternative process, the predictable quadratic variation is a local martingale. Its existence follows from the Doob–Meyer decomposition theorem and, for continuous local martingales, it is the same as the quadratic variation.