In mathematics, quadratic variation is used in the analysis of stochastic processes such as Brownian motion and other martingales.
is a real-valued stochastic process defined on a probability space
ranging over the non-negative real numbers.
Its quadratic variation is the process, written as
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 for every
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) of two processes
is The covariation may be written in terms of the quadratic variation by the polarization identity: Notation: the quadratic variation is also notated as
Such processes are very common including, in particular, all continuously differentiable functions.
This statement can be generalized to non-continuous processes.
Any càdlàg finite variation process
has quadratic variation equal to the sum of the squares of the jumps of
To state this more precisely, the left limit of
The quadratic variation of a standard Brownian motion
Any such process has quadratic variation given by 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 which can be used to compute
Alternatively this can be written as a stochastic differential equation: where
All càdlàg martingales, and local martingales have well defined quadratic variation, which follows from the fact that such processes are examples of semimartingales.
of a general locally square integrable martingale
is the unique right-continuous and increasing process starting at zero, with jumps
A useful result for square integrable martingales is the Itô isometry, which can be used to calculate the variance of Itô integrals, This result holds whenever
is a càdlàg square integrable martingale and
is a bounded predictable process, and is often used in the construction of the Itô integral.
This gives bounds for the maximum of a martingale in terms of the quadratic variation.
is a continuous local martingale, then the Burkholder–Davis–Gundy inequality holds for any
An alternative process, the predictable quadratic variation is sometimes used for locally square integrable martingales.
, and is defined to be the unique right-continuous and increasing predictable process starting at zero such that
Its existence follows from the Doob–Meyer decomposition theorem and, for continuous local martingales, it is the same as the quadratic variation.