In

mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

,

**quadratic variation** is used in the analysis of

stochastic processIn probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...

es such as

Brownian motionIn mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener. It is often called standard Brownian motion, after Robert Brown...

and

martingaleIn probability theory, a martingale is a model of a fair game where no knowledge of past events can help to predict future winnings. In particular, a martingale is a sequence of random variables for which, at a particular time in the realized sequence, the expectation of the next value in the...

s. Quadratic variation is just one kind of

variationIn mathematics, the total variation identifies several slightly different concepts, related to the structure of the codomain of a function or a measure...

of a process.

## Definition

Suppose that

*X*_{t} is a real-valued stochastic process defined on a

probability spaceIn probability theory, a probability space or a probability triple is a mathematical construct that models a real-world process consisting of states that occur randomly. A probability space is constructed with a specific kind of situation or experiment in mind...

and with time index

*t* ranging over the non-negative real numbers. Its quadratic variation is the process, written as [

*X*]

_{t}, defined as

where

*P* ranges over

partitions of the intervalIn mathematics, a partition, P of an interval [a, b] on the real line is a finite sequence of the formIn mathematics, a partition, P of an interval [a, b] on the real line is a finite sequence of the form...

[0,

*t*] and the norm of the partition

*P* is the mesh. This limit, if it exists, is defined using

convergence in probabilityIn probability theory, there exist several different notions of convergence of random variables. The convergence of sequences of random variables to some limit random variable is an important concept in probability theory, and its applications to statistics and stochastic processes...

. 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 MotionBrownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications...

.

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 identityIn mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space. Let \|x\| \, denote the norm of vector x and \langle x, \ y \rangle \, the inner product of vectors x and y...

:

## Finite variation processes

A process

*X* is said to have

*finite variation* if it has

bounded variationIn mathematical analysis, a function of bounded variation, also known as a BV function, is a real-valued function whose total variation is bounded : the graph of a function having this property is well behaved in a precise sense...

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

càdlàgIn mathematics, a càdlàg , RCLL , or corlol function is a function defined on the real numbers that is everywhere right-continuous and has left limits everywhere...

finite variation process

*X* has quadratic variation equal to the sum of the squares of the jumps of

*X*. To state this more precisely, the left limit of

*X*_{t} with respect to

*t* is denoted by

*X*_{t-}, and the jump of

*X* at time

*t* can be written as Δ

*X*_{t} =

*X*_{t} -

*X*_{t-}. Then, the quadratic variation is given by

The proof that continuous finite variation processes have zero quadratic variation follows from the following inequality. Here,

*P* is a partition of the interval [0,

*t*], and

*V*_{t}(

*X*) is the variation of

*X* over [0,

*t*].

By the continuity of

*X*, this vanishes in the limit as

goes to zero.

## Itō processes

The quadratic variation of a standard

Brownian motionIn mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener. It is often called standard Brownian motion, after Robert Brown...

*B* exists, and is given by [

*B*]

_{t} =

*t*. This generalizes to Itō processes which, by definition, can be expressed in terms of Itō integrals

where

*B* is a Brownian motion. Any such process has quadratic variation given by

## Semimartingales

Quadratic variations and covariations of all

semimartingaleIn probability theory, a real valued process X is called a semimartingale if it can be decomposed as the sum of a local martingale and an adapted finite-variation process....

s can be shown to exist. They form an important part of the theory of stochastic calculus, appearing in

Itō's lemmaIn mathematics, Itō's lemma is used in Itō stochastic calculus to find the differential of a function of a particular type of stochastic process. It is named after its discoverer, Kiyoshi Itō...

, which is the generalization of the chain rule to the Itō integral. The quadratic covariation also appears in the integration by parts formula

which can be used to compute [

*X*,

*Y*].

Alternatively this can be written as a Stochastic Differential Equation:

where

## Martingales

All

càdlàgIn mathematics, a càdlàg , RCLL , or corlol function is a function defined on the real numbers that is everywhere right-continuous and has left limits everywhere...

martingales, and

local martingaleIn mathematics, a local martingale is a type of stochastic process, satisfying the localized version of the martingale property. Every martingale is a local martingale; every bounded local martingale is a martingale; however, in general a local martingale is not a martingale, because its...

s have well defined quadratic variation, which follows from the fact that such processes are examples of semimartingales.

It can be shown that the quadratic variation [

*M*] of a general local martingale

*M* is the unique right-continuous and increasing process starting at zero, with jumps Δ[

*M*] = Δ

*M*^{2}, and such that

*M*^{2} − [

*M*] is a local martingale.

A useful result for square integrable martingales is the

Itō isometryIn mathematics, the Itō isometry, named after Kiyoshi Itō, is a crucial fact about Itō stochastic integrals. One of its main applications is to enable the computation of variances for stochastic processes....

, which can be used to calculate the variance of Ito integrals,

This result holds whenever

*M* is a càdlàg square integrable martingale and

*H* is a bounded

predictable processIn stochastic analysis, a part of the mathematical theory of probability, a predictable process is a stochastic process which the value is knowable at a prior time...

, and is often used in the construction of the Itō integral.

Another important result is the

**Burkholder–Davis–Gundy inequality**. This gives bounds for the maximum of a martingale in terms of the quadratic variation. For a local martingale

*M* starting at zero, with maximum denoted by

*M*_{t}^{*} ≡sup

_{s≤t}|

*M*_{s}|, and any real number

*p* > 0, the inequality is

Here,

*c*_{p} <

*C*_{p} are constants depending on the choice of

*p*, but not depending on the martingale

*M* or time

*t* used. If

*M* is a continuous local martingale, then the Burkholder–Davis–Gundy inequality holds for any positive value of

*p*.

An alternative process, the

**predictable quadratic variation** is sometimes used for locally square integrable martingales. This is written as <

*M*>

_{t}, and is defined to be the unique right-continuous and increasing predictable process starting at zero such that

*M*^{2} − <

*M*> 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.