Quadratic variation
Encyclopedia
In mathematics
, quadratic variation is used in the analysis of stochastic process
es such as Brownian motion
and martingale
s. Quadratic variation is just one kind of variation
of a process.
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 interval
[0,t] 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
:
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
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 Xt with respect to t is denoted by Xt-, and the jump of X at time t can be written as ΔXt = Xt - Xt-. 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 Vt(X) is the variation of X over [0,t].
By the continuity of X, this vanishes in the limit as
goes to zero.
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
s can be shown to exist. They form an important part of the theory of stochastic calculus, appearing in Itō's lemma
, 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, and local martingale
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] = ΔM2, and such that M2 − [M] is a local martingale.
A useful result for square integrable martingales is the Itō isometry
, 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 process
, 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 Mt* ≡sups≤t|Ms|, and any real number p > 0, the inequality is
Here, cp < Cp 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 M2 − <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.
Mathematics
Mathematics 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 process
Stochastic process
In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...
es such as Brownian motion
Wiener process
In 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 martingale
Martingale (probability theory)
In 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 variation
Total variation
In 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 Xt is a real-valued stochastic process defined on a probability spaceProbability space
In 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 aswhere P ranges over partitions of the interval
Partition of an interval
In 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 probability
Convergence of random variables
In 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 Motion
Brownian motion
Brownian 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 identity
Polarization identity
In 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 variationBounded variation
In 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 càdlàg
Càdlàg
In 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 Xt with respect to t is denoted by Xt-, and the jump of X at time t can be written as ΔXt = Xt - Xt-. 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 Vt(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 motionWiener process
In 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 semimartingaleSemimartingale
In 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 lemma
Ito's lemma
In 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 càdlàgCàdlàg
In 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 martingale
Local martingale
In 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] = ΔM2, and such that M2 − [M] is a local martingale.
A useful result for square integrable martingales is the Itō isometry
Ito isometry
In 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 process
Predictable process
In 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 Mt* ≡sups≤t|Ms|, and any real number p > 0, the inequality is
Here, cp < Cp 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 M2 − <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.