quadratic variation

quadratic variation

The

quadratic variation

of a process is an

extension

of the notion of the

total variation

of a

function

, but rather than

summing

the

absolute values

of the changes of a function sampled at a

sequence

of times, the squares are summed.
This has important applications in stochastic calculus, appearing in the

integration by parts

and change of variables formulas for

stochastic integration

. The quadratic variation also has applications to the study of

martingales

, occuring in the Ito isometry and Burkholder-Davis-Gundy inequalities.

An important example is for a

Brownian motion

W

. In this case, the quadratic variation is

[

W

]

t

=

t

and, by Lévy’s

theorem

, this fact characterizes Brownian motion among all

local martingales

.
Quadratic variations are well defined for all cadlag martingales and, more generally, all

semimartingales

.

In standard, non-stochastic

calculus

, quadratic variation does not play a large role. This is because it is equal to zero for all

continuously differentiable

processes and, in fact, for all

continuous

finite variation processes

. Furthermore, it can be shown if any continuous

deterministic

process has a well defined quadratic variation along all

partitions

(

http://

planetmath

.org/Partition3

), then it is zero.

In the following definitions, we assume the

existence

of a

probability space

(

Ω

,

ℱ

,

ℙ

)

.

1

discrete time processes

Let

(

X

t

)

be a

stochastic process

, with time ranging over the

nonnegative integers

t

=

0

,

1

,

2

,

…

. Then, the quadratic variation

[

X

]

is the process

[

X

]

t

=

∑

s

=

1

t

(

X

s

-

X

s

-

1

)

2

.

Similarly, for processes

X

,

Y

, the

quadratic covariation

[

X

,

Y

]

is defined by

[

X

,

Y

]

t

=

∑

s

=

1

t

(

X

s

-

X

s

-

1

)

⁢

(

Y

s

-

Y

s

-

1

)

.

Note that

[

X

,

Y

]

is

bilinear

and

symmetric

in interchanging

X

and

Y

. The quadratic variation is alternatively given by

[

X

]

=

[

X

,

X

]

, and the covariation can be written in terms of the quadratic variation by the

polarization identity

,

[

X

,

Y

]

=

(

[

X

+

Y

]

-

[

X

-

Y

]

)

/

4

.

2

quadratic variation on an interval

Now suppose that

(

X

t

)

is a stochastic process with time

t

running over the

interval

[

0

,

T

]

, for some

T

>

0

. If

P

is a partition of the interval,

P

=

{

0

=

τ

0

≤

τ

1

≤

⋯

≤

τ

m

=

T

}

then we can define the quadratic variation and covariation along the partition

P

by

[

X

]

P

≡

∑

k

=

1

m

(

X

τ

k

-

X

τ

k

-

1

)

2

,

[

X

,

Y

]

P

≡

∑

k

=

1

m

(

X

τ

k

-

X

τ

k

-

1

)

⁢

(

Y

τ

k

-

Y

τ

k

-

1

)

.

The

mesh

of the partition is

|

P

|

=

max

k

⁡

(

τ

k

-

τ

k

-

1

)

. Given a sequence of partitions

P

n

with mesh going to zero as

n

→

∞

, the quadratic variation and covariation are defined by

[

X

]

T

≡

lim

n

→

∞

⁡

[

X

]

P

n

,

[

X

,

Y

]

T

≡

lim

n

→

∞

⁡

[

X

,

Y

]

P

n

.

It is only required that these limits exist under

convergence in probability

, which is much weaker than requiring convergence for all or almost all

ω

∈

Ω

.
The limits are not guaranteed to exist. If they do, then the quadratic variation and covariation are said to exist along the sequence of partitions

P

n

.

More generally,

random partitions

can be used where the times

τ

k

are

random variables

, usually

stopping times

. In this case, the processes

X

,

Y

are required to

satisfy

additional properties in order for their values at a random time to make sense, normally that they are right-continuous or cadlag. The mesh of each partition

P

n

is then a random variable, and is only required to tend to zero in

probability

as

n

→

∞

.

If the quadratic variation (respectively, covariation) exists along all such sequences of non-stochastic, or

deterministic

, partitions then it is easily shown to be

independent

of the sequence of partitions used. In this case, we simply say that the quadratic variation

[

X

]

T

(resp. covariation

[

X

,

Y

]

T

) exists without referring to the partitions used.

3

quadratic variation as a process

Suppose that

(

X

t

)

and

(

Y

t

)

are stochastic processes with time index running over the nonnegative

real numbers

,

t

∈

ℝ

+

. In this case, the quadratic variation and covariation can be defined as as above over the interval

[

0

,

t

]

, for each

t

>

0

.

However, in continuous-time, defining it in this way does not determine the

sample paths

of the processes

[

X

]

t

,

[

X

,

Y

]

t

. This is because these random variables are only defined

ℙ

-almost everywhere at each

t

, which does not say anything about their joint properties at uncountably many times. It is possible to remedy this by requiring that the quadratic variation and covariation be right-continuous. That is, we take a right-continuous version of the processes.

Alternatively, the limits can be taken simultaneously at all times. If

P

is a partition of

ℝ

+

,

P

=

{

0

=

τ

0

≤

τ

1

≤

τ

2

≤

⋯

↑

∞

}

then the quadratic variation and covariation along the partition

P

are the processes

[

X

]

t

P

≡

∑

k

=

1

∞

(

X

τ

k

∧

t

-

X

τ

k

-

1

∧

t

)

2

,

[

X

,

Y

]

t

P

≡

∑

k

=

1

∞

(

X

τ

k

∧

t

-

X

τ

k

-

1

∧

t

)

⁢

(

Y

τ

k

∧

t

-

Y

τ

k

-

1

∧

t

)

.

Since

τ

k

→

∞

, all but finitely many terms in these sums will be zero. As above, the partition

P

may be random, in which case

τ

k

are random variables.

Now suppose that

P

n

is a sequence of such partitions whose mesh over the bounded intervals

[

0

,

t

]

,

|

P

n

t

|

≡

max

k

⁡

(

τ

k

∧

t

-

τ

k

-

1

∧

t

)

tends to zero in probability as

n

→

∞

. Then the quadratic variation and covariation along

P

n

is defined by the limits

[

X

]

≡

lim

n

→

∞

⁡

[

X

]

P

n

,

[

X

,

Y

]

≡

lim

n

→

∞

⁡

[

X

,

Y

]

P

n

.

The limit here is usually required to exist under

ucp convergence

. As a

consequence

, quadratic variations of cadlag

adapted processes

are themselves cadlag and adapted, and quadratic variations of continuous processes are continuous.

If the quadratic variation or covariation exists along all such sequences of deterministic partitions then, as above, they are simply said to exist without making reference to the partitions used.

As squares of real numbers are

positive

, it follows that the quadratic variation is an increasing process and, by the polarization identity, if

[

X

]

,

[

Y

]

,

[

X

,

Y

]

all exist then the quadratic covariation is a

difference

of increasing functions, and so is a finite variation process.

The quadratic variation and covariation are sometimes written as

[

X

]

t

=

∫

0

t

(

d

⁢

X

)

2

,

[

X

,

Y

]

t

=

∫

0

t

𝑑

X

⁢

𝑑

Y

,

and in

differential

notation,

d

⁢

[

X

]

t

=

(

d

⁢

X

t

)

2

,

d

⁢

[

X

,

Y

]

t

=

d

⁢

X

t

⁢

d

⁢

Y

t

.

If we write

Δ

⁢

X

t

≡

X

t

-

X

t

-

for the jump of a

cadlag process

at time

t

then, as a consequence of ucp convergence,

Δ

⁢

[

X

]

t

=

(

Δ

⁢

X

t

)

2

,

Δ

⁢

[

X

,

Y

]

t

=

Δ

⁢

X

t

⁢

Δ

⁢

Y

t

.

These

identities

hold simultaneously for all times

t

>

0

, with probability one.

Title

quadratic variation

Canonical name

QuadraticVariation

Date of creation

2013-03-22 18:41:12

Last modified on

2013-03-22 18:41:12

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

4

Author

gel (22282)

Entry type

Definition

Classification

msc 60G07

Classification

msc 60H10

Classification

msc 60H20

Related topic

QuadraticVariationOfASemimartingale

Related topic

QuadraticVariationOfBrownianMotion

Defines

quadratic covariation

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