Ito’s lemma

Ito’s lemma

Itô’s lemma, also known as

Itô’s

formula

, is an

extension

of the

chain rule

(

http://planetmath.org/ChainRuleSeveralVariables

) to the

stochastic integral

, and is often regarded as one of the most important results of

stochastic calculus

. The case described here applies to arbitrary

continuous

semimartingales

. For the application to Itô processes see Itô’s formula (

http://planetmath.org/ItosFormula

) or see the generalized Itô formula (

http://planetmath.org/GeneralizedItoFormula

) for noncontinuous processes.

For a

function

f

on a subset of

ℝ

n

, we write

f

,

i

for the

partial derivative

with respect to the

i

’th coordinate and

f

,

i

j

for the second order derivatives.

Theorem

(Itô).

Suppose that

X

=

(

X

1

,

…

,

X

n

)

is a continuous semimartingale taking values in an

open subset

U

of

R

n

and

f

:

U

→

R

is twice

continuously differentiable

. Then,

d

⁢

f

⁢

(

X

)

=

∑

i

=

1

n

f

,

i

⁢

(

X

)

⁢

d

⁢

X

i

+

1

2

⁢

∑

i

,

j

=

1

n

f

,

i

j

⁢

(

X

)

⁢

d

⁢

[

X

i

,

X

j

]

.

(1)

In particular, for a continuous real-valued semimartingale

X

, (

1

) becomes

d

⁢

f

⁢

(

X

)

=

f

′

⁢

(

X

)

⁢

d

⁢

X

+

1

2

⁢

f

′′

⁢

(

X

)

⁢

d

⁢

[

X

]

,

which is a form of the “change of variables formula” for stochastic calculus.
A major distinction between standard and stochastic calculus is that here we need to include the

quadratic variation

and covariation terms

[

X

]

and

[

X

i

,

X

j

]

.

Equation (

1

) results from taking a

Taylor expansion

up to

second order

which, setting

δ

⁢

f

⁢

(

x

)

≡

f

⁢

(

x

+

δ

⁢

x

)

-

f

⁢

(

x

)

, reads

δ

⁢

f

⁢

(

x

)

=

∑

i

=

1

n

f

,

i

⁢

(

x

)

⁢

δ

⁢

x

i

+

1

2

⁢

∑

i

,

j

=

1

n

f

,

i

j

⁢

(

x

)

⁢

δ

⁢

x

i

⁢

δ

⁢

x

j

+

o

⁢

(

δ

⁢

x

2

)

.

(2)

Taking the limit as

δ

⁢

x

goes to zero, all of the terms on the right hand side of (

2

), other than the first, go to zero with order (

http://planetmath.org/LandauNotation

)

O

⁢

(

δ

⁢

x

2

)

and, therefore, can be neglected in the limit. This results in the standard chain rule. However, when

δ

⁢

X

=

X

t

+

h

-

X

t

for a semimartingale

X

then the second order terms in (

2

) only go to zero at rate

O

⁢

(

h

)

and, therefore, must be retained even in the limit as

h

→

0

. This is a

consequence

of semimartingales, such as

Brownian motion

, being nowhere

differentiable

.
In fact, if

X

is a

finite variation process

, then it can be shown that the quadratic covariation terms are zero, and the standard chain rule results.

A consequence of Itô’s lemma is that if

X

is a continuous semimartingale and

f

is twice continuously differentiable, then

f

⁢

(

X

)

will be a semimartingale. However, the generalized Itô formula shows that it is not necessary to restrict this statement to continuous processes.

Title

Ito’s lemma

Canonical name

ItosLemma

Date of creation

2013-03-22 18:41:44

Last modified on

2013-03-22 18:41:44

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

5

Author

gel (22282)

Entry type

Theorem

Classification

msc 60H10

Classification

msc 60G07

Classification

msc 60H05

Synonym

Itô’s lemma

Synonym

Itö’s lemma

Synonym

Ito’s formula

Synonym

Itô’s formula

Synonym

Itö’s formula

Related topic

ItosFormula

Related topic

GeneralizedItoFormula

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