Itô’s formula

Itô’s formula

0.1

Case of single space dimension

Let

X

t

be an Itô process satisfying the

stochastic
differential equation

d

⁢

X

t

=

μ

t

⁢

d

⁢

t

+

σ

t

⁢

d

⁢

W

t

,

with

μ

t

and

σ

t

being

adapted processes

,
adapted to the same filtration as the

Brownian motion

W

t

.
Let

f

be a function with

continuous

partial derivatives

∂

⁡

f

∂

⁡

t

,

∂

⁡

f

∂

⁡

x

and

∂

2

⁡

f

∂

⁡

x

2

.

Then

Y

t

=

f

⁢

(

X

t

)

is also an Itô process, and its stochastic
differential equation
is

d

⁢

Y

t

=

∂

⁡

f

∂

⁡

t

⁢

d

⁢

t

+

∂

⁡

f

∂

⁡

x

⁢

d

⁢

X

t

+

1

2

⁢

∂

2

⁡

f

∂

⁡

x

2

⁢

(

d

⁢

X

t

)

⁢

(

d

⁢

X

t

)

=

(

∂

⁡

f

∂

⁡

t

+

∂

⁡

f

∂

⁡

x

⁢

μ

t

+

1

2

⁢

σ

t

2

)

⁢

d

⁢

t

+

∂

⁡

f

∂

⁡

x

⁢

σ

t

⁢

d

⁢

W

t

,

where all partial derivatives are to be taken at

(

t

,

X

t

)

.

0.2

Case of multiple space dimensions

There is also an analogue for multiple space

dimensions

.

Let

X

t

be a

ℝ

n

-valued Itô process satisfying the stochastic
differential equation

d

⁢

X

t

=

μ

t

⁢

d

⁢

t

+

σ

t

⁢

d

⁢

W

t

,

with

μ

t

and

σ

t

being adapted processes,
adapted to the same filtration as the

m

-dimensional Brownian motion

W

t

.

μ

t

is

ℝ

n

-valued and

σ

t

is

L

⁢

(

ℝ

m

,

ℝ

n

)

-valued.

Let

f

:

ℝ

n

×

ℝ

→

ℝ

be a function with
continuous partial derivatives.

Then

Y

t

=

f

⁢

(

X

t

)

is also an Itô process, and its stochastic
differential equation
is

d

⁢

Y

t

=

∂

⁡

f

∂

⁡

t

⁢

d

⁢

t

+

(

D

⁡

f

)

⁢

d

⁢

X

t

+

1

2

⁢

d

⁢

X

t

*

⁢

(

D

2

⁡

f

)

⁢

d

⁢

X

t

=

∂

⁡

f

∂

⁡

t

⁢

d

⁢

t

+

(

D

⁡

f

)

⁢

μ

t

⁢

d

⁢

t

+

(

D

⁡

f

)

⁢

σ

t

⁢

d

⁢

W

t

+

1

2

⁢

d

⁢

W

t

*

⁢

σ

t

*

⁢

(

D

2

⁡

f

)

⁢

σ

t

⁢

d

⁢

W

t

=

∂

⁡

f

∂

⁡

t

⁢

d

⁢

t

+

(

D

⁡

f

)

⁢

μ

t

⁢

d

⁢

t

+

(

D

⁡

f

)

⁢

σ

t

⁢

d

⁢

W

t

+

1

2

⁢

tr

⁡

(

σ

t

*

⁢

(

D

2

⁡

f

)

⁢

σ

t

)

⁢

d

⁢

t

=

(

∂

⁡

f

∂

⁡

t

+

(

D

⁡

f

)

⁢

μ

t

+

1

2

⁢

tr

⁡

(

(

σ

t

⁢

σ

t

*

)

⁢

(

D

2

⁡

f

)

)

)

⁢

d

⁢

t

+

(

D

⁡

f

)

⁢

σ

t

⁢

d

⁢

W

t

,

where

•

tr

is the trace operation;

∗

is the

transpose

•

D

⁡

f

is the

derivative

with respect to the space variables;
its value is a

linear transformation

from

L

⁢

(

ℝ

n

,

ℝ

)

•

D

2

⁡

f

is the

second derivative

with respect to space variables;
represented as the

Hessian matrix

•

the third line follows because

d

⁢

W

t

i

⁢

d

⁢

W

t

j

=

δ

i

⁢

j

⁢

d

⁢

t

.

The

quadratic form

tr

⁡

(

σ

t

⁢

σ

t

*

⁢

D

2

⁡

f

)

⁢

d

⁢

t

represents the

quadratic variation

of the process. When

σ

t

is the identity
transformation, this reduces to the

Laplacian

of

f

.

Itô’s formula in multiple dimensions can also be written with
the standard vector calculus operators.
It is in the

similar

notation typically used for the
related

parabolic partial differential equation

describing an Itô diffusion:

d

⁢

Y

t

=

(

∂

⁡

f

∂

⁡

t

+

μ

t

⋅

∇

⁡

f

+

1

2

⁢

(

∇

⋅

(

σ

t

⁢

σ

t

*

)

⁢

∇

)

⁢

f

)

⁢

d

⁢

t

+

(

σ

t

⁢

d

⁢

W

t

)

⋅

∇

⁡

f

.

References

1

Bernt Øksendal.

,
An Introduction with Applications

. 5th ed., Springer 1998.

2

Hui-Hsiung Kuo.

Introduction to

Stochastic Integration

.
Springer 2006.

Title

Itô’s formula

Canonical name

ItosFormula

Date of creation

2013-03-22 17:16:14

Last modified on

2013-03-22 17:16:14

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

13

Author

stevecheng (10074)

Entry type

Axiom

Classification

msc 60H10

Classification

msc 60H05

Synonym

Itô’s formula

Synonym

Itô’s

chain rule

Synonym

Ito’s formula

Synonym

Ito’s lemma

Synonym

Ito’s chain rule

Related topic

ItosLemma2

Related topic

GeneralizedItoFormula

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