generalized Ito formula

generalized Ito formula

The generalized Itô

formula

, or

generalized Itô’s lemma

, is an

extension

of Itô’s lemma (

http://

planetmath

.org/ItosLemma2

) that applies also to

discontinuous

processes. For a

cadlag process

X

, we write

Δ

⁢

X

t

≡

X

t

-

X

t

-

for its jump at time

t

.

Theorem

.

Suppose that

X

=

(

X

1

,

…

,

X

n

)

is a

semimartingale

taking values in an open subset

U

of

R

n

and

f

:

U

→

R

is twice

continuously differentiable

. Then,

d

⁢

f

⁢

(

X

s

)

=

∑

i

=

1

n

f

,

i

⁢

(

X

t

-

)

⁢

d

⁢

X

t

i

+

1

2

⁢

∑

i

,

j

=

1

n

f

,

i

j

⁢

(

X

t

-

)

⁢

d

⁢

[

X

i

,

X

j

]

t

c

+

(

Δ

⁢

f

⁢

(

X

t

)

-

∑

i

=

1

n

f

,

i

⁢

(

X

t

-

)

⁢

Δ

⁢

X

t

i

)

.

(1)

Here,

[

X

i

,

X

j

]

c

represents the

continuous

part of the

quadratic covariation

,

[

X

i

,

X

j

]

t

c

=

[

X

i

,

X

j

]

t

-

∑

s

≤

t

Δ

⁢

X

s

i

⁢

Δ

⁢

X

s

j

which is a continuous

finite variation process

.
The final term on the right hand side of (

1

) involving the jumps of

X

represents the

differential

d

⁢

Z

of the process

Z

t

=

∑

s

≤

t

(

Δ

⁢

f

⁢

(

X

s

)

-

∑

i

=

1

n

f

,

i

⁢

(

X

s

-

)

⁢

Δ

⁢

X

s

i

)

.

This is indeed a well defined finite variation process, as the sum of the

absolute values

∑

s

≤

t

|

Δ

⁢

f

⁢

(

X

s

)

-

∑

i

=

1

n

f

,

i

⁢

(

X

s

-

)

⁢

Δ

⁢

X

s

i

|

≤

K

⁢

∑

s

≤

t

∥

Δ

⁢

X

s

∥

2

≤

K

⁢

∑

i

=

1

n

[

X

i

]

t

is finite. Here,

K

is a finite

random variable

, and this bound follows from expanding

f

as a

Taylor series

to

second order

.

The reason for using differential notation and writing the formula in terms of the continuous part of the quadratic covariation should be clear when it is considered that writing out the expression in full gives the following rather messy formula.

f

⁢

(

X

t

)

=

f

⁢

(

X

0

)

+

∑

i

=

1

n

∫

0

t

f

,

i

⁢

(

X

s

-

)

⁢

𝑑

X

s

i

+

1

2

⁢

∑

i

,

j

=

1

n

∫

0

t

f

,

i

j

⁢

(

X

s

-

)

⁢

d

⁢

[

X

i

,

X

j

]

s

+

∑

s

≤

t

(

Δ

⁢

f

⁢

(

X

s

)

-

∑

i

=

1

n

f

,

i

⁢

(

X

s

-

)

⁢

Δ

⁢

X

s

i

-

1

2

⁢

∑

i

,

j

=

1

n

∫

0

t

f

,

i

j

⁢

(

X

s

-

)

⁢

Δ

⁢

X

s

i

⁢

Δ

⁢

X

s

j

)

.

This formula may be understood as Itô’s formula (

http://planetmath.org/ItosLemma2

) for continuous processes together with an additional term to ensure that the jumps of the right hand side are equal to

Δ

⁢

f

⁢

(

X

t

)

. The need for this adjustment term comes from the fact that Itô’s formula for continuous processes is essentially a Taylor expansion to second order, which only applies when the increments

δ

⁢

X

t

=

X

t

+

δ

⁢

t

-

X

t

vanish in the limit of small

δ

⁢

t

. This needs adjusting whenever the process jumps.

The first term on the right hand side of (

1

) is a

stochastic integral

and, hence, is a semimartingale. As the remaining terms are finite variation processes, the following

consequence

is obtained.

Corollary

.

Suppose that

X

=

(

X

1

,

…

,

X

n

)

is a semimartingale taking values in an open subset

U

of

R

n

and

f

:

U

→

R

is twice continuously differentiable. Then

f

⁢

(

X

)

is a semimartingale.

Title

generalized Ito formula

Canonical name

GeneralizedItoFormula

Date of creation

2013-03-22 18:41:47

Last modified on

2013-03-22 18:41:47

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

6

Author

gel (22282)

Entry type

Theorem

Classification

msc 60G07

Classification

msc 60H05

Classification

msc 60H10

Synonym

generalized Itô formula

Synonym

generalized Itö formula

Synonym

generalized Ito’s lemma

Synonym

generalized Itô’s lemma

Synonym

generalized Itö’s lemma

Related topic

ItosLemma2

Related topic

ItosFormula

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