stochastic integration by parts

stochastic integration by parts

The

stochastic integral

satisfies

a version of the classical

integration by parts

formula

, which is just the

integral

version of the

product rule

. The only

difference

here is the

existence

of a

quadratic covariation

term.

Theorem

.

Let

X

,

Y

be

semimartingales

. Then,

X

t

⁢

Y

t

=

X

0

⁢

Y

0

+

∫

0

t

X

s

-

⁢

𝑑

Y

s

+

∫

0

t

Y

s

-

⁢

𝑑

X

s

+

[

X

,

Y

]

t

.

(1)

Alternatively, in

differential

notation, this reads

d

⁢

(

X

t

⁢

Y

t

)

=

X

t

-

⁢

d

⁢

Y

t

+

Y

t

-

⁢

d

⁢

X

t

+

d

⁢

[

X

,

Y

]

t

.

The existence of the quadratic covariation term

[

X

,

Y

]

in the integration by parts formula, and also in Itô’s lemma, is an important difference between standard

calculus

and

stochastic calculus

. To see the need for this term, consider the following. Choosing any

h

>

0

, write the increment of a process over a time step of size

h

as

δ

⁢

X

t

≡

X

t

+

h

-

X

t

. The increment of a

product

of processes satisfies the following simple

identity

,

δ

⁢

(

X

⁢

Y

)

t

=

X

t

⁢

δ

⁢

Y

t

+

Y

t

⁢

δ

⁢

X

t

+

δ

⁢

X

t

⁢

δ

⁢

Y

t

.

(2)

As we let

h

tend to zero, for

differentiable

processes the final term of (

2

) is of order (

http://

planetmath

.org/LandauNotation

)

O

⁢

(

h

2

)

, so can be neglected in the limit. However, when

X

and

Y

are semimartingales, such as

Brownian motion

, the final term will be of order

h

, and needs to be retained even in the limit.

The proof of equation (

1

) is given by the proof of the existence of the

quadratic variation of semimartingales

(

http://planetmath.org/QuadraticVariationOfASemimartingale

) and, in particular, is just a rearrangement of the formula given for the quadratic covariation of semimartingales. Whenever either of

X

or

Y

is a

continuous

finite variation process

, the quadratic covariation term

[

X

,

Y

]

is zero, so (

1

) becomes the standard integration by parts formula. More generally, for noncontinuous processes we have the following.

Corollary

.

Let

X

be a semimartingale and

Y

be an

adapted

finite variation process. Then,

X

t

⁢

Y

t

=

X

0

⁢

Y

0

+

∫

0

t

X

s

⁢

𝑑

Y

s

+

∫

0

t

Y

s

-

⁢

𝑑

X

s

.

(3)

As

Y

is a finite variation process, the first integral on the right hand side of (

3

) makes sense as a

Lebesgue-Stieltjes integral

.
Equation (

3

) follows from the integration by parts formula by first substituting the following formula for the covariation whenever

Y

has finite variation into (

1

)

[

X

,

Y

]

t

=

∑

s

≤

t

Δ

⁢

X

s

⁢

Δ

⁢

Y

s

and then using the following identity

∫

0

t

X

s

⁢

𝑑

Y

s

-

∫

0

t

X

s

-

⁢

𝑑

Y

s

=

∫

0

t

Δ

⁢

X

s

⁢

𝑑

Y

s

=

∫

0

t

∑

u

Δ

⁢

X

u

⁢

1

{

u

=

s

}

⁢

d

⁢

Y

s

=

∑

u

≤

t

Δ

⁢

X

u

⁢

Δ

⁢

Y

u

.

Title

stochastic integration by parts

Canonical name

StochasticIntegrationByParts

Date of creation

2013-03-22 18:41:35

Last modified on

2013-03-22 18:41:35

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

4

Author

gel (22282)

Entry type

Theorem

Classification

msc 60H05

Classification

msc 60G07

Classification

msc 60H10

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