Itô integral

Itô integral

1

The need for a stochastic integral

Suppose that we wish to model a system which is continuously subject to random shocks. Then, we may want to consider a

differential equation

of the form

d

⁢

X

t

d

⁢

t

=

b

⁢

(

X

t

)

+

a

⁢

(

X

t

)

⁢

ξ

t

.

(1)

Here,

X

t

is the

stochastic process

describing the state of the system at each time

t

≥

0

,

b

describes the behavior in the absence of any random shocks and

a

is the sensitivity to the the random noise

ξ

t

. The

random variables

ξ

t

are assumed to be

independent and identically distributed

with mean zero for each

t

≥

0

, and defined on an underlying

probability space

(

Ω

,

ℱ

,

ℙ

)

.

The first problem is that it is not clear how to define

ξ

continuously in time such that (

1

) can be solved. Let

W

be the

integral

W

t

=

∫

0

t

ξ

s

⁢

𝑑

s

.

Then,

W

t

should be

continuous

in

t

and, for each

h

>

0

, the

sequence

of increments

(

W

(

n

+

1

)

⁢

h

-

W

n

⁢

h

)

n

=

0

,

1

,

…

should be independent and identically distributed random variables. A possible candidate for

W

is

Brownian motion

, and it can be shown that, up to a

constant

scaling factor, this is the only possibility.
However, Brownian motion is nowhere

differentiable

and

ξ

t

=

d

⁢

W

t

/

d

⁢

t

cannot exist. We therefore re-express (

1

) in terms of

W

by integrating it

X

t

=

X

0

+

∫

0

t

b

⁢

(

X

s

)

⁢

𝑑

s

+

∫

0

t

a

⁢

(

X

s

)

⁢

𝑑

W

s

.

(2)

The idea of Itô integration is to give meaning to the final integral on the right hand side of (

2

).
Defining the integral with respect to

piecewise constant functions

of the form

α

t

=

∑

k

=

1

n

c

k

⁢

1

{

t

k

-

1

<

t

≤

t

k

}

(3)

is easy. Here,

0

≤

t

0

≤

t

1

≤

⋯

≤

t

n

is an increasing sequence of times and

c

k

are random variables. Then, the integral of

α

with respect to

W

is

∫

0

t

α

s

⁢

𝑑

W

s

=

∑

k

=

1

n

c

k

⁢

(

W

t

k

∧

t

-

W

t

k

-

1

∧

t

)

.

(4)

We could attempt to define the integral with respect to

W

on the right hand side of (

2

) by approximating

a

⁢

(

X

t

)

by such piecewise constant functions. However, the

sample paths

of Brownian motion are of

infinite

total variation

over all

intervals

. This means that if we let

α

n

be

α

t

n

=

∑

k

=

1

n

sign

⁡

(

W

k

/

n

-

W

(

k

-

1

)

/

n

)

⁢

1

{

(

k

-

1

)

/

n

<

t

≤

k

/

n

}

then the integrals

∫

0

1

α

n

⁢

𝑑

W

=

∑

k

=

1

n

|

W

k

/

n

-

W

(

k

-

1

)

/

n

|

will tend to

infinity

as

n

→

∞

, despite the fact that

α

n

are all bounded by one. This means that even if we approximate

a

⁢

(

X

t

)

as closely as we like by piecewise constant functions, the integral can

diverge

to infinity.

2

The Itô integral

Itô’s solution was to first note that if we can define the integral in a way such that (

2

) has a unique solution, then

X

t

should only depend on the values of

W

s

for

s

≤

t

. So, define

ℱ

t

to be the

collection

of events in the probability space observable up until time

t

. This is the smallest

σ

-

algebra

with respect to which

W

s

is

measurable

for

s

≤

t

,

ℱ

t

=

σ

(

W

s

:

s

≤

t

)

.

Then,

(

ℱ

t

)

t

∈

ℝ

+

is a

filtration

of

σ

-algebras (

http://planetmath.org/FiltrationOfSigmaAlgebras

) to which

W

is

adapted

.
If we assume that the

piecewise

constant process

α

t

is adapted, then the

variables

c

k

in (

1

) will be

ℱ

t

k

-

1

-measurable. Such processes are called

elementary predictable

, and the following

identity

is obtained

𝔼

⁢

[

(

∫

0

t

α

⁢

𝑑

W

)

2

]

=

𝔼

⁢

[

∫

0

t

α

s

2

⁢

𝑑

s

]

.

(5)

This is known as the

Itô isometry

. It ensures that, for bounded and adapted

integrands

α

, the integral with respect to

W

cannot become too large, on

average

. Furthermore, the right hand side is defined for all jointly measurable processes

α

, and allows us extend the integral with respect to

W

to integrands which can be approximated by piecewise constant and adapted processes. Such processes are called

predictable

and include all continuous and adapted processes, so it gives meaning to (

2

). If we define

∥

α

∥

2

,

t

≡

𝔼

⁢

[

∫

0

t

α

s

2

⁢

𝑑

s

]

1

2

then this is a norm on the set of predictable processes — in fact, it is the

L

2

⁢

(

ℙ

×

λ

)

-norm, where

λ

is the

Lebesgue measure

on

[

0

,

t

]

. Equation (

5

) shows that the map taking the adapted and piecewise constant function

α

to

∫

0

t

α

⁢

𝑑

W

is an

isometry

with respect to the norms

∥

⋅

∥

2

,

t

and the

L

2

⁢

(

ℙ

)

-norm

∥

⋅

∥

2

. Furthermore, it can be shown, by using the

functional monotone class theorem

, that every predictable process

α

with finite

∥

⋅

∥

2

,

t

norm can be approximated in this norm by a sequence elementary predictable processes

α

n

. That is, the elementary predictable processes are dense (

http://planetmath.org/Dense

) under the

∥

⋅

∥

2

,

t

norm, and the Itô integral is defined to be the unique continuous

extension

(

http://planetmath.org/BoundedLinearExtension

) from the elementary predictable processes. So, if

∥

α

-

α

n

∥

2

,

t

→

0

then,

∫

0

t

α

⁢

𝑑

W

=

lim

n

→

∞

⁡

∫

0

t

α

n

⁢

𝑑

W

where convergence is in the

L

2

⁢

(

ℙ

)

norm.

3

Extension of the integral

The

definition

above gives a construction of the Itô integral which is useful in many applications.
However, the

restriction

that

∥

α

∥

2

,

t

must be finite is too restrictive for some situations.
For example, it is possible to construct solutions to the

stochastic differential equation

d

⁢

X

=

X

c

⁢

d

⁢

W

for any constant

c

>

0

and any given initial value

X

0

>

0

. When

c

>

1

then it is known that

X

t

2

⁢

c

does not have finite

expectation

for

t

>

0

and, therefore,

X

c

does not have finite

∥

⋅

∥

2

,

t

norm.

The Itô integral can be extended to all predictable processes

α

such that

∫

0

t

α

s

2

⁢

𝑑

s

<

∞

with

probability

one, for each

t

>

0

. Such processes are said to be

W

-

integrable

.
Given any such

α

, the functional monotone class theorem can be used to show that there exists a sequence of elementary predictable processes

α

n

such that

∫

0

t

(

α

s

n

-

α

s

)

2

⁢

𝑑

s

→

0

in probability (

http://planetmath.org/ConvergenceInProbability

) for every

t

>

0

. Then, using the Itô isometry it can be shown that the limit

∫

0

t

α

⁢

𝑑

W

≡

lim

n

→

∞

⁡

∫

0

t

α

n

⁢

𝑑

W

exists, where convergence is in probability.
Finally, we note that that this only defines the integral

ℙ

-almost everywhere, at each time. However, as

probability measures

only

satisfy

countable additivity

, this only simultaneously defines the values of the integral at different times

t

on

countable

subsets of

ℝ

+

. The additional constraint is added that the sample paths

t

↦

∫

0

t

α

⁢

𝑑

W

are continuous, and it can be shown that it is possible to take such continuous versions of the integral. This defines the integral simultaneously at all times up to a zero probability set.

We have presented a definition of the Itô integral with respect to a Brownian motion

W

. Then, an important result for manipulating such integrals is Itô’s lemma (

http://planetmath.org/ItosFormula

).
Furthermore, it is sometimes necessary to define integration for more general processes. See

stochastic integral

(

http://planetmath.org/StochasticIntegration

) for a definition of the integral with respect to general

semimartingales

, which includes

discontinuous

processes, such as Lévy processes.

References

1

Bernt Øksendal.

,
An

Introduction

with Applications.

5th ed. Springer, 1998.

Title

Itô integral

Canonical

name

ItoIntegral

Date of creation

2013-03-22 16:13:29

Last modified on

2013-03-22 16:13:29

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

21

Author

stevecheng (10074)

Entry type

Topic

Classification

msc 60H10

Synonym

Ito integral

Synonym

stochastic integral

Related topic

StochasticCalculusAndSDE

Related topic

StochasticIntegration

Defines

Itô integral

Generated on Sat Feb 10 12:00:22 2018 by

LaTeXML