stochastic integral

stochastic integral

We present a definition of the

stochastic integral

of a

predictable process

with respect to a general real-valued

semimartingale

.
In the literature on stochastic calculus there are actually several different different definitions available, often based on specific constructions of the

integral

in terms of decompositions of the semimartingale into a sum of a

local martingale

and a

finite variation process

.
The approach taken here is to instead define the integral in terms of its most basic properties — in particular that it

satisfies

the expected

generalizations

of standard non-stochastic integration, namely linearity and dominated convergence. Unlike in standard

calculus

, where the value of an integral is a

real number

, here the value is a

random variable

.

Many

stochastic processes

, such as

Brownian motion

, have paths which are nowhere

differentiable

and have

infinite

total variation

over finite time intervals. In such cases, standard definitions of integration, such as the

Riemann-Stieltjes integral

, cannot be used. However, by only considering predictable integrands and by relaxing properties such as dominated convergence to only require

convergence in probability

, the stochastic integral is a well-defined quantity.

Stochastic integration as described here is sometimes referred to as the Itö or forward integral, in order to distinguish it from the backward and Stratonovich integrals.

Let

X

be a semimartingale defined with respect to a

filtered probability space

(

Ω

,

ℱ

,

(

ℱ

t

)

t

∈

ℝ

+

,

ℙ

)

.
Then, for a predictable process

ξ

, the stochastic integral of

ξ

with respect to

X

is a càdlàg process

t

↦

∫

0

t

ξ

⁢

𝑑

X

.

For each fixed time

t

, this is a random variable defined on the

measurable space

(

Ω

,

ℱ

)

.

For

bounded

integrands, the integral satisfies the following properties.

1.

(

Elementary

integrands)
For any time

T

∈

ℝ

+

and bounded,

ℱ

T

-

measurable

random variable

A

, a predictable process satisfying

ξ

t

=

1

{

t

>

T

}

⁢

A

over

t

>

0

has the integral

∫

0

t

ξ

⁢

𝑑

X

=

A

⁢

1

{

t

>

T

}

⁢

(

X

t

-

X

T

)

⁢

(almost surely)

.

2.

(Linearity) If

α

,

β

are bounded and predictable processes and

λ

,

μ

∈

ℝ

then

∫

0

t

(

λ

⁢

α

+

μ

⁢

β

)

⁢

𝑑

X

=

λ

⁢

∫

0

t

α

⁢

𝑑

X

+

μ

⁢

∫

0

t

β

⁢

𝑑

X

⁢

(almost surely).

(1)

3.

(Bounded convergence in probability)
If

(

ξ

n

)

n

∈

ℕ

is a

sequence

of predictable processes such that

|

ξ

n

|

≤

1

and

ξ

n

→

0

as

n

tends to

infinity

then,

∫

0

t

ξ

n

⁢

𝑑

X

→

0

in

probability

as

n

→

∞

, for each

t

≥

0

.

These three properties uniquely define the integration for bounded integrands.
By linearity, property

1

above is

equivalent

to stating that the integral agrees with the explicit expression for

elementary predictable

integrands and then, by bounded convergence, that it agrees with the explicit expression for all simple predictable integrands.

The stochastic integral can be extended to more general

unbounded

predictable integrands.
A predictable process

ξ

is

X

-

integrable

if the set of random variables

{

∫

0

t

α

⁢

𝑑

X

:

|

α

|

≤

|

ξ

|

⁢

is bounded and predictable

}

is bounded in probability for every

t

∈

ℝ

+

.
The set of all

X

-integrable processes is sometimes denoted by

L

1

⁢

(

X

)

. By bounded convergence in probability, this contains all bounded predictable processes, and it is easily shown that the set of

X

-integrable processes are

closed under

taking linear combinations.
Furthermore, regardless of the specific semimartingale under consideration, every

locally bounded

predictable process will be

X

-integrable.

The stochastic integral of arbitrary integrands in

L

1

⁢

(

X

)

is the unique

extension

from bounded predictable integrands described above such that linearity (

1

) holds, and the following dominated convergence result holds. If

ξ

is

X

-integrable and

(

ξ

n

)

n

∈

ℕ

is a sequence of

X

-integrable processes such that

|

ξ

n

|

≤

|

ξ

|

and

ξ

n

→

0

then

∫

0

t

ξ

n

⁢

𝑑

X

→

0

in probability as

n

→

∞

, for each

t

∈

ℝ

+

.

For any semimartingale

X

and

X

-integrable process

ξ

, the integral is sometimes denoted by

ξ

⋅

X

,

(

ξ

⋅

X

)

t

≡

∫

0

t

ξ

⁢

𝑑

X

.

Alternatively, stochastic integrals are often written in differential form. That is,

d

⁢

Y

=

ξ

⁢

d

⁢

X

is equivalent to stating that

Y

t

-

Y

0

=

∫

0

t

ξ

⁢

𝑑

X

for each

t

>

0

.

References

1

K. Bichteler,

Stochastic integration with jumps

. Encyclopedia of Mathematics and its Applications, 89. Cambridge University Press, 2002.

2

Sheng-we He, Jia-gang Wang, Jia-an Yan,

Semimartingale theory and stochastic calculus.

Kexue Chubanshe (Science Press), CRC Press, 1992.

3

Olav Kallenberg,

Foundations

of modern probability

, Second edition. Probability and its Applications. Springer-Verlag, 2002.

4

Philip E. Protter,

Stochastic integration and differential equations.

Second edition. Applications of Mathematics, 21. Stochastic Modelling and Applied Probability. Springer-Verlag, 2004.

5

L.C.G. Rogers & David Williams,

Diffusions,

Markov processes

, and

martingales

. Vol. 2. Itô calculus.

Reprint of the second edition. Cambridge Mathematical Library. Cambridge University Press, 2000.

Title

stochastic integral

Canonical name

StochasticIntegral

Date of creation

2013-03-22 18:36:45

Last modified on

2013-03-22 18:36:45

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

9

Author

gel (22282)

Entry type

Definition

Classification

msc 60G07

Classification

msc 60H10

Classification

msc 60H05

Synonym

stochastic integration

Related topic

ItoIntegral

Related topic

PropertiesOfXIntegrableProcesses

Related topic

StochasticIntegrationAsALimitOfRiemannSums

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