aarjaneiro/planetmath huggingface dataset:
Bichteler-Dellacherie theorem

Bichteler-Dellacherie theorem

The

Bichteler-Dellacherie theorem

is an important result in stochastic calculus, and states the equivalence of two very different

definitions

of

semimartingales

.
The result also goes under other names, such as the

Dellacherie-Meyer-Mokobodzky theorem

.
Prior to its discovery, a theory of

stochastic integration

had been developed for

local martingales

. As standard Lebesgue-Stieltjes integration can be applied to

finite variation processes

, this allowed an integral to be defined with respect to sums of local martingales and finite variation processes, known as a semimartingales. The Bichteler-Dellacherie theorem then shows that, as long as we require stochastic integration to

satisfy

bounded

convergence, then semimartingales are actually the most general objects which can be used.

We consider a real valued

stochastic process

X

adapted

to a

filtered probability space

(

Ω

,

ℱ

,

(

ℱ

t

)

t

∈

ℝ

+

,

ℙ

)

. Then, the integral

∫

0

t

ξ

⁢

𝑑

X

can be written out explicitly for any

simple predictable process

ξ

.

Theorem

(Bichteler-Dellacherie).

Let

X

be a cadlag adapted stochastic process. Then, the following are

equivalent

.

1.

For every

t

>

0

, the set

{

∫

0

t

ξ

⁢

𝑑

X

:

|

ξ

|

≤

1

⁢

is simple predictable

}

is bounded in

probability

.

2.

A decomposition

X

=

M

+

V

exists, where

M

is a local martingale and

V

is a finite variation process.

3.

A decomposition

X

=

M

+

V

exists, where

M

is locally a uniformly bounded

martingale

and

V

is a finite variation process.

Condition

1

is equivalent to stating that if

ξ

n

is a

sequence

of simple predictable processes converging uniformly to zero, then the integrals

∫

0

t

ξ

n

⁢

𝑑

X

tend to zero in probability as

n

→

∞

, which is a weak form of bounded convergence for stochastic integration.

Conditions

1

and

2

are the two definitions often used for the process

X

to be a semimartingale.
However, condition

3

gives a

stronger

decomposition which is often more useful in practise. The

property

that

M

is locally a uniformly bounded martingale means that there exists a sequence of

stopping times

τ

n

, almost surely increasing to

infinity

, such that the

stopped processes

M

τ

n

are uniformly bounded martingales.

References

1

Philip E. Protter,

Stochastic integration and

differential equations

.

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

Title

Bichteler-Dellacherie theorem

Canonical

name

BichtelerDellacherieTheorem

Date of creation

2013-03-22 18:36:48

Last modified on

2013-03-22 18:36:48

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

8

Author

gel (22282)

Entry type

Theorem

Classification

msc 60G48

Classification

msc 60H05

Classification

msc 60G07

Synonym

Dellacherie-Meyer-Mokobodzky theorem

Generated on Sat Feb 10 11:59:38 2018 by

LaTeXML
random walk

random walk

Definition

. Let

(

Ω

,

ℱ

,

𝐏

)

be a

probability space

and

{

X

i

}

a discrete-time
stochastic process defined on

(

Ω

,

ℱ

,

𝐏

)

,
such that the

X

i

are iid real-valued

random variables

, and

i

∈

ℕ

, the set of natural numbers. The

random
walk

defined on

X

i

is the sequence of partial sums, or partial
series

S

n

:=

∑

i

=

1

n

X

i

.

If

X

i

∈

{

-

1

,

1

}

, then the random walk defined on

X

i

is called a

simple random walk

. A

symmetric simple random walk

is
a simple random walk such that

𝐏

(

X

i

=

1

)

=

1

/

2

.

The above defines random walks in one-dimension. One can easily
generalize to define higher dimensional random walks, by requiring
the

X

i

to be vector-valued (in

ℝ

n

), instead of

ℝ

.

Remarks

.

1.

Intuitively, a random walk can be viewed as movement in space
where the length and the direction of each step are random.

2.

It can be shown that, the limiting case of a random walk is a

Brownian motion

(with some conditions imposed on the

X

i

so as to

satisfy

part of the defining conditions of a Brownian motion). By
limiting case we mean, loosely speaking, that the lengths of the
steps are very small, approaching 0, while the total lengths of the
walk remains a

constant

(so that the number of steps is very large,
approaching

∞

).

3.

If the random variables

X

i

defining the random walk

w

i

are integrable with zero mean

E

⁡

[

X

i

]

=

0

,

S

i

is a

martingale

.

Title

random walk

Canonical name

RandomWalk

Date of creation

2013-03-22 14:59:22

Last modified on

2013-03-22 14:59:22

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 60G50

Classification

msc 82B41

Defines

simple random walk

Defines

symmetric simple random walk

Generated on Sat Feb 10 11:59:42 2018 by

LaTeXML
counting process

counting process

A

stochastic process

{

X

⁢

(

t

)

∣

t

∈

ℝ

+

∪

{

0

}

}

is called a

counting process

if, for each outcome

ω

in the

sample space

Ω

,

1.

X

⁢

(

t

)

∈

ℤ

+

∪

{

0

}

for all

t

,

2.

X

⁢

(

t

)

⁢

(

ω

)

is

piecewise

constant

,

3.

X

⁢

(

t

)

⁢

(

ω

)

is non-decreasing,

4.

X

⁢

(

t

)

⁢

(

ω

)

is right

continuous

(continuous from the right), and

5.

for any

t

, there is an

s

∈

ℝ

such that

t

<

s

and

X

⁢

(

t

)

⁢

(

ω

)

+

1

=

X

⁢

(

s

)

⁢

(

ω

)

.

Remark

. For any

t

, the

random variable

X

⁢

(

t

)

is usually called the number of

occurrences

of some event by time

t

. Then, for

s

<

t

,

X

⁢

(

t

)

-

X

⁢

(

s

)

is the number of occurrences in the

half-open interval

(

s

,

t

]

.

Title

counting process

Canonical name

CountingProcess

Date of creation

2013-03-22 15:01:19

Last modified on

2013-03-22 15:01:19

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

5

Author

CWoo (3771)

Entry type

Definition

Classification

msc 60G51

Generated on Sat Feb 10 11:59:44 2018 by

LaTeXML
independent increment

independent increment

A

stochastic process

{

X

⁢

(

t

)

∣

t

∈

T

}

of real-valued

random variables

X

⁢

(

t

)

, where

T

is

linearly ordered

, is said have

independent increments

if for any

a

,

b

,

c

,

d

∈

T

such that

a

<

b

<

c

<

d

,

X

⁢

(

a

)

-

X

⁢

(

b

)

and

X

⁢

(

c

)

-

X

⁢

(

d

)

are

independent

random
variables.

Remark

. In case when

X

⁢

(

t

)

is monotonically non-decreasing,
as in the case of a

counting process

, it is customary to write

X

⁢

(

b

)

-

X

⁢

(

a

)

and

X

⁢

(

d

)

-

X

⁢

(

c

)

instead of the above to emphasize the
comparison of two

positive

quantities (for example, the numbers of

occurrences

of a certain event in some time intervals).

Title

independent increment

Canonical name

IndependentIncrement

Date of creation

2013-03-22 15:01:22

Last modified on

2013-03-22 15:01:22

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

4

Author

CWoo (3771)

Entry type

Definition

Classification

msc 60G51

Generated on Sat Feb 10 11:59:46 2018 by

LaTeXML
Poisson process

Poisson process

A

counting process

{

X

⁢

(

t

)

∣

t

∈

ℝ

+

∪

{

0

}

}

is called a

simple Poisson

, or simply a

Poisson process

with
parameter

λ

, also known as the

intensity

, if

1.

X

⁢

(

0

)

=

0

,

2.

{

X

⁢

(

t

)

}

has

stationary independent increments

,

3.

P

(

X

(

t

)

=

1

)

=

λ

t

+

o

(

t

)

,

4.

P

(

X

(

t

)

>

1

)

=

o

(

t

)

,

where

o

⁢

(

t

)

is the O notation.

Remarks

.

•

The intensity

λ

is assumed to be a constant in terms of

t

.

•

Condition 3 above says that the

rate

in which the an event occurs once in time interval

t

, as

t

approaches 0, is

λ

. Condition 4 says that the event occurs more than once is very unlikely (the rate approaches zero as the time interval shrinks to zero).

•

It can be shown that

X

⁢

(

t

)

has a

Poisson distribution

(hence the name of the

stochastic process

) with parameter

λ

⁢

t

:

P

(

X

(

t

)

=

n

)

=

e

-

λ

⁢

t

(

λ

⁢

t

)

n

n

!

.

•

Therefore,

E

⁡

[

X

⁢

(

t

)

]

=

λ

⁢

t

.

Title

Poisson process

Canonical name

PoissonProcess

Date of creation

2013-03-22 15:01:29

Last modified on

2013-03-22 15:01:29

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 60G51

Synonym

homogeneous Poisson process

Defines

simple Poisson process

Defines

intensity

Generated on Sat Feb 10 11:59:48 2018 by

LaTeXML
stationary increment

stationary increment

A

stochastic process

{

X

⁢

(

t

)

∣

t

∈

T

}

of real-valued

random variables

X

⁢

(

t

)

, where

T

is a subset of

ℝ

, is
said have

stationary increments

if the

probability
distribution function

for

X

⁢

(

s

+

t

)

-

X

⁢

(

s

)

is fixed (the same) for all

s

∈

T

such that

s

+

t

∈

T

. In other words, the

distribution

for

X

⁢

(

s

+

t

)

-

X

⁢

(

s

)

is a

function

of “how long” or

t

, not “when” or

s

.

A stochastic process that possesses both stationary increments and

independent increments

is said to have

stationary independent
increments

.

Title

stationary increment

Canonical name

StationaryIncrement

Date of creation

2013-03-22 15:01:25

Last modified on

2013-03-22 15:01:25

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Definition

Classification

msc 60G51

Defines

stationary independent increment

Generated on Sat Feb 10 11:59:49 2018 by

LaTeXML
eigenvalues of stochastic matrix

eigenvalues of stochastic matrix

Theorem

:
The

spectrum

of a

stochastic matrix

is

contained

in the unit disc in the

complex plane

.

Proof.

Let

A

be a stochastic matrix and let

m

be an

eigenvalue

of

A

, with

v

eigenvector

; then, for any

self-consistent matrix norm

∥

.

∥

, we have:

|

m

|

⁢

∥

v

∥

=

∥

m

⁢

v

∥

=

∥

A

⁢

v

∥

≤

∥

A

∥

⁢

∥

v

∥

,

that is, since

v

is nonzero,

|

m

|

≤

∥

A

∥

.

Now, for a (doubly) stochastic matrix,

∥

A

∥

1

=

max

j

⁡

(

∑

i

|

a

i

⁢

j

|

)

=

1

whence the

conclusion

.
∎

Title

eigenvalues of stochastic matrix

Canonical name

EigenvaluesOfStochasticMatrix

Date of creation

2013-03-22 16:18:02

Last modified on

2013-03-22 16:18:02

Owner

Andrea Ambrosio (7332)

Last modified by

Andrea Ambrosio (7332)

Numerical id

7

Author

Andrea Ambrosio (7332)

Entry type

Theorem

Classification

msc 60G99

Classification

msc 15A51

Generated on Sat Feb 10 11:59:51 2018 by

LaTeXML
stochastic matrix

stochastic matrix

Definition

Let

I

be a finite or countable set, and let

𝐏

=

(

p

i

⁢

j

:

i

,

j

∈

I

)

be a matrix and let all

p

i

⁢

j

be nonnegative. We say

𝐏

is

stochastic

if

∑

i

∈

I

p

i

⁢

j

=

1

for every

j

∈

I

. We call

𝐏

doubly stochastic

if, in

addition

,

∑

j

∈

I

p

i

⁢

j

=

1

for all

i

∈

I

.
Equivalently,

𝐏

is stochastic if every column is a

distribution

, and doubly stochastic if, in addition, every row is a distribution.

Stochastic and

doubly stochastic matrices

are common in discussions of

random processes

, particularly

Markov chains

.

Title

stochastic matrix

Canonical

name

StochasticMatrix

Date of creation

2013-03-22 12:37:29

Last modified on

2013-03-22 12:37:29

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

9

Author

mathwizard (128)

Entry type

Definition

Classification

msc 60G99

Classification

msc 15A51

Related topic

Distribution

Related topic

Matrix

Defines

doubly stochastic

Defines

stochastic matrix

Generated on Sat Feb 10 11:59:53 2018 by

LaTeXML
associativity of stochastic integration

associativity of stochastic integration

The

chain rule

for expressing the

derivative

of a variable

z

with respect to

x

in terms of a third variable

y

is

d

⁢

z

d

⁢

x

=

d

⁢

z

d

⁢

y

⁢

d

⁢

y

d

⁢

x

.

Equivalently, if

d

⁢

y

=

α

⁢

d

⁢

x

and

d

⁢

z

=

β

⁢

d

⁢

y

then

d

⁢

z

=

β

⁢

α

⁢

d

⁢

x

.
The following

theorem

shows that the

stochastic integral

satisfies

a

generalization

of this.

Theorem

.

Let

X

be a

semimartingale

and

α

be an

X

-

integrable

process. Setting

Y

=

∫

α

⁢

𝑑

X

then

Y

is a semimartingale. Furthermore, a

predictable process

β

is

Y

-integrable if and only if

β

⁢

α

is

X

-integrable, in which case

∫

β

⁢

𝑑

Y

=

∫

β

⁢

α

⁢

𝑑

X

.

(1)

Note that expressed in alternative notation, (

1

) becomes

β

⋅

(

α

⋅

X

)

=

(

β

⁢

α

)

⋅

X

or, in

differential

notional,

β

⁢

(

α

⁢

d

⁢

X

)

=

(

β

⁢

α

)

⁢

d

⁢

X

.

That is, stochastic integration is associative.

Title

associativity of stochastic integration

Canonical name

AssociativityOfStochasticIntegration

Date of creation

2013-03-22 18:41:06

Last modified on

2013-03-22 18:41:06

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

Generated on Sat Feb 10 11:59:55 2018 by

LaTeXML
semimartingale convergence implies ucp convergence

semimartingale convergence implies ucp convergence

Let

(

Ω

,

ℱ

,

(

ℱ

t

)

t

∈

𝔽

,

ℙ

)

be a

filtered probability space

. On the space of cadlag

adapted processes

, the

semimartingale topology

is

stronger

than

ucp convergence

.

Theorem

.

Let

X

n

be a sequence of cadlag adapted processes converging to

X

in the semimartingale topology. Then,

X

n

converges ucp to

X

.

To show this, suppose that

X

n

→

X

in the semimartingale topology, and define the

stopping times

τ

n

by

τ

n

=

inf

⁡

{

t

≥

0

:

|

X

t

n

-

X

t

|

≥

ϵ

}

(1)

(

hitting times are stopping times

).
Then, letting

ξ

t

n

be the

simple predictable process

1

{

t

≤

τ

n

}

,

X

τ

n

∧

t

n

-

X

τ

n

∧

t

=

X

0

n

-

X

0

+

∫

0

t

ξ

n

⁢

𝑑

X

n

-

∫

0

t

ξ

n

⁢

𝑑

X

→

0

in

probability

as

n

→

∞

. However, note that whenever

|

X

s

n

-

X

s

|

>

ϵ

for some

s

<

t

then

τ

≤

s

<

t

and

|

X

τ

n

n

-

X

τ

n

|

≥

ϵ

. So

ℙ

(

sup

s

<

t

|

X

s

n

-

X

s

|

>

ϵ

)

≤

ℙ

(

τ

n

≤

t

)

≤

ℙ

(

|

X

τ

n

∧

t

n

-

X

τ

n

∧

t

|

≥

ϵ

)

→

0

as

n

→

∞

, proving ucp convergence.

As a minor technical point, note that the result that the

hitting times

τ

n

are stopping times requires the

filtration

to be at least

universally complete

. However, this condition is not needed. It is easily shown that semimartingale convergence is not affected by passing to the

completion

(

http://planetmath.org/CompleteMeasure

) of the filtered probability space or, alternatively, it is enough to define the stopping times in (

1

) by restricting

τ

n

to finite but suitably

dense subsets

of

[

0

,

t

]

and using the right-continuity of the processes.

Title

semimartingale convergence implies ucp convergence

Canonical name

SemimartingaleConvergenceImpliesUcpConvergence

Date of creation

2013-03-22 18:40:44

Last modified on

2013-03-22 18:40:44

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

5

Author

gel (22282)

Entry type

Theorem

Classification

msc 60H05

Classification

msc 60G48

Classification

msc 60G07

Generated on Sat Feb 10 11:59:57 2018 by

LaTeXML
stochastic integration as a limit of Riemann sums

stochastic integration as a limit of Riemann sums

As with the Riemann (

http://planetmath.org/RiemannIntegral

) and

Riemann-Stieltjes integrals

, the

stochastic integral

can be calculated as a limit of approximations computed on

partitions

(

http://planetmath.org/Partition3

), called

Riemann sums

.

Let

P

n

be a

sequence

of partitions of

ℝ

+

,

P

n

=

{

0

=

τ

0

n

≤

τ

1

n

≤

⋯

↑

∞

}

where,

τ

k

n

can, in general, be

stopping times

. Suppose that the mesh

|

P

n

t

|

=

sup

k

⁡

(

τ

k

n

∧

t

-

τ

k

-

1

n

∧

t

)

tends to zero in

probability

(

http://planetmath.org/ConvergenceInProbability

) as

n

→

∞

, for each time

t

>

0

.

The stochastic integral of a process

Y

with respect to

X

can be defined on each of the partitions,

I

t

n

⁢

(

Y

,

X

)

≡

∑

k

Y

τ

k

-

1

n

⁢

(

X

τ

k

n

∧

t

-

X

τ

k

-

1

n

∧

t

)

.

Since the times

τ

k

n

tend to

infinity

as

k

→

∞

, all but finitely many terms are zero.
Note that here, the process

Y

is sampled at

τ

k

-

1

n

, which are the left hand points of the

intervals

. For this reason, the stochastic integral is sometimes called the forward

integral

. Alternatively, the backward integral can be computed by sampling

Y

at time

t

k

and the Stratonovich integral takes the

average

of

Y

t

k

-

1

and

Y

t

k

. However, these alternative integrals are less general and may not exist even when

Y

is a

continuous

and

adapted process

.

For left-continuous integrands, the approximations do indeed

converge

to the stochastic integral.

Theorem 1

.

Suppose that

X

is a

semimartingale

and

Y

is an adapted, left-continuous and

locally bounded process

. Then,

I

t

n

⁢

(

Y

,

X

)

→

∫

0

t

Y

⁢

𝑑

X

in probability as

n

→

∞

. Furthermore, this

converges ucp

and in the

semimartingale topology

.

Similarly, convergence is also obtained for cadlag integrands. However, in this case, it is necessary to use the left limit

Y

s

-

in the integral. The integral of

Y

does not even exist when it is a general cadlag adapted process, as it might not be

predictable

.

Theorem 2

.

Suppose that

X

is a semimartingale and

Y

is a cadlag adapted process. Then,

I

t

n

⁢

(

Y

,

X

)

→

∫

0

t

Y

s

-

⁢

𝑑

X

s

in probability as

n

→

∞

. Furthermore, this converges ucp and in the semimartingale topology.

Title

stochastic integration as a limit of Riemann sums

Canonical name

StochasticIntegrationAsALimitOfRiemannSums

Date of creation

2013-03-22 18:41:33

Last modified on

2013-03-22 18:41:33

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

Related topic

StochasticIntegration

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analytic solution to Ornstein-Uhlenbeck SDE

analytic solution to Ornstein-Uhlenbeck SDE

This entry derives
the analytical solution
to the

stochastic differential equation

for the

Ornstein-Uhlenbeck process

:

d

⁢

X

t

=

κ

⁢

(

θ

-

X

t

)

⁢

d

⁢

t

+

σ

⁢

d

⁢

W

t

,

(1)

where

W

t

is a standard

Brownian motion

,
and

κ

>

0

,

θ

, and

σ

>

0

are
constants.

Motivated by the observation
that

θ

is supposed to be the long-term mean
of the process

X

t

,
we can simplify the SDE (

1

)
by introducing the change of variable

Y

t

=

X

t

-

θ

that subtracts off the mean.
Then

Y

t

satisfies the SDE:

d

⁢

Y

t

=

d

⁢

X

t

=

-

κ

⁢

Y

t

⁢

d

⁢

t

+

σ

⁢

d

⁢

W

t

.

(2)

In SDE (

2

), the process

Y

t

is seen to have a drift
towards the value zero, at an

exponential

rate

κ

. This motivates
the change of variables

Y

t

=

e

-

κ

⁢

t

⁢

Z

t

⇔

Z

t

=

e

κ

⁢

t

⁢

Y

t

,

which should remove the drift.
A calculation with the

product rule

for Itô

integrals

shows that this is so:

d

⁢

Z

t

=

κ

⁢

e

κ

⁢

t

⁢

Y

t

⁢

d

⁢

t

+

e

κ

⁢

t

⁢

d

⁢

Y

t

=

κ

⁢

e

κ

⁢

t

⁢

Y

t

⁢

d

⁢

t

+

e

κ

⁢

t

⁢

(

-

κ

⁢

Y

t

⁢

d

⁢

t

+

σ

⁢

d

⁢

W

t

)

=

0

⁢

d

⁢

t

+

σ

⁢

e

κ

⁢

t

⁢

d

⁢

W

t

.

The solution for

Z

t

is immediately obtained
by Itô-integrating both sides from

s

to

t

:

Z

t

=

Z

s

+

σ

⁢

∫

s

t

e

κ

⁢

u

⁢

𝑑

W

u

.

Reversing the changes of variables, we have:

Y

t

=

e

-

κ

⁢

t

⁢

Z

t

=

e

-

κ

⁢

(

t

-

s

)

⁢

Y

s

+

σ

⁢

e

-

κ

⁢

t

⁢

∫

s

t

e

κ

⁢

u

⁢

𝑑

W

u

,

and

X

t

=

Y

t

+

θ

=

θ

+

e

-

κ

⁢

(

t

-

s

)

⁢

(

X

s

-

θ

)

+

σ

⁢

∫

s

t

e

-

κ

⁢

(

t

-

u

)

⁢

𝑑

W

u

.

Title

analytic solution to Ornstein-Uhlenbeck SDE

Canonical name

AnalyticSolutionToOrnsteinUhlenbeckSDE

Date of creation

2013-03-22 17:19:29

Last modified on

2013-03-22 17:19:29

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

4

Author

stevecheng (10074)

Entry type

Derivation

Classification

msc 60H10

Classification

msc 60-00

Generated on Sat Feb 10 12:00:08 2018 by

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dominated convergence for stochastic integration

dominated convergence for stochastic integration

The

dominated convergence theorem

for standard integration states that if a

sequence

of

measurable functions

converge to a limit, and are dominated by an integrable function, then their

integrals

converge to the integral of the limit. That is, the limit commutes with integration. A similar result holds for

stochastic integration

with respect to a

semimartingale

X

, except the integrals are

random variables

, and the integrals

converge in probability

.

Theorem

(Dominated convergence).

If

ξ

n

are

predictable processes

converging pointwise to

ξ

, and

|

ξ

n

|

≤

α

for every

n

and some

X

-integrable process

α

, then

∫

0

t

ξ

n

⁢

𝑑

X

→

∫

0

t

ξ

⁢

𝑑

X

(1)

in

probability

as

n

→

∞

. Furthermore,

ucp convergence

and

semimartingale convergence

hold.

Note that as

ξ

and

ξ

n

are

bounded

by an

X

-integrable process, they are guaranteed to also be

X

-integrable. Convergence in probability for each

t

was taken as part of the definition of the stochastic integral, but the dominated convergence theorem stated here says that the stronger ucp and semimartingale convergence also hold.

If

α

is a

locally bounded

(

http://planetmath.org/LocalPropertiesOfProcesses

) predictable process, then it is automatically

X

-integrable for any semimartingale

X

. It follows that if

ξ

n

are predictable processes converging to

ξ

and if

sup

n

⁡

|

ξ

n

|

is locally bounded then the limit (

1

) holds. This result is sometimes known as the

locally bounded convergence theorem

.

To prove this result, it is enough to show that semimartingale convergence holds, as

semimartingale convergence implies ucp convergence

.
So, let

|

α

n

|

≤

1

be a sequence of

simple predictable processes

and set

Y

n

=

∫

ξ

n

⁢

𝑑

X

,

Y

=

∫

ξ

⁢

𝑑

X

.

Associativity of stochastic integration

gives

∫

0

t

α

n

⁢

𝑑

Y

n

-

∫

0

t

α

n

⁢

𝑑

Y

=

∫

0

t

α

n

⁢

(

ξ

n

-

ξ

)

⁢

𝑑

X

However,

|

α

n

⁢

(

ξ

n

-

ξ

)

|

≤

2

⁢

α

, which is

X

-integrable. So, this converges to zero in probability by the definition of the stochastic integral, and

Y

n

→

Y

in the semimartingale topology.

Title

dominated convergence for stochastic integration

Canonical name

DominatedConvergenceForStochasticIntegration

Date of creation

2013-03-22 18:41:03

Last modified on

2013-03-22 18:41:03

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

8

Author

gel (22282)

Entry type

Theorem

Classification

msc 60H10

Classification

msc 60G07

Classification

msc 60H05

Defines

locally bounded convergence theorem

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