local properties of processes

local properties of processes

Many properties of

stochastic processes

, such as the

martingale

property, can be generalized to a corresponding

local property

. The local properties can be more useful than the original property because they are often preserved under certain

transformations

of processes, such as random time changes and

stochastic integration

.

Let

(

Ω

,

ℱ

,

(

ℱ

t

)

t

∈

𝕋

,

ℙ

)

be a

filtered probability space

and

π

be a property of stochastic processes with time

index set

𝕋

⊆

ℝ

.
The property

π

is said to hold

locally

for a process

X

if there exists a

sequence

of

stopping times

(

τ

n

)

n

∈

ℤ

+

taking values in

𝕋

∪

{

∞

}

and almost surely increasing to

infinity

, such that the

stopped processes

X

τ

n

have property

π

for each

n

.

Often, the index set

𝕋

has a

minimal element

t

0

, in which case it is convenient to extend the

concept

of localization slightly so that

π

holds locally if there is a sequence of stopping times

τ

n

almost surely increasing to infinity and such that

1

{

τ

n

>

t

0

}

⁢

X

τ

n

have property

π

.

The property of locally satisfying

π

is often denoted as

π

loc

. Similarly, if

π

is a class of processes then the processes which are locally in

π

is denoted by

π

loc

.
Letting

τ

n

be the stopping times taking the

constant

value

∞

shows that every process in

π

is also locally in

π

, so

π

⊆

π

loc

.

In most cases where localization is used, such as with the class of right-continuous martingales, for any process

X

in

π

and stopping time

τ

then

1

{

τ

>

t

0

}

⁢

X

τ

is also in

π

. If this is the case then it is easily shown that a process is locally in

π

loc

if and only if it is locally in

π

. So,

(

π

loc

)

loc

=

π

loc

.

Examples of commonly used local properties are as follows.

1.

A process

X

is said to be a

local martingale

if it is locally a right-continuous martingale. That is, if there is a sequence of stopping times

τ

n

almost surely increasing to infinity and such that

1

{

τ

n

>

t

0

}

⁢

X

τ

n

∧

t

is integrable and,

1

{

τ

n

>

t

0

}

X

τ

n

∧

s

=

𝔼

[

1

{

τ

n

>

t

0

}

X

τ

n

∧

t

∣

ℱ

s

]

for all

s

<

t

∈

𝕋

.
In the discrete-time case where

𝕋

=

ℤ

+

then it can be shown that a local martingale

X

is a martingale if and only if

𝔼

⁢

[

|

X

t

|

]

<

∞

for every

t

∈

ℤ

+

.
More generally, in continuous-time where

𝕋

is an

interval

of the

real numbers

, then the

stronger

property that

{

X

τ

:

τ

≤

t

⁢

is a stopping time

}

is uniformly integrable for every

t

∈

𝕋

gives a necessary and

sufficient

condition for a local martingale to be a martingale.

Local martingales form a very important class of processes in the theory of stochastic calculus. This is because the local martingale property is preserved by the stochastic integral, but the martingale property is not.
Examples of local martingales which are not proper martingales are given by solutions to the

stochastic differential equation

d

⁢

X

=

X

α

⁢

d

⁢

W

where

X

is a nonnegative process,

W

is a

Brownian motion

and

α

>

1

is a fixed real number.

An alternative definition of local martingales which is sometimes used requires

X

τ

n

to be a martingale for each

n

. This definition is slightly more restrictive, and is

equivalent

to the definition given above together with the condition that

X

t

0

must be integrable.

2.

A

local submartingale

(resp.

local supermartingale

) is a right-continuous process which is locally a submartingale (resp. supermartingale). A local submartingale can be shown to be a submartingale if and only if

X

t

0

is integrable and the set

{

X

τ

∨

0

:

τ

≤

t

⁢

is a stopping time

}

is locally integrable for every

t

∈

𝕋

. In particular, every nonpositive local submartingale

X

for which

X

t

0

is integrable is a submartingale. Similarly every nonnegative supermartingale

X

such that

X

t

0

is integrable is a supermartingale.

3.

An increasing and non-negative process

X

is

locally integrable

if it is locally an integrable process. That is, there is a sequence of stopping times

τ

n

increasing to infinity and such that

𝔼

⁢

[

1

{

τ

n

>

t

0

}

⁢

|

X

τ

n

∧

t

|

]

<

∞

for every

n

∈

ℤ

+

and

t

∈

𝕋

. By

monotonicity

of

X

, this is equivalent to

𝔼

⁢

[

1

{

τ

n

>

t

0

}

⁢

|

X

τ

n

|

]

<

∞

. For example, the maximum process

X

t

*

≡

sup

s

≤

t

⁡

|

X

s

|

of a local martingale

X

is locally integrable.

4.

A process

X

is said to be

locally bounded

if there is a sequence of stopping times

τ

n

almost surely increasing to infinity and such that

1

{

τ

n

≥

t

0

}

⁢

X

τ

n

are uniformly bounded processes. For example, in discrete-time so

𝕋

=

ℤ

+

, then every

predictable process

is locally bounded.

Similarly, in continuous-time, if

π

is a property of stochastic processes and

X

is a stochastic process such that the left limits of

X

t

with respect to

t

exist everywhere, then

X

is said to

prelocally

satisfy

π

if there is a sequence of stopping times

τ

n

almost surely increasing to infinity and such that the prestopped processes

1

{

τ

n

>

t

0

}

⁢

X

τ

n

-

satisfy

π

.

Title

local properties of processes

Canonical name

LocalPropertiesOfProcesses

Date of creation

2013-03-22 18:38:50

Last modified on

2013-03-22 18:38:50

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

5

Author

gel (22282)

Entry type

Definition

Classification

msc 60G05

Classification

msc 60G40

Classification

msc 60G48

Related topic

LocalMartingale

Defines

local property

Defines

local submartingale

Defines

local martingale

Defines

locally integrable process

Defines

locally bounded process

Defines

prelocalization

Defines

prelocal property

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