aarjaneiro/planetmath huggingface dataset:
progressively measurable process

progressively measurable process

A

stochastic process

(

X

t

)

t

∈

ℤ

+

is said to be

adapted

to a

filtration

(

http://planetmath.org/FiltrationOfSigmaAlgebras

)

(

ℱ

t

)

on the

measurable space

(

Ω

,

ℱ

)

if

X

t

is an

ℱ

t

-measurable

random variable

for each

t

=

0

,

1

,

…

. However, for continuous-time processes, where the time

t

ranges over an arbitrary

index set

𝕋

⊆

ℝ

, the property of being adapted is too weak to be helpful in many situations. Instead, considering the process as a map

X

:

𝕋

×

Ω

→

ℝ

,

(

t

,

ω

)

↦

X

t

⁢

(

ω

)

it is useful to consider the measurability of

X

.

The process

X

is

progressive

or

progressively measurable

if, for every

s

∈

𝕋

, the

stopped process

X

t

s

≡

X

min

⁡

(

s

,

t

)

is

ℬ

⁢

(

𝕋

)

⊗

ℱ

s

-measurable. In particular, every progressively measurable process will be adapted and jointly measurable. In discrete time, when

𝕋

is

countable

, the

converse

holds and every adapted process is progressive.

A set

S

⊆

𝕋

×

Ω

is said to be progressive if its

characteristic function

1

S

is progressive. Equivalently,

S

∩

(

(

-

∞

,

s

]

×

Ω

)

∈

ℬ

⁢

(

𝕋

)

⊗

ℱ

s

for every

s

∈

𝕋

. The progressively measurable sets form a

σ

-

algebra

, and a stochastic process is progressive if and only if it is measurable with respect to this

σ

-algebra.

Title

progressively measurable process

Canonical name

ProgressivelyMeasurableProcess

Date of creation

2013-03-22 18:37:31

Last modified on

2013-03-22 18:37:31

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

4

Author

gel (22282)

Entry type

Definition

Classification

msc 60G05

Synonym

progressive process

Related topic

PredictableProcess

Related topic

OptionalProcess

Defines

progressive

Defines

progressively measurable

Generated on Sat Feb 10 11:57:24 2018 by

LaTeXML
proof of measurability of stopped processes

proof of measurability of stopped processes

Let

(

ℱ

t

)

t

∈

𝕋

be a

filtration

(

http://planetmath.org/FiltrationOfSigmaAlgebras

) on the

measurable space

(

Ω

,

ℱ

)

,

τ

be a

stopping time

, and

X

be a

stochastic process

. We prove the following measurability properties of the

stopped process

X

τ

.

If

X

is jointly measurable then so is

X

τ

.

Suppose first that

X

is a process of the form

X

t

=

1

A

⁢

1

{

t

≥

s

}

for some

A

∈

ℱ

and

t

∈

𝕋

. Then,

X

t

τ

=

1

A

∩

{

τ

≥

s

}

⁢

1

{

t

≥

s

}

is

ℬ

⁢

(

𝕋

)

⊗

ℱ

-measurable. By the

functional monotone class theorem

, it follows that

X

τ

is a

bounded

ℬ

⁢

(

𝕋

)

⊗

ℱ

-measurable process whenever

X

is. By taking limits of bounded processes, this generalizes to all jointly measurable processes.

If

X

is

progressively measurable

then so is

X

τ

.

For any given

t

∈

𝕋

, let

Y

be the

ℬ

⁢

(

𝕋

)

⊗

ℱ

t

-measurable process

Y

s

=

X

s

t

=

X

s

∧

t

. As

τ

∧

t

is

ℱ

t

-measurable, the result proven above says that

(

X

τ

)

t

=

Y

τ

∧

t

will also be

ℬ

⁢

(

𝕋

)

⊗

ℱ

t

-measurable, so

X

τ

is progressive.

If

X

is

optional

then so is

X

τ

.

As the optional processes are

generated by

the right-continuous and

adapted processes

then it is enough to prove this result when

X

is right-continuous and adapted. Clearly,

X

τ

will be right-continuous. Also,

X

will be progressive (see

measurability of stochastic processes

) and, by the result proven above, it follows that

X

τ

is progressive and, in particular, is adapted.

If

X

is

predictable

then so is

X

τ

.

By the definition of predictable processes, it is enough to prove the result in the cases where

X

t

=

1

A

⁢

1

{

t

>

s

}

for some

s

∈

𝕋

and

A

∈

ℱ

s

, and

X

t

=

1

A

⁢

1

{

t

=

t

0

}

where

t

0

is the

minimal element

of

𝕋

and

A

∈

ℱ

t

0

.

In the first case,

X

t

τ

=

1

A

∩

{

τ

>

s

}

⁢

1

{

t

>

s

}

is predictable and, in the second case,

X

t

τ

=

1

A

⁢

1

{

t

=

t

0

}

+

1

A

∩

{

τ

=

t

0

}

⁢

1

{

t

>

t

0

}

is predictable.

Title

proof of measurability of stopped processes

Canonical name

ProofOfMeasurabilityOfStoppedProcesses

Date of creation

2013-03-22 18:39:03

Last modified on

2013-03-22 18:39:03

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

4

Author

gel (22282)

Entry type

Proof

Classification

msc 60G05

Generated on Sat Feb 10 11:57:29 2018 by

LaTeXML
stochastic process

stochastic process

Let

(

Ω

,

ℱ

,

𝐏

)

be a

probability space

. A

stochastic process

is a

collection

{

X

t

∣

t

∈

T

}

of

random variables

X

t

defined on

(

Ω

,

ℱ

,

𝐏

)

, where

T

is a set, called the

index set

of the process

{

X

t

∣

t

∈

T

}

.

T

is
usually (but not always) a subset of

ℝ

.

X

is sometimes known as a

random function

.

Given any

t

, the possible values of

X

t

are called the

states

of the process at

t

. The set of all states (for all

t

) of a stochastic process is called its

state space

.

If

T

is discrete, then the stochastic process is a

discrete-time process

. If

T

is an interval of

ℝ

,
then

{

X

t

∣

t

∈

T

}

is a

continuous-time
process

. If

T

can be

linearly ordered

, then

t

is also known as
the

time

.

A stochastic process

X

with state space

S

can be thought of in either of following three ways.

•

As a collection of random variables,

X

t

, for each

t

in the index set

T

.

•

As a

function

in two variables

t

∈

T

and

ω

∈

Ω

,

X

:

T

×

Ω

→

S

,

(

t

,

ω

)

↦

X

t

⁢

(

ω

)

.

The process is said to be measurable, or, jointly measurable if it is

ℬ

⁢

(

T

)

⊗

ℱ

/

ℬ

⁢

(

S

)

-measurable. Here,

ℬ

⁢

(

T

)

and

ℬ

⁢

(

S

)

are the Borel

σ

-algebras on

T

and

S

respectively.

•

In terms of the

sample paths

. Each

ω

∈

Ω

maps to a function

T

→

S

,

t

↦

X

t

⁢

(

ω

)

.

Many common examples of stochastic processes have sample paths which are either

continuous

or cadlag.

Examples

. The following list is some of the most common and
important stochastic processes:

1.

a

random walk

, as well as its limiting case, a

Brownian motion

, or a

Wiener process

2.

Poisson process

3.

Markov process

; a

Markov chain

is a Markov process whose state space is discrete

4.

renewal process

Remarks

.

•

Sometimes, a stochastic process is also called a

random process

, although a stochastic process is generally
linked to any “time” dependent process. In a random process, the
index set may not be linearly ordered, as in the case of a random
field, where the index set may be, for example, the unit sphere

S

2

⊆

ℝ

3

.

•

In

statistics

, a stochastic process is often known as a

time series

, where the index set is a finite (or at most

countable

) ordered

sequence

of real numbers.

Title

stochastic process

Canonical name

StochasticProcess

Date of creation

2013-03-22 14:39:10

Last modified on

2013-03-22 14:39:10

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

14

Author

gel (22282)

Entry type

Definition

Classification

msc 60G05

Classification

msc 60G60

Synonym

random process

Related topic

DistributionsOfAStochasticProcess

Defines

discrete-time process

Defines

continuous-time process

Defines

state

Defines

time series

Defines

state space

Defines

random function

Defines

jointly measurable

Generated on Sat Feb 10 11:57:31 2018 by

LaTeXML
càdlàg process

càdlàg process

A càdlàg process

X

is a

stochastic process

for which the paths

t

↦

X

t

are right-continuous with left limits everywhere, with

probability

one. The word

càdlàg

is an acronym from the French for “continu à droite, limites à gauche”.
Such processes are widely used in the theory of noncontinuous stochastic processes. For example,

semimartingales

are càdlàg, and continuous-time

martingales

and many types of

Markov processes

have càdlàg

modifications

.

Given a càdlàg process

X

t

with time index

t

ranging over the nonnegative real numbers, its left limits are often denoted by

X

t

-

=

lim

s

→

t

,

s

<

t

⁡

X

s

for every

t

>

0

. Also, the jump at time

t

is written as

Δ

⁢

X

t

=

X

t

-

X

t

-

.

Alternative terms used to refer to a càdlàg process are

rcll

(right-continuous with left limits),

R-process

and

right-process

.

Although used less frequently, a process whose paths are almost surely left-continuous with right limits everywhere are known as

càglàd

,

lcrl

or

L-processes

.

Title

càdlàg process

Canonical name

CadlagProcess

Date of creation

2013-03-22 18:36:36

Last modified on

2013-03-22 18:36:36

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

7

Author

gel (22282)

Entry type

Definition

Classification

msc 60G07

Synonym

cadlag process

Synonym

rcll process

Synonym

R-process

Synonym

right-process

Related topic

UcpConvergenceOfProcesses

Defines

cadlag

Defines

rcll

Defines

R-process

Defines

right-process

Defines

càglàd

Defines

lcrl

Defines

L-process

Generated on Sat Feb 10 11:57:36 2018 by

LaTeXML
completeness of semimartingale convergence

completeness of semimartingale convergence

Theorem

.

Let

(

Ω

,

F

,

(

F

t

)

t

∈

R

+

,

P

)

be a

filtered probability space

. Then, the space of

semimartingales

S

forms a complete (

http://planetmath.org/Complete

)

topological vector space

under the

semimartingale topology

.

That is, semimartingale convergence is a vector topology (

http://planetmath.org/TopologicalVectorSpace

) and, for any sequence

X

n

∈

𝒮

such that

X

n

-

X

m

→

0

as

m

,

n

→

∞

then there exists an

X

∈

𝒮

with

X

n

→

X

.

Title

completeness of semimartingale convergence

Canonical name

CompletenessOfSemimartingaleConvergence

Date of creation

2013-03-22 18:40:47

Last modified on

2013-03-22 18:40:47

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

4

Author

gel (22282)

Entry type

Theorem

Classification

msc 60G07

Classification

msc 60G48

Classification

msc 60H05

Generated on Sat Feb 10 11:57:38 2018 by

LaTeXML
completeness under ucp convergence

completeness under ucp convergence

Let

(

Ω

,

ℱ

,

(

ℱ

t

)

t

∈

ℝ

+

,

ℙ

)

be a

filtered probability space

. Then, under the

ucp topology

, various classes of

stochastic processes

form

complete

(

http://planetmath.org/Complete

) vector spaces.

For a

σ

-algebra

ℳ

on

ℝ

+

×

Ω

, we can look at the processes which are

ℳ

-

measurable

when regarded as a map from

ℝ

+

×

Ω

to

ℝ

. As the ucp topology is only defined for jointly measurable processes, we restrict attention to sub-

σ

-algebras of the

product

ℬ

⁢

(

ℝ

+

)

⊗

ℱ

.

Theorem 1

.

Let

M

be a sub-

σ

-algebra of

B

⁢

(

R

+

)

⊗

F

. Then, the set of

M

-measurable processes is complete under ucp convergence.

That is, if

X

n

is a

sequence

of

ℳ

-measurable processes such that

X

n

-

X

m

→

ucp

0

as

m

,

n

→

∞

then

X

n

→

ucp

X

for some

ℳ

-measurable process

X

.

In particular, the spaces of jointly measurable,

progressive

,

optional

and

predictable processes

are each complete under ucp convergence.
We can also look at the properties of the

sample paths

of the processes.

Theorem 2

.

Let

S

be any set of

functions

R

+

→

R

which is closed (

http://planetmath.org/Closed

) under

uniform convergence

on

compacts

(the

compact-open topology

). Then, the set of jointly measurable processes whose sample paths are almost surely in

S

is complete under ucp convergence.

So, for example, the

continuous

, right-continuous, left-continuous and

cadlag processes

are each complete under the ucp topology. Furthermore, combining

theorems

1

and

2

, and using the fact that a cadlag process is

adapted

if and only if it is jointly measurable (see

measurability of stochastic processes

(

http://planetmath.org/MeasurabilityOfStochasticProcesses

)), the following useful result is obtained.

Corollary

.

The space of cadlag adapted processes is complete under ucp convergence.

Title

completeness under ucp convergence

Canonical name

CompletenessUnderUcpConvergence

Date of creation

2013-03-22 18:40:33

Last modified on

2013-03-22 18:40:33

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

5

Author

gel (22282)

Entry type

Theorem

Classification

msc 60G07

Classification

msc 60G05

Generated on Sat Feb 10 11:57:40 2018 by

LaTeXML
finite variation process

finite variation process

In the theory of

stochastic processes

, the term

finite-variation process

is used to refer to a process

X

t

whose paths are right-continuous and have finite

total variation

over every compact time interval, with

probability

one. See, for example, the

Poisson process

.

It can be shown that any function on the

real numbers

with finite total variation has left and right limits everywhere. Consequently,

finite variation processes

are always cadlag.

Title

finite variation process

Canonical name

FiniteVariationProcess

Date of creation

2013-03-22 18:36:41

Last modified on

2013-03-22 18:36:41

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

5

Author

gel (22282)

Entry type

Definition

Classification

msc 60G07

Generated on Sat Feb 10 11:57:44 2018 by

LaTeXML
independent stochastic processes

independent stochastic processes

Two

stochastic processes

{

X

⁢

(

t

)

∣

t

∈

T

}

and

{

Y

⁢

(

t

)

∣

t

∈

T

}

are said to be

if for

any

positive integer

n

<

∞

, and

any

sequence

t

1

,

…

,

t

n

∈

T

, the random vectors

𝑿

:=

(

X

⁢

(

t

1

)

,

…

,

X

⁢

(

t

n

)

)

and

𝒀

:=

(

Y

⁢

(

t

1

)

,

…

,

Y

⁢

(

t

n

)

)

are

independent

. This means,
for any two

n

-dimensional Borel sets

A

,

B

⊆

ℝ

n

,
we have

P

⁢

[

𝑿

-

1

⁢

(

A

)

∩

𝒀

-

1

⁢

(

B

)

]

=

P

⁢

[

𝑿

-

1

⁢

(

A

)

]

⁢

P

⁢

[

𝒀

-

1

⁢

(

B

)

]

.

Title

independent stochastic processes

Canonical name

IndependentStochasticProcesses

Date of creation

2013-03-22 15:24:36

Last modified on

2013-03-22 15:24:36

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Definition

Classification

msc 60G07

Generated on Sat Feb 10 11:57:50 2018 by

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Kolmogorov’s continuity theorem

Kolmogorov’s continuity theorem

Let

X

=

{

X

t

}

t

≥

0

be a process satisfying the following condition

:

For all

T

>

0

there exist positive constants

α

,

β

,

D

such that

E

⁢

[

|

X

t

-

X

s

|

α

]

≤

D

⁢

|

t

-

s

|

1

+

β

⁢

0

≤

s

,

t

≤

T

.

Then there exists a

continuous

modification of

X

.

Title

Kolmogorov’s continuity theorem

Canonical name

KolmogorovsContinuityTheorem

Date of creation

2013-03-22 15:43:43

Last modified on

2013-03-22 15:43:43

Owner

georgiosl (7242)

Last modified by

georgiosl (7242)

Numerical id

9

Author

georgiosl (7242)

Entry type

Theorem

Classification

msc 60G07

Related topic

DistributionsOfAStochasticProcess

Generated on Sat Feb 10 11:57:51 2018 by

LaTeXML
Kolmogorov’s extension theorem

Kolmogorov’s extension theorem

For all

t

1

,

⋯

,

t

k

,

k

∈

ℕ

, let

v

t

1

,

⋯

,

t

k

be

probability measures

on

ℝ

n

⁢

k

satisfying the following properties (consistency conditions):

1.

v

t

σ

⁢

(

1

)

,

⋯

,

t

σ

⁢

(

k

)

⁢

(

F

1

×

⋯

×

F

k

)

=

v

t

1

,

⋯

⁢

t

k

⁢

(

F

σ

-

1

⁢

(

1

)

×

⋯

⁢

F

σ

-

1

⁢

(

k

)

)

for all

permutations

σ

of

{

1

,

2

,

⋯

,

k

}

and for all

Borel sets

F

i

of

ℝ

n

2.

v

t

1

,

⋯

,

t

k

⁢

(

F

1

×

⋯

×

F

k

)

=

v

t

1

,

⋯

,

t

k

,

t

k

+

1

,

⋯

⁢

t

k

+

m

⁢

(

F

1

×

⋯

×

F

k

×

ℝ

n

×

⋯

×

ℝ

n

)

for all

m

∈

ℕ

and for all Borel sets

F

i

of

ℝ

n

Then there exists a

probability space

(

Ω

,

ℱ

,

P

)

and a

stochastic process

X

t

on

Ω

, indexed by

T

, taking values in

ℝ

n

such that

v

t

1

,

⋯

,

t

k

(

F

1

×

⋯

×

F

k

)

=

P

(

X

t

1

∈

F

1

,

⋯

,

X

t

k

∈

F

k

)

for all

t

i

∈

T

,

k

∈

ℝ

n

and all Borel sets

F

i

of

ℝ

n

Title

Kolmogorov’s extension theorem

Canonical name

KolmogorovsExtensionTheorem

Date of creation

2013-04-12 21:33:32

Last modified on

2013-04-12 21:33:32

Owner

Filipe (28191)

Last modified by

Filipe (28191)

Numerical id

3

Author

Filipe (28191)

Entry type

Theorem

Classification

msc 60G07

Generated on Sat Feb 10 11:57:54 2018 by

LaTeXML
local martingale

local martingale

Let

(

Ω

,

ℱ

,

(

ℱ

t

)

t

∈

𝕋

,

ℙ

)

be a

filtered probability space

, where the time

index set

𝕋

⊆

ℝ

has

minimal element

t

0

. The most common cases are discrete-time, with

𝕋

=

ℤ

+

, and

continuous

time where

𝕋

=

ℝ

+

, in which case

t

0

=

0

.

A process

X

is said to be a

local martingale

if it is locally (

http://planetmath.org/LocalPropertiesOfProcesses

) a right-continuous

martingale

. That is, if there is a

sequence

of

stopping times

τ

n

almost surely increasing to

infinity

and such that the

stopped processes

1

{

τ

n

>

t

0

}

⁢

X

τ

n

are martingales. Equivalently,

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.

Title

local martingale

Canonical name

LocalMartingale

Date of creation

2013-03-22 15:12:43

Last modified on

2013-03-22 15:12:43

Owner

skubeedooo (5401)

Last modified by

skubeedooo (5401)

Numerical id

8

Author

skubeedooo (5401)

Entry type

Definition

Classification

msc 60G07

Classification

msc 60G48

Related topic

Martingale

Related topic

LocalPropertiesOfProcesses

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Martingale criterion (continuous time)

Martingale criterion (continuous time)

Theorem 1

.

Let

X

be a

local martingale

. Then

X

is a

continuous

martingale

iff the set

{

X

τ

:

τ

⁢

stopping time

,

τ

≤

c

}

is

uniformly integrable

for each

c

≥

0

.

Title

Martingale criterion (continuous time)

Canonical name

MartingaleCriterioncontinuousTime

Date of creation

2013-03-22 18:54:26

Last modified on

2013-03-22 18:54:26

Owner

karstenb (16623)

Last modified by

karstenb (16623)

Numerical id

4

Author

karstenb (16623)

Entry type

Theorem

Classification

msc 60G07

Classification

msc 60G48

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

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Martingale criterion (discrete time)

Martingale criterion (discrete time)

Theorem

.

Let

X

=

(

X

n

,

F

n

)

be a

local martingale

. If there is an

n

0

∈

N

such that

∀

n

≥

n

0

,

n

∈

N

either:

E

⁢

X

n

-

<

∞

or:

E

⁢

X

n

+

<

∞

Then

X

is a

martingale

.

Title

Martingale criterion (discrete time)

Canonical name

MartingaleCriteriondiscreteTime

Date of creation

2013-03-22 18:34:48

Last modified on

2013-03-22 18:34:48

Owner

karstenb (16623)

Last modified by

karstenb (16623)

Numerical id

5

Author

karstenb (16623)

Entry type

Theorem

Classification

msc 60G07

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

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modification of a stochastic process

modification of a stochastic process

Let

{

X

t

}

t

≥

0

,

{

Y

t

}

t

≥

0

be

stochastic processes

on

(

Ω

,

ℱ

,

P

)

.

{

X

t

}

t

≥

0

is a

modification

of

{

Y

t

}

t

≥

0

if

P

⁢

[

{

ω

:

X

t

⁢

(

ω

)

=

Y

t

⁢

(

ω

)

}

]

=

1

for all

t

∈

[

0

,

∞

)

.

References

1

Bernt Øksendal.

, 5th ed Springer 1998.

Title

modification of a stochastic process

Canonical name

ModificationOfAStochasticProcess

Date of creation

2013-03-22 16:09:44

Last modified on

2013-03-22 16:09:44

Owner

georgiosl (7242)

Last modified by

georgiosl (7242)

Numerical id

9

Author

georgiosl (7242)

Entry type

Definition

Classification

msc 60G07

Classification

msc 60G05

Related topic

DistributionsOfAStochasticProcess

Defines

modification

Generated on Sat Feb 10 11:58:01 2018 by

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optional process

optional process

Suppose we are given a

filtration

(

http://planetmath.org/FiltrationOfSigmaAlgebras

)

(

ℱ

)

t

∈

𝕋

on a

measurable space

(

Ω

,

ℱ

)

. A

stochastic process

is said to be

adapted

if

X

t

is

ℱ

t

-measurable for every time

t

in the

index set

𝕋

. For an arbitrary,

uncountable

, index set

𝕋

⊆

ℝ

, this

property

is too restrictive to be useful. Instead, we can impose measurability conditions on

X

considered as a map from

𝕋

×

Ω

to

ℝ

.
For

instance

, we could require

X

to be

progressively measurable

, but that is still too weak a condition for many purposes. A

stronger

condition is for

X

to be

optional

. The index set

𝕋

is assumed to be a

closed subset

of

ℝ

in the following

definition

.

The class of optional processes forms the smallest set containing all adapted and right-continuous processes, and which is

closed under

taking limits of sequences of processes.

The

σ

-

algebra

,

𝒪

, on

𝕋

×

Ω

generated by

the right-continuous and adapted processes is called the

optional

σ

-algebra. Then, a process is optional if and only if it is

𝒪

-measurable.

Alternatively, the optional

σ

-algebra may be defined as

𝒪

=

σ

⁢

(

{

[

T

,

∞

)

:

T

⁢

is a stopping time

}

)

.

Here,

[

T

,

∞

)

is a stochastic interval, consisting of the pairs

(

t

,

ω

)

∈

𝕋

×

Ω

such that

T

⁢

(

ω

)

≤

t

.
In continuous-time, the equivalence of these two definitions for

𝒪

does require mild conditions on the filtration — it is enough for

ℱ

t

to be

universally complete

.

In the discrete-time case where the index set

𝕋

countable

, then the definitions above imply that a process

X

t

is optional if and only if it is adapted.

Title

optional process

Canonical

name

OptionalProcess

Date of creation

2013-03-22 18:37:34

Last modified on

2013-03-22 18:37:34

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

5

Author

gel (22282)

Entry type

Definition

Classification

msc 60G07

Related topic

ProgressivelyMeasurableProcess

Related topic

PredictableProcess

Defines

optional

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