σ-algebra at a stopping time

σ

-algebra at a stopping time

Let

(

ℱ

t

)

t

∈

𝕋

be a

filtration

(

http://planetmath.org/FiltrationOfSigmaAlgebras

) on a

measurable space

(

Ω

,

ℱ

)

. For every

t

∈

𝕋

, the

σ

-algebra

ℱ

t

represents the

collection

of events which are observable up until time

t

. This

concept

can be generalized to any

stopping time

τ

:

Ω

→

𝕋

∪

{

∞

}

.

For a stopping time

τ

, the collection of events observable up until time

τ

is denoted by

ℱ

τ

and is

generated by

sampling

progressively measurable processes

ℱ

τ

=

σ

⁢

(

{

X

τ

∧

t

:

X

⁢

is progressive,

⁢

t

∈

𝕋

}

)

.

The reason for sampling

X

at time

τ

∧

t

rather than at

τ

is to include the possibility that

τ

=

∞

, in which case

X

τ

is not defined.

A

random variable

V

is

ℱ

τ

-measurable if and only if it is

ℱ

∞

-measurable and the process

X

t

≡

1

{

τ

≤

t

}

⁢

V

is

adapted

.

This can be shown as follows. If

X

is a progressively measurable process, then the

stopped process

X

τ

∧

s

is also progressive. In particular,

V

≡

X

τ

∧

s

=

X

s

τ

∧

s

is

ℱ

∞

-measurable and

1

{

τ

≤

t

}

⁢

V

=

1

{

τ

≤

t

}

⁢

X

t

τ

∧

s

is

ℱ

t

-measurable.

Conversely

, if

V

is

ℱ

t

-measurable then

X

s

≡

1

{

s

>

t

}

⁢

V

is a progressive process and

1

{

τ

>

t

}

⁢

V

=

X

τ

∧

t

is

ℱ

τ

-measurable. By letting

t

increase to

infinity

, it follows that

1

{

τ

=

∞

}

⁢

V

is

ℱ

τ

-measurable for every

ℱ

∞

-measurable random variable

V

.
Now suppose also that

X

t

≡

1

{

τ

≤

t

}

⁢

V

is adapted, and hence progressive. Then,

1

{

τ

≤

t

}

⁢

V

=

X

τ

∧

t

is

ℱ

τ

-measurable. Letting

t

increase to infinity shows that

V

=

1

{

τ

<

∞

}

⁢

V

+

1

{

τ

=

∞

}

⁢

V

is

ℱ

τ

-measurable.

As a set

A

is

ℱ

τ

-measurable if and only if

1

A

is an

ℱ

τ

-measurable random variable, this gives the following alternative definition,

ℱ

τ

=

{

A

∈

ℱ

∞

:

A

∩

{

τ

≤

t

}

∈

ℱ

t

for all

t

∈

𝕋

}

.

From this, it is not difficult to show that the following properties are satisfied

1.

Any stopping time

τ

is

ℱ

τ

-measurable.

2.

If

τ

⁢

(

ω

)

=

t

∈

𝕋

∪

{

∞

}

for all

ω

∈

Ω

then

ℱ

τ

=

ℱ

t

.

3.

If

σ

,

τ

are stopping times and

A

∈

ℱ

σ

then

A

∩

{

σ

≤

τ

}

∈

ℱ

τ

. In particular, if

σ

≤

τ

then

ℱ

σ

⊆

ℱ

τ

.

4.

If

σ

,

τ

are stopping times and

A

∈

ℱ

σ

then

A

∩

{

σ

=

τ

}

∈

ℱ

τ

.

5.

if the filtration

(

ℱ

t

)

is right-continuous and

τ

n

≥

τ

are stopping times with

τ

n

→

τ

then

ℱ

τ

=

⋂

n

ℱ

τ

n

. More generally, if

τ

n

=

τ

eventually then this is true irrespective of whether the filtration is right-continuous.

6.

If

τ

n

are stopping times with

τ

n

=

τ

eventually then

ℱ

τ

n

→

ℱ

τ

. That is,

ℱ

τ

=

⋂

n

σ

⁢

(

⋃

m

≥

n

ℱ

τ

m

)

.

In continuous-time, for any stopping time

τ

the

σ

-algebra

ℱ

τ

+

is the set of events observable up until time

t

with respect to the right-continuous filtration

(

ℱ

t

+

)

. That is,

ℱ

τ

+

=

{

A

∈

ℱ

∞

:

A

∩

{

τ

≤

t

}

∈

ℱ

t

+

for every

t

∈

𝕋

}

=

{

A

∈

ℱ

∞

:

A

∩

{

τ

<

t

}

∈

ℱ

t

for every

t

∈

𝕋

}

.

If

τ

n

≥

τ

are stopping times with

τ

n

>

τ

whenever

τ

<

∞

is not a

maximal element

of

𝕋

, and

τ

n

→

τ

then,

ℱ

τ

+

=

⋂

n

ℱ

τ

n

=

⋂

n

ℱ

τ

n

+

.

The

σ

-algebra of events observable up until just before time

τ

is denoted by

ℱ

τ

-

and is generated by sampling

predictable processes

ℱ

τ

-

=

σ

⁢

(

{

X

τ

∧

t

:

X

⁢

is predictable,

⁢

t

∈

𝕋

}

)

.

Suppose that the

index set

𝕋

⊆

ℝ

has minimal element

t

0

.
As the predictable

σ

-algebra is generated by sets of the form

(

s

,

∞

)

×

A

for

s

∈

𝕋

and

A

∈

ℱ

s

, and

{

t

0

}

×

A

for

A

∈

ℱ

t

0

, the definition above can be rewritten as,

ℱ

τ

-

=

σ

(

{

A

∩

{

τ

>

s

}

:

s

∈

𝕋

,

A

∈

ℱ

s

}

∪

ℱ

t

0

)

.

Clearly,

ℱ

τ

-

⊆

ℱ

τ

⊆

ℱ

τ

+

. Furthermore, for any stopping times

σ

,

τ

then

ℱ

σ

+

⊆

ℱ

τ

-

when restricted to the set

{

σ

<

τ

}

.

If

τ

n

is a

sequence

of stopping times announcing (

http://planetmath.org/PredictableStoppingTime

)

τ

, so that

τ

is predictable, then

ℱ

τ

-

=

σ

⁢

(

⋃

n

ℱ

τ

n

)

.

Title

σ

-algebra at a stopping time

Canonical name

sigmaalgebraAtAStoppingTime

Date of creation

2013-03-22 18:38:56

Last modified on

2013-03-22 18:38:56

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

6

Author

gel (22282)

Entry type

Definition

Classification

msc 60G40

Related topic

DoobsOptionalSamplingTheorem

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