existence of the conditional expectation

existence of the conditional expectation

Let

(

Ω

,

ℱ

,

ℙ

)

be a

probability space

and

X

be a

random variable

. For any

σ

-algebra

𝒢

⊆

ℱ

, we show the existence of the

conditional expectation

𝔼

[

X

∣

𝒢

]

. Although it is possible to do this using the

Radon-Nikodym theorem

, a different approach is used here which relies on the completeness of the

vector space

L

2

.
The defining property of the conditional expectation

Y

=

𝔼

[

X

∣

𝒢

]

is

𝔼

⁢

[

1

G

⁢

Y

]

=

𝔼

⁢

[

1

G

⁢

X

]

(1)

for sets

G

∈

𝒢

. We shall prove the existence of the conditional expectation for all nonnegative random variables and, more generally, whenever

𝔼

[

|

X

|

∣

𝒢

]

is almost surely finite.

First, the conditional expectation of every square-integrable random variable exists.

Theorem 1

.

Suppose that

E

⁢

[

X

2

]

<

∞

. Then there is a

G

-

measurable

random variable

Y

satisfying

E

⁢

[

Y

2

]

<

∞

and equation (

1

) is satisfied for all

G

∈

G

.

Proof.

Consider the norm

∥

Y

∥

2

≡

𝔼

⁢

[

Y

2

]

1

/

2

on the vector space

V

=

L

2

⁢

(

Ω

,

ℱ

,

ℙ

)

of real valued random variables

Y

satisfying

𝔼

⁢

[

Y

2

]

<

∞

(up to

ℙ

almost everywhere equivalence). This is given by the following

inner product

⟨

Y

1

,

Y

2

⟩

≡

𝔼

⁢

[

Y

1

⁢

Y

2

]

.

As

L

p

-spaces are

complete

, this makes

V

into a

Hilbert space

(see also,

L

2

-spaces are Hilbert spaces (

http://planetmath.org/L2SpacesAreHilbertSpaces

)). Then,

U

≡

L

2

⁢

(

Ω

,

𝒢

,

ℙ

)

is a complete, and hence closed,

subspace

of

V

.

By the existence of

orthogonal projections

(

http://planetmath.org/ProjectionsAndClosedSubspaces

) onto closed subspaces of Hilbert spaces, there is an orthogonal projection

π

:

V

→

U

. In particular,

⟨

π

⁢

Y

-

Y

,

Z

⟩

=

0

for all

Y

∈

V

and

Z

∈

U

. Setting

Y

=

π

⁢

X

gives

𝔼

⁢

[

1

G

⁢

Y

]

-

𝔼

⁢

[

1

G

⁢

X

]

=

⟨

1

G

,

π

⁢

X

-

X

⟩

=

0

as required.
∎

We can now prove the

existence of conditional expectations

of nonnegative random variables. Note that here there are no integrability conditions on

X

.

Theorem 2

.

Let

X

be a nonnegative random variable taking values in

R

∪

{

∞

}

. Then, there exists a nonnegative

G

-measurable random variable

Y

taking values in

R

∪

{

∞

}

and satisfying (

1

) for all

G

∈

G

. Furthermore,

Y

is uniquely defined

P

-almost everywhere (

http://planetmath.org/AlmostSurely

).

Proof.

First, let

X

n

=

min

⁡

(

n

,

X

)

. As this is

bounded

, theorem

1

says that the conditional expectations

Y

n

=

𝔼

[

Y

n

∣

𝒢

]

exist. Furthermore, as

X

0

=

0

, we may take

Y

0

=

0

.
For any

n

, setting

G

=

{

Y

n

+

1

<

Y

n

}

∈

𝒢

gives

𝔼

⁢

[

1

G

⁢

(

Y

n

-

Y

n

+

1

)

]

=

𝔼

⁢

[

1

G

⁢

(

X

n

-

X

n

+

1

)

]

≤

0

.

So

1

G

⁢

(

Y

n

-

Y

n

+

1

)

is a nonnegative random variable with nonpositive

expectation

, hence is almost surely equal to zero. Therefore,

Y

n

+

1

≥

Y

n

(almost surely) and, by replacing

Y

n

with the

maximum

of

Y

1

,

…

⁢

Y

n

we may suppose that

(

Y

n

)

is an

increasing sequence

of random variables. Setting

Y

=

sup

n

⁡

Y

n

, the

monotone convergence theorem

gives

𝔼

⁢

[

1

G

⁢

Y

]

=

lim

n

→

∞

⁡

𝔼

⁢

[

1

G

⁢

Y

n

]

=

lim

n

→

∞

⁡

𝔼

⁢

[

1

G

⁢

X

n

]

=

𝔼

⁢

[

1

G

⁢

X

]

as required.

Finally, suppose that

Y

~

is also a nonnegative

𝒢

-measurable random variable satisfying (

1

). For any

x

∈

ℝ

, setting

G

=

{

Y

~

>

Y

,

x

>

Y

}

then

1

G

⁢

Y

is bounded and,

𝔼

⁢

[

1

G

⁢

(

Y

~

-

Y

)

]

=

𝔼

⁢

[

1

G

⁢

X

]

-

𝔼

⁢

[

1

G

⁢

X

]

=

0

showing that

ℙ

⁢

(

G

)

=

0

. Letting

x

increase to

infinity

gives

Y

~

≤

Y

(almost surely) and, similarly,

Y

≤

Y

~

so that

Y

=

Y

~

almost surely.
∎

Finally, we show existence of the conditional expectation of every random variable

X

satisfying

𝔼

[

|

X

|

∣

𝒢

]

<

∞

almost surely. Note, in particular, that this is satisfied whenever

X

is

integrable

, as

𝔼

[

𝔼

[

|

X

|

∣

𝒢

]

]

=

𝔼

[

|

X

|

]

<

∞

.

Theorem 3

.

Let

X

be a random variable such that

E

[

|

X

|

∣

G

]

<

∞

almost surely. Then, there exists a

G

-measurable random variable

Y

such that

E

⁢

[

1

G

⁢

|

Y

|

]

<

∞

and (

1

) is satisfied for every

G

∈

G

with

E

⁢

[

1

G

⁢

|

X

|

]

<

∞

.

Furthermore,

Y

is uniquely defined up to

P

-a.e. equivalence.

Proof.

The positive and negative parts

X

+

,

X

-

of

X

satisfy

𝔼

[

X

+

∣

𝒢

]

+

𝔼

[

X

-

∣

𝒢

]

=

𝔼

[

|

X

|

∣

𝒢

]

<

∞

almost surely. We can therefore set

Y

±

≡

𝔼

[

X

±

∣

𝒢

]

and

Y

=

Y

+

-

Y

-

.

If

G

∈

𝒢

satisfies

𝔼

⁢

[

1

G

⁢

|

X

|

]

<

∞

then

𝔼

⁢

[

1

G

⁢

Y

±

]

=

𝔼

⁢

[

1

G

⁢

X

±

]

<

∞

, so

𝔼

⁢

[

1

G

⁢

|

Y

|

]

<

∞

and,

𝔼

⁢

[

1

G

⁢

Y

]

=

𝔼

⁢

[

1

G

⁢

Y

+

]

-

𝔼

⁢

[

1

G

⁢

Y

-

]

=

𝔼

⁢

[

1

G

⁢

X

+

]

-

𝔼

⁢

[

1

G

⁢

X

-

]

=

𝔼

⁢

[

1

G

⁢

X

]

as required.

Finally, suppose that

Y

~

satisfies the same conditions as

Y

. For any

x

≥

0

set

G

=

{

Y

+

+

Y

-

≤

x

,

Y

~

>

Y

}

∈

𝒢

.
Then,

𝔼

⁢

[

1

G

⁢

|

X

|

]

=

𝔼

⁢

[

1

G

⁢

(

Y

+

+

Y

-

)

]

≤

x

<

∞

.

So,

𝔼

⁢

[

1

G

⁢

|

Y

|

]

and

𝔼

⁢

[

1

G

⁢

|

Y

~

|

]

are finite, hence (

1

) gives

𝔼

⁢

[

1

G

⁢

(

Y

~

-

Y

)

]

=

𝔼

⁢

[

1

G

⁢

X

]

-

𝔼

⁢

[

1

G

⁢

X

]

=

0

.

So

ℙ

⁢

(

G

)

=

0

and, letting

x

increase to infinity,

Y

~

≤

Y

almost surely. Similarly,

Y

≤

Y

~

and therefore

Y

~

=

Y

almost surely.
∎

Title

existence of the conditional expectation

Canonical name

ExistenceOfTheConditionalExpectation

Date of creation

2013-03-22 18:39:28

Last modified on

2013-03-22 18:39:28

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

5

Author

gel (22282)

Entry type

Theorem

Classification

msc 60A10

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