proof of completeness under ucp convergence

proof of completeness under ucp convergence

Let

(

Ω

,

ℱ

,

(

ℱ

t

)

t

∈

ℝ

+

,

ℙ

)

be a

filtered probability space

,

ℳ

be a sub-

σ

-algebra of

ℬ

⁢

(

ℝ

+

)

⊗

ℱ

, and

S

be a set of real valued

functions

on

ℝ

+

which is closed (

http://planetmath.org/Closed

) under

uniform convergence

on

compacts

. We show that both the set of

ℳ

-measurable processes and the set of

jointly measurable

processes with

sample paths

almost surely in

S

are

complete

(

http://planetmath.org/Complete

) under

ucp convergence

. The method used will be to show that we can pass to a subsequence which almost surely

converges uniformly

on compacts.

We start by writing out the metric generating the topology of uniform convergence on compacts (

compact-open topology

) for functions

ℝ

+

→

ℝ

. This is the same as uniform convergence on each of the

bounded intervals

[

0

,

n

)

for positive integers

n

,

d

⁢

(

X

)

≡

∑

n

=

1

∞

2

-

n

⁢

min

⁡

(

1

,

sup

t

<

n

⁡

|

X

t

|

)

.

Then, the metric is

(

X

,

Y

)

↦

d

⁢

(

X

-

Y

)

.

Convergence

under the ucp topology is given by

D

ucp

⁢

(

X

)

=

𝔼

⁢

[

d

⁢

(

X

)

]

for any jointly measurable

stochastic process

X

, with the (pseudo)metric being

(

X

,

Y

)

↦

D

ucp

⁢

(

X

-

Y

)

.

Now, suppose that

X

n

is a

sequence

of jointly measurable processes such that

X

n

-

X

m

→

ucp

0

as

m

,

n

→

∞

. Then,

D

ucp

⁢

(

X

n

-

X

m

)

→

0

and we may pass to a subsequence

X

n

k

satisfying

D

ucp

⁢

(

X

n

j

-

X

n

k

)

≤

2

-

j

whenever

k

>

j

. So,

𝔼

⁢

[

∑

k

d

⁢

(

X

n

k

-

X

n

k

+

1

)

]

=

∑

k

D

ucp

⁢

(

X

n

k

-

X

n

k

+

1

)

≤

∑

k

2

-

k

=

1

.

In particular, this shows that

∑

k

d

⁢

(

X

n

k

-

X

n

k

+

1

)

is almost surely finite and, therefore,

d

⁢

(

X

n

j

-

X

n

k

)

≤

∑

i

=

j

k

-

1

d

⁢

(

X

n

i

-

X

n

i

+

1

)

→

0

as

k

>

j

→

∞

, with

probability

one.

So, the sequence

X

n

k

is almost surely Cauchy (

http://planetmath.org/CauchySequence

), under the topology of uniform convergence on compacts. We set

X

t

⁢

(

ω

)

≡

{

lim

k

→

∞

⁡

X

t

n

k

⁢

(

ω

)

,

if the limit exists

,

0

,

otherwise

.

As measurability of real valued functions is preserved under

pointwise convergence

, it follows that if

X

n

are

ℳ

-measurable, then so is

X

. In particular,

X

is a jointly measurable process.
Furthermore, since convergence is almost surely uniform on compacts, if

X

n

have sample paths in

S

with probability one then so does

X

.

It only remains to show that

X

n

→

ucp

X

. However, we have already shown that

d

⁢

(

X

n

k

-

X

)

→

0

with probability one, hence

D

ucp

⁢

(

X

n

k

-

X

)

→

0

.

D

ucp

⁢

(

X

n

-

X

)

≤

D

ucp

⁢

(

X

n

-

X

n

k

)

+

D

ucp

⁢

(

X

n

k

-

X

)

.

Letting

k

go to

infinity

, this is

bounded

by

sup

m

>

n

⁡

D

ucp

⁢

(

X

n

-

X

m

)

, which goes to zero as

n

→

∞

.

Title

proof of completeness under ucp convergence

Canonical name

ProofOfCompletenessUnderUcpConvergence

Date of creation

2013-03-22 18:40:35

Last modified on

2013-03-22 18:40:35

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

5

Author

gel (22282)

Entry type

Proof

Classification

msc 60G07

Classification

msc 60G05

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