ucp convergence of processes

ucp convergence of processes

Let

(

Ω

,

ℱ

,

(

ℱ

t

)

t

∈

ℝ

+

,

ℙ

)

be a

filtered probability space

. Then a

sequence

of

stochastic processes

(

X

t

n

)

t

∈

ℝ

+

is said to

converge

to the process

(

X

t

)

in the

ucp topology

(uniform convergence on

compacts

in

probability

) if

sup

s

<

t

⁡

|

X

s

n

-

X

s

|

→

0

(1)

in probability (

http://planetmath.org/ConvergenceInProbability

) as

n

→

∞

, for every

t

>

0

. That is, if

ℙ

(

sup

s

<

t

|

X

s

n

-

X

s

|

>

ϵ

)

→

0

for all

ϵ

,

t

>

0

.
The notation

X

n

→

ucp

X

is sometimes used, and

X

n

is said to converge ucp to

X

.
This mode of convergence occurs

frequently in

the theory of continuous-time stochastic processes, and some examples are given below.

Note that the expression on the left hand side of (

1

) is a

supremum

of an

uncountable set

of

random variables

and, therefore, need not be a

measurable

quantity in general.
If, however, the processes have left or right-continuous

sample paths

, then the supremum can be restricted to

rational

times

sup

s

<

t

⁡

|

X

s

n

-

X

s

|

=

sup

s

<

t

,

s

∈

ℚ

+

⁡

|

X

s

n

-

X

s

|

and is measurable. Typically, it is only required that the sample paths are left or right-continuous almost surely, so the above equality holds on a set of probability one.
More generally, if the processes are jointly measurable, then the expression on the left hand side of (

1

) will be a measurable random variable in the

completion

(

http://planetmath.org/CompleteMeasure

) of the

probability space

(by the

measurable projection theorem

).
This is enough to ensure that the definition above is meaningful, and gives a well defined

topology

on the space of jointly measurable processes.

The ucp topology can be

generated by

a

pseudometric

. For example, setting

D

ucp

⁢

(

X

)

=

∑

n

=

1

∞

2

-

n

⁢

𝔼

⁢

[

min

⁡

(

1

,

sup

s

<

n

⁡

|

X

s

|

)

]

then

(

X

,

Y

)

↦

D

ucp

⁢

(

X

-

Y

)

is a pseudometric on the space of measurable processes such that

X

n

→

ucp

X

if and only if

D

ucp

⁢

(

X

n

-

X

)

→

0

as

n

→

∞

.
Furthermore, this becomes a metric under the identification of processes with almost surely identical sample paths.

If

X

n

→

ucp

X

then we may pass to a subsequence satisfying

D

⁢

(

X

n

k

-

X

)

<

2

-

k

. The

Borel-Cantelli lemma

then implies that

X

n

k

→

X

uniformly on all compact

intervals

, with probability one.
So, any sequence converging in the ucp topology has a subsequence converging uniformly on compacts, with probability one. Consequently, given any property of the sample paths which is preserved under uniform convergence on compacts, then it is also preserved under ucp convergence with probability one. In particular, ucp limits of

cadlag processes

are themselves cadlag.

Some examples of ucp convergence are given below.

1.

If

X

n

,

X

are cadlag

martingales

such that

𝔼

⁢

[

|

X

t

n

-

X

t

|

]

→

0

for every

t

>

0

then, Doob’s

inequality

ℙ

(

sup

s

≤

t

|

X

s

n

-

X

s

|

>

ϵ

)

≤

ϵ

-

1

𝔼

[

|

X

t

n

-

X

t

|

]

shows that

X

n

converges ucp to

X

.

2.

Let

X

be a

semimartingale

and

ξ

n

be

predictable processes

converging

pointwise

to

ξ

such that

sup

n

⁡

|

ξ

n

|

is

X

-

integrable

. Then, the

dominated convergence theorem

for

stochastic integration

(

http://planetmath.org/DominatedConvergenceForStochasticIntegration

) gives

∫

ξ

n

⁢

𝑑

X

→

ucp

∫

ξ

⁢

dX

.

3.

Suppose that the

stochastic differential equation

d

⁢

X

=

a

⁢

(

X

)

⁢

d

⁢

W

+

b

⁢

(

X

)

⁢

d

⁢

t

for

continuous functions

a

,

b

:

ℝ

→

ℝ

and

Brownian motion

W

has a unique solution with

X

0

=

0

. For any

partition

0

=

t

0

≤

t

1

≤

⋯

↑

∞

, the following

discrete

approximation can be constructed

X

~

0

=

0

,

X

~

t

k

+

1

=

X

~

t

k

+

a

⁢

(

X

~

t

k

)

⁢

(

W

t

k

+

1

-

W

t

k

)

+

b

⁢

(

X

~

t

k

)

⁢

(

t

k

+

1

-

t

k

)

.

Setting

X

~

t

=

X

~

t

k

for

t

∈

(

t

k

,

t

k

+

1

)

then these discrete approximations converge to

X

in the ucp topology as the partition mesh goes to zero.

References

1

Philip E. Protter,

Stochastic integration and

differential equations

.

Second edition. Applications of Mathematics, 21. Stochastic Modelling and Applied Probability. Springer-Verlag, 2004.

Title

ucp convergence of processes

Canonical name

UcpConvergenceOfProcesses

Date of creation

2013-03-22 18:39:48

Last modified on

2013-03-22 18:39:48

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

6

Author

gel (22282)

Entry type

Definition

Classification

msc 60G05

Classification

msc 60G07

Related topic

CadlagProcess

Related topic

SemimartingaleTopology

Defines

ucp convergence

Defines

ucp topology

Defines

converges ucp

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