semimartingale topology

semimartingale topology

Let

(

Ω

,

ℱ

,

(

ℱ

t

)

t

∈

ℝ

+

)

,

ℙ

)

be a

filtered probability space

and

(

X

t

n

)

,

(

X

t

)

be cadlag

adapted processes

.
Then,

X

n

is said to

converge

to

X

in the

semimartingale topology

if

X

0

n

→

X

0

in

probability

and

∫

0

t

ξ

n

⁢

𝑑

X

n

-

∫

0

t

ξ

n

⁢

𝑑

X

→

0

in probability as

n

→

∞

, for every

t

>

0

and sequence of

simple predictable processes

|

ξ

n

|

≤

1

.

This

topology

occurs with

stochastic calculus

where, according to the

dominated convergence theorem

(

http://planetmath.org/DominatedConvergenceForStochasticIntegration

),

stochastic integrals

converge in the semimartingale topology.
Furthermore, stochastic integration with respect to any

locally bounded

(

http://planetmath.org/LocalPropertiesOfProcesses

)

predictable process

ξ

is

continuous

under the semimartingale topology. That is, if

X

n

are

semimartingales

converging to

X

then

∫

ξ

⁢

𝑑

X

n

converges to

∫

ξ

⁢

𝑑

X

, a fact which does not hold under weaker topologies such as

ucp convergence

.

Also, for cadlag

martingales

,

L

1

convergence implies semimartingale convergence.

It can be shown that

semimartingale convergence implies ucp convergence

. Consequently,

X

n

converges to

X

in the semimartingale topology if and only if

X

0

n

-

X

0

+

∫

ξ

n

⁢

𝑑

X

n

-

∫

ξ

n

⁢

𝑑

X

→

ucp

0

for all sequences of simple predictable processes

|

ξ

n

|

≤

1

.

The topology is described by a metric as follows. First, let

D

ucp

⁢

(

X

-

Y

)

be a metric defining the ucp topology. For example,

D

ucp

⁢

(

X

)

=

∑

n

=

1

∞

2

-

n

⁢

𝔼

⁢

[

min

⁡

(

1

,

sup

t

<

n

⁡

|

X

t

|

)

]

.

Then, a metric

D

s

⁢

(

X

-

Y

)

for semimartingale convergence is given by

D

s

⁢

(

X

)

=

sup

⁡

{

D

ucp

⁢

(

X

0

+

ξ

⋅

X

)

:

|

ξ

|

≤

1

⁢

is simple previsible

}

(

ξ

⋅

X

denotes the

integral

∫

ξ

⁢

𝑑

X

). This is a proper metric under identification of processes with almost surely

equivalent

sample paths

, otherwise it is a

pseudometric

.

If

λ

n

≠

0

is a sequence of

real numbers

converging to zero and

X

is a cadlag adapted process then

λ

n

⁢

X

→

0

in the semimartingale topology if and only if

λ

n

⁢

∫

0

t

ξ

n

⁢

𝑑

X

→

0

in probability, for every

t

>

0

and simple predictable processes

|

ξ

n

|

≤

1

.
By the

sequential characterization of boundedness

(

http://planetmath.org/SequentialCharacterizationOfBoundedness

), this is equivalent to the statement that

{

∫

0

t

ξ

⁢

𝑑

X

:

|

ξ

|

≤

1

⁢

is simple predictable

}

is

bounded

in probability for every

t

>

0

.
So,

λ

n

⁢

X

→

0

in the semimartingale topology if and only if

X

is a semimartingale. It follows that semimartingale convergence only becomes a vector topology (

http://planetmath.org/TopologicalVectorSpace

) when restricted to the space of semimartingales. Then, it can be shown that the set of semimartingales is a complete topological vector space (

http://planetmath.org/CompletenessOfSemimartingaleConvergence

).

Title

semimartingale topology

Canonical

name

SemimartingaleTopology

Date of creation

2013-03-22 18:40:41

Last modified on

2013-03-22 18:40:41

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

6

Author

gel (22282)

Entry type

Definition

Classification

msc 60G48

Classification

msc 60G07

Classification

msc 60H05

Synonym

semimartingale convergence

Related topic

UcpConvergence

Related topic

UcpConvergenceOfProcesses

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