proof of completeness of semimartingale convergence

proof of completeness of semimartingale convergence

We start by showing that

semimartingale convergence

is a vector topology (

http://planetmath.org/TopologicalVectorSpace

) on the space of

semimartingales

. That is,

addition

of processes and multiplication by real numbers are

continuous

(

http://planetmath.org/Continuous

)

operations

.

The continuity of addition follows immediately from the fact that the topology is given by a translation-invariant metric

D

s

; if

X

n

→

X

and

Y

n

→

Y

then

D

s

⁢

(

X

n

+

Y

n

-

X

-

Y

)

≤

D

s

⁢

(

X

n

-

X

)

+

D

s

⁢

(

Y

n

-

Y

)

→

0

so

X

n

+

Y

n

→

X

+

Y

.
Also, suppose that

λ

n

are real numbers converging to

λ

. Then, it is easily shown that

D

s

⁢

(

λ

n

⁢

Y

)

≤

max

⁡

(

|

λ

n

|

,

1

)

⁢

D

s

⁢

(

Y

)

for all processes

Y

and, therefore,

D

s

⁢

(

λ

n

⁢

X

n

-

λ

⁢

X

)

≤

D

s

⁢

(

λ

n

⁢

(

X

n

-

X

)

)

+

D

s

⁢

(

(

λ

n

-

λ

)

⁢

X

)

≤

max

⁡

(

|

λ

n

|

,

1

)

⁢

D

s

⁢

(

X

n

-

X

)

+

D

s

⁢

(

(

λ

n

-

λ

)

⁢

X

)

.

As was noted (in semimartingale convergence (

http://planetmath.org/SemimartingaleConvergence

)), the statement that

D

⁢

(

(

λ

n

-

λ

)

⁢

X

)

→

0

whenever

λ

n

→

λ

is

equivalent

to the statement that

X

is a semimartingale, so

λ

n

⁢

X

n

→

λ

⁢

X

and

D

s

does indeed generate a vector topology.

It only remains to show that the topology is

complete

.
So, suppose that

X

n

-

X

m

→

0

under the semimartingale topology. Then, we also have

ucp convergence

(see

semimartingale convergence implies ucp convergence

) and,

X

n

→

ucp

X

for a cadlag

adapted process

X

(see

completeness under ucp convergence

). We need to show that this also

converges

under the semimartingale topology.

For any

simple predictable process

ξ

,

∫

0

t

ξ

⁢

𝑑

X

n

→

∫

0

t

ξ

⁢

𝑑

X

in

probability

. Therefore, for any

sequence

of simple predictable processes

ξ

n

,

∫

0

t

ξ

n

⁢

𝑑

X

n

-

∫

0

t

ξ

n

⁢

𝑑

X

=

lim

m

→

∞

⁡

(

∫

0

t

ξ

n

⁢

𝑑

X

n

-

∫

0

t

ξ

n

⁢

𝑑

X

m

)

where the limit is taken in probability. If

|

ξ

n

|

≤

1

then the condition that

X

n

-

X

m

goes to zero in the semimartingale topology implies that the

right hand side

tends to zero in probability as

n

goes to

infinity

and,

X

n

→

X

in the semimartingale topology.

This shows that semimartingale convergence is complete on the space of cadlag adapted processes. To show that it is complete on the set of semimartingales, we just need to show that the process

X

above is a semimartingale whenever

X

n

are.
However, for any sequence of real numbers

λ

n

→

0

then,

D

s

⁢

(

λ

n

⁢

X

)

≤

D

s

⁢

(

λ

n

⁢

X

m

)

+

max

⁡

(

|

λ

n

|

,

1

)

⁢

D

s

⁢

(

X

-

X

m

)

.

As noted previously, the condition that

X

m

are semimartingales gives

λ

n

⁢

X

m

→

0

as

n

tends to infinity.

lim sup

n

→

∞

⁡

D

s

⁢

(

λ

n

⁢

X

)

≤

sup

n

⁡

max

⁡

(

|

λ

n

|

,

1

)

⁢

D

s

⁢

(

X

-

X

m

)

.

Taking the limit

m

→

∞

shows that

λ

n

⁢

X

tends to zero in the semimartingale topology and, consequently,

X

is a semimartingale.

Title

proof of completeness of semimartingale convergence

Canonical name

ProofOfCompletenessOfSemimartingaleConvergence

Date of creation

2013-03-22 18:40:54

Last modified on

2013-03-22 18:40:54

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

4

Author

gel (22282)

Entry type

Proof

Classification

msc 60G07

Classification

msc 60G48

Classification

msc 60H05

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