aarjaneiro/planetmath huggingface dataset:
proof of Martingale criterion

proof of Martingale criterion

Let

(

τ

k

)

k

≥

1

be a localizing

sequence

of

stopping times

for

X

. Then:

Λ

{

τ

k

≥

n

}

↑

Λ

Ω

⁢

a.s.

⁢

k

→

∞

,

∀

n

∈

ℕ

since

{

τ

k

≥

n

}

↑

k

⋃

1

∞

{

τ

k

≥

n

}

∀

n

∈

ℕ

.

⋃

k

=

1

∞

{

τ

k

≥

n

}

=

Ω

⁢

a.s., since

⁢

τ

k

→

∞

,

a.s.

Now assume

E

⁢

X

n

-

<

∞

,

∀

n

≥

n

0

(the case

E

⁢

X

n

+

<

∞

being analogous).

1)

We have

E

⁢

X

n

-

<

∞

⁢

∀

n

∈

ℕ

.

We proceed by (backward)

induction

. For

n

=

n

0

the statement holds.

n

↦

n

-

1

:

(

X

τ

k

)

-

=

(

X

τ

k

∧

n

-

)

n

∈

ℕ

⁢

submartingale

We have:

∫

{

τ

k

≥

n

}

X

n

-

1

-

⁢

𝑑

P

=

∫

{

τ

k

≥

n

}

X

τ

k

∧

(

n

-

1

)

-

⁢

𝑑

P

≤

∫

{

τ

k

≥

n

}

X

τ

k

∧

n

-

⁢

𝑑

P

=

∫

{

τ

k

≥

n

}

X

n

-

⁢

𝑑

P

≤

∫

X

n

-

⁢

𝑑

P

<

∞

Where the first to second line is the

submartingale

property

and the last line follows by

induction hypothesis

.

Using Fatou we get:

∫

X

n

-

1

-

⁢

𝑑

P

=

∫

lim

k

→

∞

⁡

X

n

-

1

-

⁢

Λ

{

τ

k

≥

n

}

⁢

d

⁢

P

≤

lim inf

k

→

∞

⁡

∫

X

n

-

1

-

⁢

Λ

{

τ

k

≥

n

}

⁢

𝑑

P

≤

∫

X

n

-

⁢

𝑑

P

<

∞

2)

We have

X

n

∈

ℒ

1

(

n

∈

ℕ

)

.

We have

X

τ

k

∧

n

+

→

X

n

+

a.s.,

k

→

∞

,

∀

n

∈

ℕ

. With Fatou we get:

E

⁢

X

n

+

≤

lim inf

k

→

∞

⁡

E

⁢

X

τ

k

∧

n

+

=

E

⁢

X

0

+

lim inf

k

→

∞

⁡

E

⁢

X

τ

k

∧

n

-

=

E

⁢

X

0

+

lim inf

k

→

∞

⁡

E

⁢

(

∑

j

=

0

n

-

1

X

j

-

⁢

Λ

{

τ

k

=

j

}

+

X

n

-

⁢

Λ

{

τ

k

≥

n

}

)

≤

E

⁢

X

0

+

∑

j

=

0

n

E

⁢

X

j

-

<

∞

With

1)

X

n

∈

ℒ

1

follows.

3)

X

is a

martingale

, because

X

n

τ

k

→

X

n

a.s.

k

→

∞

and:

|

X

n

τ

k

|

≤

∑

j

=

0

n

|

X

j

|

∈

ℒ

1

⁢

(

⁢

ℒ

1

⁢

-bound)

Thus

X

n

τ

k

⟶

L

1

X

n

,

k

→

∞

⁢

∀

n

∈

ℕ

by bounded convergence

theorem

.
Hence

X

must be martingale and we are done. ∎

Title

proof of Martingale criterion

Canonical

name

ProofOfMartingaleCriterion

Date of creation

2013-03-22 18:34:51

Last modified on

2013-03-22 18:34:51

Owner

karstenb (16623)

Last modified by

karstenb (16623)

Numerical id

5

Author

karstenb (16623)

Entry type

Proof

Classification

msc 60G07

Generated on Sat Feb 10 11:58:15 2018 by

LaTeXML
proof of Martingale criterion (continuous time)

proof of Martingale criterion (continuous time)

Proof.

1.

Let

X

be a

martingale

. By the

optional

sampling theorem

we have

E

(

X

c

|

ℱ

τ

)

=

X

c

∧

τ

=

X

τ

∀

τ

≤

c

. Since

conditional expectations are uniformly integrable

the first direction follows.

2.

Let

(

τ

k

)

k

≥

1

be a local sequence of

stopping times

(i.e.

τ

k

↑

∞

a.s. and

X

τ

k

martingale

∀

k

∈

ℕ

).
For each

t

∈

ℝ

+

we have

X

τ

k

∧

t

→

X

t

,

k

→

∞

almost surely.
The set

{

X

τ

k

∧

t

:

k

∈

ℕ

}

⊂

{

X

τ

:

τ

⁢

stopping time

,

τ

≤

c

}

is

uniformly integrable

(take

c

=

t

). It follows that

X

t

τ

k

⟶

ℒ

1

X

t

,

k

→

∞

. Since the martingale property is stable under

ℒ

1

convergence,

X

is a martingale.
∎

Title

proof of Martingale criterion (continuous time)

Canonical name

ProofOfMartingaleCriterioncontinuousTime

Date of creation

2013-03-22 18:54:28

Last modified on

2013-03-22 18:54:28

Owner

karstenb (16623)

Last modified by

karstenb (16623)

Numerical id

4

Author

karstenb (16623)

Entry type

Proof

Classification

msc 60G07

Classification

msc 60G48

Generated on Sat Feb 10 11:58:17 2018 by

LaTeXML
quadratic variation of a semimartingale

quadratic variation of a semimartingale

Given any

semimartingale

X

, its

quadratic variation

[

X

]

exists and, for any two semimartingales

X

,

Y

, their quadratic covariation

[

X

,

Y

]

exists. This is a consequence of the existence of the

stochastic integral

, and the covariation can be expressed by the

integration by parts

formula

[

X

,

Y

]

t

=

X

t

⁢

Y

t

-

X

0

⁢

Y

0

-

∫

0

t

X

s

-

⁢

𝑑

Y

s

-

∫

0

t

Y

s

-

⁢

𝑑

X

s

.

Furthermore, suppose that

P

n

is a

sequence

of

partitions

(

http://planetmath.org/Partition3

) of

ℝ

+

,

P

n

=

{

0

=

τ

0

n

≤

τ

1

n

≤

⋯

↑

∞

}

where,

τ

k

n

can, in general, be

stopping times

. Suppose that the mesh

|

P

n

t

|

=

sup

k

⁡

(

τ

k

n

∧

t

-

τ

k

-

1

n

∧

t

)

tends to zero in

probability

as

n

→

∞

, for each time

t

>

0

.
Then, the

approximations

[

X

,

Y

]

P

n

to the quadratic covariation

converge ucp

(

http://planetmath.org/UcpConvergence

) to

[

X

,

Y

]

and, convergence also holds in the

semimartingale topology

.

A consequence of ucp convergence is that the jumps of the quadratic variation and covariation satisfy

Δ

⁢

[

X

]

=

(

Δ

⁢

X

)

2

,

Δ

⁢

[

X

,

Y

]

=

Δ

⁢

X

⁢

Δ

⁢

Y

at all times.
In particular,

[

X

,

Y

]

is

continuous

whenever

X

or

Y

is continuous.
As quadratic variations are increasing processes, this shows that the sum of the squares of the jumps of a semimartingale is finite over any bounded interval

∑

s

≤

t

(

Δ

⁢

X

s

)

2

≤

[

X

]

t

<

∞

.

Given any two semimartingales

X

,

Y

, the

polarization identity

[

X

,

Y

]

=

(

[

X

+

Y

]

-

[

X

-

Y

]

)

/

4

expresses the covariation as a

difference

of increasing processes and, therefore is of finite variation (

http://planetmath.org/FiniteVariationProcess

), So, the continuous part of the covariation

[

X

,

Y

]

t

c

≡

[

X

,

Y

]

t

-

∑

s

≤

t

Δ

⁢

X

s

⁢

Δ

⁢

Y

s

is well defined and continuous.

Title

quadratic variation of a semimartingale

Canonical name

QuadraticVariationOfASemimartingale

Date of creation

2013-03-22 18:41:21

Last modified on

2013-03-22 18:41:21

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

4

Author

gel (22282)

Entry type

Theorem

Classification

msc 60G07

Classification

msc 60G48

Classification

msc 60H05

Related topic

QuadraticVariation

Generated on Sat Feb 10 11:58:23 2018 by

LaTeXML
semimartingale

semimartingale

Semimartingales

are

adapted

stochastic processes

which can be used as

integrators

in the general theory of

stochastic integration

. Examples of semimartingales include

Brownian motion

, all

local martingales

,

finite variation processes

and

Levy processes

.

Given a

filtered probability space

(

Ω

,

ℱ

,

(

ℱ

t

)

t

∈

ℝ

+

,

ℙ

)

, we consider real-valued stochastic processes

X

t

with time index

t

ranging over the nonnegative

real numbers

.
Then, semimartingales have historically been defined as follows.

Definition

.

A semimartingale

X

is a cadlag adapted process having the decomposition

X

=

M

+

V

for a local martingale

M

and a finite variation process

V

.

More recently, the following alternative definition has also become common.
For

simple predictable

integrands

ξ

, the

stochastic integral

∫

ξ

⁢

𝑑

X

is easily defined for any process

X

. The following definition characterizes semimartingales as processes for which this

integral

is well behaved.

Definition

.

A semimartingale

X

is a cadlag adapted process such that

{

∫

0

t

ξ

⁢

𝑑

X

:

|

ξ

|

≤

1

⁢

is simple predictable

}

is

bounded

in

probability

for each

t

∈

R

+

.

Writing

∥

ξ

∥

for the supremum norm of a process

ξ

, this definition characterizes semimartingales as processes for which

∫

0

t

ξ

n

⁢

𝑑

X

→

0

in probability as

n

→

∞

for each

t

>

0

, where

ξ

n

is any sequence of simple predictable processes satisfying

∥

ξ

n

∥

→

0

. This property is necessary and, as it turns out, sufficient for the

development

of a theory of stochastic integration for which results such as bounded convergence holds.

The equivalence of these two definitions of semimartingales is stated by the

Bichteler-Dellacherie theorem

.

A stochastic process

X

t

=

(

X

t

1

,

X

t

2

,

…

,

X

t

n

)

taking values in

ℝ

n

is said to be a semimartingale if

X

t

k

is a semimartingale for each

k

=

1

,

2

,

…

,

n

.

Title

semimartingale

Canonical name

Semimartingale

Date of creation

2013-03-22 18:36:38

Last modified on

2013-03-22 18:36:38

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

8

Author

gel (22282)

Entry type

Definition

Classification

msc 60G07

Classification

msc 60G48

Classification

msc 60H05

Generated on Sat Feb 10 11:58:25 2018 by

LaTeXML
simple predictable process

simple predictable process

Simple predictable processes

are a particularly simple class of

stochastic processes

, for which the

Ito integral

can be defined directly. They are often used as the starting point for defining

stochastic integrals

of more general

predictable

integrands.

Suppose we are given a

filtration

(

http://planetmath.org/FiltrationOfSigmaAlgebras

)

(

ℱ

t

)

on the

measurable space

(

Ω

,

ℱ

)

with time index

t

ranging over the nonnegative

real numbers

.
A simple predictable process

ξ

is a left-continuous and

adapted process

which can be written as

ξ

t

=

1

{

t

=

0

}

⁢

A

0

+

∑

k

=

1

n

1

{

S

k

<

t

≤

T

k

}

⁢

A

k

for some

n

≥

0

,

stopping times

S

k

<

T

k

,

ℱ

0

-

measurable

and

bounded

random variable

A

0

and

ℱ

S

k

-measurable and bounded random variables

A

k

.
In the case where

S

k

and

T

k

are

deterministic

times, then

ξ

is called an

elementary predictable

process.

The stochastic integral of the simple predictable process

ξ

with respect to a stochastic process

X

t

can then be written as

∫

0

t

ξ

⁢

𝑑

X

=

∑

k

=

1

n

1

{

t

>

S

k

}

⁢

A

k

⁢

(

X

min

⁡

(

t

,

T

k

)

-

X

S

k

)

.

Title

simple predictable process

Canonical name

SimplePredictableProcess

Date of creation

2013-03-22 18:36:33

Last modified on

2013-03-22 18:36:33

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

7

Author

gel (22282)

Entry type

Definition

Classification

msc 60G07

Defines

simple predictable

Defines

elementary predictable

Generated on Sat Feb 10 11:58:27 2018 by

LaTeXML
exchangeable random variables

exchangeable random variables

A

finite set

of

random variables

{

X

1

,

…

,

X

n

}

defined on a common probablility space

(

Ω

,

ℱ

,

P

)

is said to be

exchangeable

if

P

(

(

X

1

∈

B

1

)

∩

⋯

∩

(

X

n

∈

B

n

)

)

=

P

(

(

X

σ

⁢

(

1

)

∈

B

1

)

∩

⋯

∩

(

X

σ

⁢

(

n

)

∈

B

n

)

)

for every set of Borel sets

{

B

1

,

…

,

B

n

}

, and every

permutation

σ

∈

S

n

. In other words,

X

1

,

…

,

X

n

are exchangeable if their joint probability distribution

function

is the same regardless of their order.

A

stochastic process

{

X

i

}

is said to be

exchangeable

if every finite subset of

{

X

i

}

is exchangeable.

Remarks

•

If

S

=

{

X

1

,

…

,

X

n

}

is exchangeable, then every subset of

S

is exchangeable (by picking suitable

B

i

and

σ

). In particular, all

X

i

are

identically distributed

, for

P

(

X

i

∈

B

)

=

P

(

(

X

i

∈

B

)

∩

(

X

j

∈

ℝ

)

)

=

P

(

(

X

j

∈

B

)

∩

(

X

i

∈

ℝ

)

)

=

P

(

X

j

∈

B

)

.

•

If

S

=

{

X

1

,

…

,

X

n

}

is iid, then

S

is exchangeable, since the

joint distribution

of

X

i

is the

product

of the

distributions

of

X

i

:

P

(

(

X

1

∈

B

1

)

∩

⋯

∩

(

X

n

∈

B

n

)

)

=

P

(

X

σ

⁢

(

1

)

∈

B

1

)

)

⋯

P

(

X

σ

⁢

(

n

)

∈

B

n

)

)

.

Title

exchangeable random variables

Canonical name

ExchangeableRandomVariables

Date of creation

2013-03-22 16:25:53

Last modified on

2013-03-22 16:25:53

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

5

Author

CWoo (3771)

Entry type

Definition

Classification

msc 60G09

Synonym

exchangeable stochastic process

Defines

exchangeable

Defines

exchangeable process

Generated on Sat Feb 10 11:58:39 2018 by

LaTeXML
autocovariance function

autocovariance function

Let

{

X

t

∣

t

∈

T

}

be a

stochastic process

such that

Var

⁡

[

X

t

]

<

∞

∀

t

∈

T

. The

autocovariance function

of

{

X

t

}

is

γ

X

⁢

(

r

,

s

)

:=

Cov

⁡

(

X

r

,

X

s

)

=

E

⁢

[

(

X

r

-

E

⁢

[

X

r

]

)

⁢

(

X

s

-

E

⁢

[

X

s

]

)

]

∀

r

,

s

∈

T

.

Title

autocovariance function

Canonical name

AutocovarianceFunction

Date of creation

2013-03-22 15:10:08

Last modified on

2013-03-22 15:10:08

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 60G10

Synonym

covariance kernel

Generated on Sat Feb 10 11:58:41 2018 by

LaTeXML
autoregressive model

autoregressive model

The

autoregressive model

of order

p

, denoted

AR

(

p

), is a

random process

model described by

y

t

=

∑

i

=

1

p

a

i

⁢

y

t

-

i

+

c

+

e

t

,

t

=

1

,

2

,

…

(1)

where

a

i

are model parameters,

y

t

is the model output in discrete time instant

t

. Term

c

is an absolute term (constant) and

e

t

denotes

discrete white noise

.

A first-order autoregression model

AR

(1) in the form

y

t

=

a

⁢

y

t

-

1

+

c

+

e

t

is one major example.

Title

autoregressive model

Canonical name

AutoregressiveModel

Date of creation

2013-03-22 18:33:37

Last modified on

2013-03-22 18:33:37

Owner

camillio (22337)

Last modified by

camillio (22337)

Numerical id

5

Author

camillio (22337)

Entry type

Definition

Classification

msc 60G10

Classification

msc 62J05

Synonym

AR

Generated on Sat Feb 10 11:58:42 2018 by

LaTeXML
discrete white noise

discrete white noise

The

stochastic process

{

Z

t

,

t

∈

T

}

,where T is the set of natural integers

ℕ

or the

set of all integers

ℤ

, is said to be white noise with mean 0 and

variance

σ

2

, written

{

Z

t

}

∼

W

⁢

N

⁢

(

0

,

σ

2

)

, if and only if

{

Z

t

}

has zero mean and

autocovariance function

γ

⁢

(

h

)

=

{

σ

2

if

⁢

h

=

0

0

if

⁢

h

≠

0

Title

discrete white noise

Canonical name

DiscreteWhiteNoise

Date of creation

2013-03-22 15:20:24

Last modified on

2013-03-22 15:20:24

Owner

georgiosl (7242)

Last modified by

georgiosl (7242)

Numerical id

8

Author

georgiosl (7242)

Entry type

Definition

Classification

msc 60G10

Classification

msc 60H40

Generated on Sat Feb 10 11:58:44 2018 by

LaTeXML
mean square convergence of the sample mean of a stationary process

mean square convergence of the sample mean of a stationary process

If

{

X

t

,

t

∈

T

}

is a

stationary process

with mean

μ

and

autocovariance function

γ

⁢

(

⋅

)

, then as

n

→

∞

we have the
following:

•

var

⁡

[

X

¯

n

]

=

E

⁢

[

(

X

¯

n

-

μ

)

2

]

→

0

if

γ

⁢

(

n

)

→

0

•

n

⁢

E

⁢

[

(

X

¯

n

-

μ

)

2

]

→

∑

h

=

-

∞

∞

γ

⁢

(

h

)

if

∑

h

=

-

∞

∞

|

γ

⁢

(

h

)

|

<

∞

where

X

¯

n

=

1

n

⁢

∑

k

=

1

n

X

k

is the

sample mean

which is a natural

unbiased estimator

of the mean

μ

of the stationary process

{

X

t

}

.

References

1

Peter J. Brockwell G., Richard A. Davis ,

Time Series

:

Theory and Methods

.

Title

mean square convergence of the sample mean of a stationary process

Canonical name

MeanSquareConvergenceOfTheSampleMeanOfAStationaryProcess

Date of creation

2013-03-22 15:20:52

Last modified on

2013-03-22 15:20:52

Owner

georgiosl (7242)

Last modified by

georgiosl (7242)

Numerical id

5

Author

georgiosl (7242)

Entry type

Theorem

Classification

msc 60G10

Generated on Sat Feb 10 11:58:46 2018 by

LaTeXML
proof of mean square convergence of the sample mean of a stationary process

proof of mean square convergence of the sample mean of a stationary process

n

⁢

var

⁡

(

X

¯

n

)

=

1

n

⁢

∑

i

=

1

n

∑

j

=

1

n

cov

⁡

(

X

i

,

X

j

)

=

∑

|

h

|

<

n

(

1

-

|

h

|

n

)

⁢

γ

⁢

(

h

)

≤

∑

|

h

|

<

n

|

γ

⁢

(

h

)

|

If

γ

⁢

(

n

)

→

0

as

n

→

∞

then

lim

n

→

∞

⁡

1

n

⁢

∑

|

h

|

<

n

|

γ

⁢

(

h

)

|

=

2

⁢

lim

n

→

∞

⁡

|

γ

⁢

(

n

)

|

=

0

, whence

var

⁡

[

X

¯

n

]

→

0

.

If

∑

h

=

-

∞

∞

|

γ

⁢

(

h

)

|

<

∞

then the

dominated Convergence theorem

gives

lim

n

→

∞

⁡

∑

|

h

|

<

n

(

1

-

|

h

|

n

)

⁢

γ

⁢

(

h

)

=

∑

h

=

-

∞

∞

γ

⁢

(

h

)

.

Title

proof of mean square convergence of the

sample mean

of a

stationary process

Canonical name

ProofOfMeanSquareConvergenceOfTheSampleMeanOfAStationaryProcess

Date of creation

2013-03-22 15:22:19

Last modified on

2013-03-22 15:22:19

Owner

georgiosl (7242)

Last modified by

georgiosl (7242)

Numerical id

6

Author

georgiosl (7242)

Entry type

Proof

Classification

msc 60G10

Generated on Sat Feb 10 11:58:48 2018 by

LaTeXML
stationary process

stationary process

Let

{

X

⁢

(

t

)

∣

t

∈

T

}

be a

stochastic process

where

T

⊆

ℝ

and has the property that

s

+

t

∈

T

whenever

s

,

t

∈

T

. Then

{

X

⁢

(

t

)

}

is said to be a

strictly stationary process

of order n

if for a given
positive integer

n

<

∞

, any

t

1

,

…

,

t

n

and

s

∈

T

, the

random vectors

(

X

⁢

(

t

1

)

,

…

,

X

⁢

(

t

n

)

)

and

(

X

⁢

(

t

1

+

s

)

,

…

,

X

⁢

(

t

n

+

s

)

)

have identical

joint distributions

.

{

X

⁢

(

t

)

}

is said to be a

strictly stationary
process

if it is a strictly stationary process of order

n

for all
positive integers

n

. Alternatively,

{

X

⁢

(

t

)

∣

t

∈

T

}

is strictly stationary if

{

X

⁢

(

t

)

}

and

{

X

⁢

(

t

+

s

)

}

are

identically distributed stochastic
processes

for all

s

∈

T

.

A weaker form of the above is the concept of a

covariance
stationary process

, or simply, a

stationary process

{

X

⁢

(

t

)

}

. Formally, a stochastic process

{

X

⁢

(

t

)

∣

t

∈

T

}

is stationary if, for any positive integer

n

<

∞

, any

t

1

,

…

,

t

n

and

s

∈

T

, the joint distributions of the random
vectors

(

X

⁢

(

t

1

)

,

…

,

X

⁢

(

t

n

)

)

and

(

X

⁢

(

t

1

+

s

)

,

…

,

X

⁢

(

t

n

+

s

)

)

have identical means (mean vectors) and identical

covariance matrices

.

So a strictly stationary process is a stationary process. A non-stationary process is sometimes called an

evolutionary process

.

Title

stationary process

Canonical name

StationaryProcess

Date of creation

2013-03-22 15:22:42

Last modified on

2013-03-22 15:22:42

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Definition

Classification

msc 60G10

Defines

strictly stationary process

Defines

covariance stationary process

Defines

evolutionary process

Generated on Sat Feb 10 11:58:50 2018 by

LaTeXML
Gaussian process

Gaussian process

A

stochastic process

{

X

⁢

(

t

)

∣

t

∈

T

}

is said to be
a

Gaussian process

if all of the members of its f.f.d.
(family of

finite dimensional distributions

) are

joint normal
distributions

. In other words, for any

positive integer

n

<

∞

,
and any

t

1

,

…

,

t

n

∈

T

, the

joint distribution

of

random
variables

X

⁢

(

t

1

)

,

…

,

X

⁢

(

t

n

)

is jointly normal.

As an example, any

Wiener process

is

Gaussian

.

Remark

. Sometimes, a Gaussian process is known as a

Gaussian random field

if

T

is a subset, usually an embedded manifold, of

ℝ

m

, with

m

>

1

.

Title

Gaussian process

Canonical name

GaussianProcess

Date of creation

2013-03-22 15:22:48

Last modified on

2013-03-22 15:22:48

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

5

Author

CWoo (3771)

Entry type

Definition

Classification

msc 60G15

Classification

msc 60G60

Synonym

Gaussian random field

Defines

normal process

Generated on Sat Feb 10 11:58:52 2018 by

LaTeXML
Wiener measure

Wiener measure

Definition 1

.

The

Wiener space

W

⁢

(

ℝ

)

is just the set of all

continuous

paths

ω

:

[

0

,

∞

)

→

ℝ

satisfying

ω

⁢

(

0

)

=

0

. It may be made into a

measurable space

by equipping it with the

σ

-algebra

ℱ

generated by

all

projection maps

ω

↦

ω

⁢

(

t

)

(or the

completion

of this under

Wiener measure

, see below).

Thus, an

ℝ

-valued continuous-time stochastic process

X

t

with continuous

sample paths

can be thought of as a

random variable

taking its values in

W

⁢

(

ℝ

)

.

Definition 2

.

In the case where

X

t

=

W

t

is

Brownian motion

, the

distribution

measure

P

induced on

W

⁢

(

ℝ

)

is called the

Wiener measure

. That is,

P

is the unique

probability measure

on

W

⁢

(

ℝ

)

such that for any

finite sequence

of times

0

<

t

1

<

…

<

t

n

and

Borel sets

A

1

,

…

,

A

n

⊂

ℝ

P

⁢

(

{

ω

:

ω

⁢

(

t

1

)

∈

A

1

,

…

,

ω

⁢

(

t

n

)

∈

A

n

}

)

=

∫

A

1

⋯

⁢

∫

A

n

p

⁢

(

t

1

,

0

,

x

1

)

⁢

p

⁢

(

t

2

-

t

1

,

x

1

,

x

2

)

⁢

⋯

(2)

⋯

⁢

p

⁢

(

t

n

-

t

n

-

1

,

x

n

-

1

,

x

n

)

⁢

d

⁢

x

1

⁢

⋯

⁢

d

⁢

x

n

,

where

p

⁢

(

t

,

x

,

y

)

=

1

2

⁢

π

⁢

t

⁢

exp

⁡

(

-

(

x

-

y

)

2

2

⁢

t

)

defined for any

x

,

y

∈

ℝ

and

t

>

0

.

This of course corresponds to the defining property of Brownian motion. The other properties carry over as well; for

instance

, the set of paths in

W

⁢

(

ℝ

)

which are nowhere

differentiable

is of

P

-measure

1

.

The Wiener space

W

⁢

(

ℝ

d

)

and corresponding Wiener measure are defined similarly, in which case

P

is the distribution of a

d

-dimensional Brownian motion.

Title

Wiener measure

Canonical name

WienerMeasure

Date of creation

2013-03-22 15:55:53

Last modified on

2013-03-22 15:55:53

Owner

neldredge (4974)

Last modified by

neldredge (4974)

Numerical id

7

Author

neldredge (4974)

Entry type

Definition

Classification

msc 60G15

Related topic

BrownianMotion

Related topic

CameronMartinSpace

Defines

Wiener space

Defines

Wiener measure

Generated on Sat Feb 10 11:58:54 2018 by

LaTeXML
sample function

sample function

Let

{

X

⁢

(

t

)

∣

t

∈

T

}

be a

stochastic process

, where

X

⁢

(

t

)

is a

random variable

on the

probability space

(

Ω

,

ℱ

,

𝐏

)

. Writing

X

⁢

(

t

)

as

X

⁢

(

t

,

ω

)

,
where

t

∈

T

and

ω

∈

Ω

, we see that if we fix the
sample point

ω

, we have a

function

in

t

:

X

ω

⁢

(

t

)

:

t

↦

X

⁢

(

t

)

. This function

X

ω

⁢

(

t

)

of

t

is
called a

sample function

, or

sample path

of the
stochastic process.

Title

sample function

Canonical name

SampleFunction

Date of creation

2013-03-22 15:21:15

Last modified on

2013-03-22 15:21:15

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

5

Author

gel (22282)

Entry type

Definition

Classification

msc 60G17

Classification

msc 60G05

Defines

sample path

Generated on Sat Feb 10 11:58:56 2018 by

LaTeXML