aarjaneiro/planetmath huggingface dataset:
properties of X-integrable processes

properties of

X

-integrable processes

Let

X

be a

semimartingale

. Then a

predictable process

ξ

is

X

-

integrable

if the

stochastic integral

∫

ξ

⁢

𝑑

X

is defined, which is

equivalent

to the set

{

∫

0

t

α

⁢

𝑑

X

:

|

α

|

≤

|

ξ

|

⁢

is predictable

}

being

bounded

in

probability

, for each

t

>

0

. We list some properties of

X

-integrable processes.

1.

Every

locally bounded

predictable process is

X

-integrable.

2.

The

X

-integrable processes are

closed under

linear combinations

. That is, if

α

,

β

are

X

-integrable and

λ

,

μ

∈

ℝ

, then

λ

⁢

α

+

μ

⁢

β

is

X

-integrable.

3.

If

|

α

|

≤

|

β

|

are predictable processes and

β

is

X

-integrable, then so is

α

.

4.

A process is

X

-integrable if it is locally

X

-integrable. That is, if there are

stopping times

τ

n

almost surely

increasing

to

infinity

and such that

1

{

t

≤

τ

n

}

⁢

ξ

t

is

X

-integrable, then

ξ

is

X

-integrable.

Title

properties of

X

-integrable processes

Canonical name

PropertiesOfXintegrableProcesses

Date of creation

2013-03-22 18:40:59

Last modified on

2013-03-22 18:40:59

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

5

Author

gel (22282)

Entry type

Theorem

Classification

msc 60H10

Classification

msc 60G07

Classification

msc 60H05

Related topic

StochasticIntegration

Generated on Sat Feb 10 12:00:32 2018 by

LaTeXML
quadratic variation of Brownian motion

quadratic variation of Brownian motion

Theorem

.

Let

(

W

t

)

t

∈

R

+

be a standard

Brownian motion

. Then, its

quadratic variation

exists and is given by

[

W

]

t

=

t

.

As Brownian motion is a

martingale

and, in particular, is a

semimartingale

then its quadratic variation must exist (

http://planetmath.org/QuadraticVariationOfASemimartingale

). We just need to compute its value along a sequence of

partitions

.

If

P

=

{

0

=

t

0

≤

t

1

≤

⋯

≤

t

m

=

t

}

is a partition (

http://planetmath.org/SubintervalPartition

) of the

interval

[

0

,

t

]

, then the quadratic variation on

P

is

[

W

]

P

=

∑

k

=

1

m

(

W

t

k

-

W

t

k

-

1

)

2

.

Using the property that the increments

W

t

k

-

W

t

k

-

1

are

independent

normal random variables

with mean zero and

variance

t

k

-

t

k

-

1

, the mean and variance of

[

W

]

P

are

𝔼

⁢

[

[

W

]

P

]

=

∑

k

=

1

m

𝔼

⁢

[

(

W

t

k

-

W

t

k

-

1

)

2

]

=

∑

k

=

1

m

(

t

k

-

t

k

-

1

)

=

t

,

Var

⁡

[

[

W

]

P

]

=

∑

k

=

1

m

Var

⁡

[

(

W

t

k

-

W

t

k

-

1

)

2

]

=

∑

k

=

1

m

2

⁢

(

t

k

-

t

k

-

1

)

2

≤

2

⁢

|

P

|

⁢

∑

k

=

1

m

(

t

k

-

t

k

-

1

)

=

2

⁢

|

P

|

⁢

t

.

Here,

|

P

|

=

max

k

⁡

(

t

k

-

t

k

-

1

)

is the mesh of the partition.
If

(

P

n

)

n

=

1

,

2

,

…

is a sequence of partitions of

[

0

,

t

]

with mesh going to zero as

n

→

∞

then,

𝔼

⁢

[

(

[

W

]

P

n

-

t

)

2

]

≤

2

⁢

|

P

n

|

⁢

t

→

0

as

n

→

∞

. This shows that

[

W

]

P

n

→

t

in the

L

2

norm and, in particular,

converges in probability

. So,

[

W

]

t

=

t

.

Title

quadratic variation of Brownian motion

Canonical name

QuadraticVariationOfBrownianMotion

Date of creation

2013-03-22 18:41:25

Last modified on

2013-03-22 18:41:25

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

6

Author

gel (22282)

Entry type

Theorem

Classification

msc 60H10

Classification

msc 60J65

Related topic

QuadraticVariation

Generated on Sat Feb 10 12:00:35 2018 by

LaTeXML
Cameron-Martin space

Cameron-Martin space

Definition 1

.

Let

W

⁢

(

ℝ

d

)

be

Wiener space

. The

Cameron-Martin space

H

⁢

(

ℝ

d

)

is the

subspace

of

W

⁢

(

ℝ

d

)

consisting of all paths

ω

such that

ω

is

absolutely continuous

and

∫

0

∞

|

ω

′

⁢

(

s

)

|

2

⁢

𝑑

s

<

∞

. (Note that if

ω

is absolutely continuous, then it is almost everywhere

differentiable

, so the

integral

makes sense.)

This can be thought of as the set of paths with “finite energy.”

Note that

H

⁢

(

ℝ

d

)

has

Wiener measure

0

, since

sample paths

of

Brownian motion

are nowhere differentiable, whereas a path from

H

⁢

(

ℝ

d

)

is almost everywhere differentiable.

Title

Cameron-Martin space

Canonical name

CameronMartinSpace

Date of creation

2013-03-22 15:55:56

Last modified on

2013-03-22 15:55:56

Owner

neldredge (4974)

Last modified by

neldredge (4974)

Numerical id

6

Author

neldredge (4974)

Entry type

Definition

Classification

msc 60H99

Related topic

WienerMeasure

Defines

Cameron-Martin space

Generated on Sat Feb 10 12:00:40 2018 by

LaTeXML
class structure

class structure

Let

(

X

n

)

n

≥

1

be a

stationary

Markov chain

and let

i

and

j

be states in the

indexing set

. We say that

i

leads to

j

or

j

is

accessible

from

i

, and write

i

→

j

, if it is possible for the chain to get from state

i

to state

j

:

i

→

j

⇔

P

(

X

n

=

j

:

X

0

=

i

)

>

0

for some

n

≥

0

If

i

→

j

and

j

→

i

we say

i

communicates with

j

and write

i

↔

j

.

↔

is an

equivalence relation

(easy to prove). The

equivalence classes

of this

relation

are the

communicating classes

of the chain. If there is just one class, we say the chain is an

irreducible chain

.

A class

C

is a

closed class

if

i

∈

C

and

i

→

j

implies that

j

∈

C

“Once the chain enters a closed class, it cannot leave it”

A state

i

is an

absorbing state

if

{

i

}

is a closed class.

Title

class structure

Canonical name

ClassStructure

Date of creation

2013-03-22 14:18:21

Last modified on

2013-03-22 14:18:21

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

12

Author

CWoo (3771)

Entry type

Definition

Classification

msc 60J10

Related topic

MarkovChain

Defines

communicating class

Defines

irreducible chain

Defines

closed class

Defines

absorbing state

Generated on Sat Feb 10 12:00:41 2018 by

LaTeXML
hitting time

hitting time

Let

(

X

n

)

n

≥

0

be a

Markov Chain

. Then the

hitting time

for a subset

A

of

I

(the

indexing set

) is the

random variable

:

H

A

=

inf

⁡

{

n

≥

0

:

X

n

∈

A

}

(set

inf

⁡

∅

=

∞

).

This can be thought of as the time before the chain is first in a state that is a member of

A

.

Wite

h

i

A

for the

probability

that, starting from

i

∈

I

the chain ever hits the set A:

h

i

A

=

P

(

H

A

<

∞

:

X

0

=

i

)

When A is a

closed class

,

h

i

A

is the

absorption probability

.

Title

hitting time

Canonical name

HittingTime

Date of creation

2013-03-22 14:18:18

Last modified on

2013-03-22 14:18:18

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

8

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 60J10

Related topic

MarkovChain

Related topic

MeanHittingTime

Defines

absorption probability

Generated on Sat Feb 10 12:00:43 2018 by

LaTeXML
Markov chain

Markov chain

Definition

We begin with a

probability space

(

Ω

,

ℱ

,

ℙ

)

. Let

I

be a

countable set

,

(

X

n

:

n

∈

ℤ

)

be a

collection

of

random variables

taking values in

I

,

𝐓

=

(

t

i

⁢

j

:

i

,

j

∈

I

)

be a

stochastic matrix

, and

λ

be a

distribution

. We call

(

X

n

)

n

≥

0

a

Markov chain

with initial distribution

λ

and

transition matrix

𝐓

if:

1.

X

0

has distribution

λ

.

2.

For

n

≥

0

,

ℙ

(

X

n

+

1

=

i

n

+

1

|

X

0

=

i

0

,

…

,

X

n

=

i

n

)

=

t

i

n

⁢

i

n

+

1

.

That is, the next value of the chain depends only on the current value, not any previous values. This is often summed up in the pithy phrase, “Markov chains have no memory.”

As a special case of (2) we have that

ℙ

(

X

n

+

1

=

i

|

X

n

=

j

)

=

t

i

⁢

j

whenever

i

,

j

∈

I

. The values

t

i

⁢

j

are therefore called

transition probabilities

for the Markov chain.

Discussion

Markov chains are arguably the simplest examples of

random processes

. They come in discrete and

continuous

versions; the discrete version is presented above.

Title

Markov chain

Canonical name

MarkovChain

Date of creation

2013-03-22 12:37:32

Last modified on

2013-03-22 12:37:32

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

5

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 60J10

Related topic

HittingTime

Related topic

MarkovChainsClassStructure

Related topic

MemorylessRandomVariable

Related topic

LeslieModel

Defines

Markov chain

Defines

transition matrix

Defines

transition probability

Generated on Sat Feb 10 12:00:45 2018 by

LaTeXML
mean hitting time

mean hitting time

Let

(

X

n

)

n

≥

0

be a

Markov chain

with transition probabilities

p

i

⁢

j

where

i

,

j

are states in an

indexing set

I

. Let

H

A

be the

hitting time

of

(

X

n

)

n

≥

0

for a subset

A

⊆

I

. That is,

H

A

is the

random variable

of the time it takes for the to first reach a in

A

.

Define the

mean hitting time

of

A

given the starts in state

i

to be

k

i

A

:=

E

(

H

A

|

X

0

=

i

)

.

Proposition 1

.

The mean hitting times are the

minimal

non negative

solution

to:

k

i

A

=

{

0

i

∈

A

1

+

∑

j

∈

I

p

i

⁢

j

⁢

k

j

A

i

∉

A

Remark

. In this case, a solution is minimal if for any non negative solution

{

y

i

|

i

∈

I

}

we have

y

i

≥

k

i

A

for all

i

∈

I

.

Proof.

If

i

∈

A

, then

H

A

=

inf

⁡

{

n

≥

0

∣

X

n

∈

A

}

≡

0

, which means

k

i

A

=

0

(the is certain to be in a state in

A

at step

n

=

0

).

If

i

∉

A

we condition on the first step:

k

i

A

=

E

(

H

A

∣

X

0

=

i

)

=

∑

j

∈

I

P

(

X

1

=

j

|

X

0

=

i

)

E

(

H

A

|

X

0

=

i

,

X

1

=

j

)

=

∑

j

∈

I

p

i

⁢

j

E

(

H

A

|

X

1

=

j

)

(by the Markov property)

=

∑

j

∈

I

p

i

⁢

j

⁢

(

1

+

k

j

A

)

=

1

+

∑

j

∈

I

p

i

⁢

j

⁢

k

j

A

So the

k

i

A

satisfy

the given equations.

Now suppose that

{

y

i

|

y

∈

I

}

is any non-negative solution to the equations. Then for

i

∈

A

we have

k

i

A

=

y

i

=

0

. If

i

∉

A

, then

y

i

=

1

+

∑

j

∈

I

p

i

⁢

j

⁢

y

j

=

1

+

∑

j

∉

A

p

i

⁢

j

⁢

(

1

+

∑

k

∉

A

p

j

⁢

k

⁢

y

k

)

=

1

+

∑

j

∉

A

p

i

⁢

j

+

∑

j

∉

A

∑

k

∉

A

p

i

⁢

j

⁢

p

j

⁢

k

⁢

y

k

=

1

+

q

1

+

q

2

+

⋯

+

q

n

+

∑

⋯

⁢

∑

p

i

⁢

j

⁢

⋯

⁢

p

u

⁢

v

⁢

y

v

,

where

q

n

=

P

(

X

1

∉

A

,

X

1

∉

A

,

⋯

,

X

n

∉

A

|

X

0

=

i

)

is the

probability

that the chain

X

does not enter

A

in the first

n

steps after the initial state

i

.

y

i

is non negative by

assumption

, therefore so is the final term, and so

y

i

≥

1

+

q

1

+

q

2

+

⋯

+

q

n

.

Since

n

is arbitrary, by taking the limit

n

→

∞

, we have that

y

i

≥

lim

n

→

∞

⁡

(

1

+

q

1

+

q

2

+

⋯

+

q

n

)

≥

k

i

A

.

So

y

i

≥

k

i

A

for all

i

∈

I

and therefore

{

k

i

A

|

i

∈

I

}

is the minimal solution.
∎

Title

mean hitting time

Canonical name

MeanHittingTime

Date of creation

2013-03-22 14:20:12

Last modified on

2013-03-22 14:20:12

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

24

Author

CWoo (3771)

Entry type

Theorem

Classification

msc 60J10

Related topic

HittingTime

Defines

mean hitting time

Generated on Sat Feb 10 12:00:55 2018 by

LaTeXML
periodicity of a Markov chain

periodicity of a Markov chain

Let

{

X

n

}

be a

stationary

(

http://planetmath.org/StationaryProcess

)

Markov chain

with

state space

I

. Let

P

i

⁢

j

n

be the

n

-step transition probability that the process goes from state

i

at time

0

to state

j

at time

n

:

P

i

⁢

j

n

=

P

(

X

n

=

j

∣

X

0

=

i

)

.

Given any state

i

∈

I

, define the set

N

⁢

(

i

)

:=

{

n

≥

1

∣

P

i

⁢

i

n

>

0

}

.

It is not hard to see that if

n

,

m

∈

N

⁢

(

i

)

, then

n

+

m

∈

N

⁢

(

i

)

. The

period

of

i

, denoted by

d

⁢

(

i

)

, is defined as

d

⁢

(

i

)

:=

{

0

if

⁢

N

⁢

(

i

)

=

∅

,

gcd

⁡

(

N

⁢

(

i

)

)

otherwise

,

where

gcd

⁡

(

N

⁢

(

i

)

)

is the

greatest common divisor

of all

positive integers

in

N

⁢

(

i

)

.

A state

i

∈

I

is said to be

aperiodic

if

d

⁢

(

i

)

=

1

. A Markov chain is called

aperiodic

if every state is aperiodic.

Property

. If states

i

,

j

∈

I

communicate (

http://planetmath.org/MarkovChainsClassStructure

), then

d

⁢

(

i

)

=

d

⁢

(

j

)

.

Proof.

We will employ a common inequality involving the

n

-step transition probabilities:

P

i

⁢

j

m

+

n

≥

P

i

⁢

k

m

⁢

P

k

⁢

j

n

for any

i

,

j

,

k

∈

I

and non-negative

integers

m

,

n

.

Suppose first that

d

⁢

(

i

)

=

0

. Since

i

↔

j

,

P

i

⁢

j

n

>

0

and

P

j

⁢

i

m

>

0

for some

n

,

m

≥

0

. This implies that

P

i

⁢

i

m

+

n

>

0

, which forces

m

+

n

=

0

or

m

=

n

=

0

, and hence

j

=

i

.

Next, assume

d

⁢

(

i

)

>

0

, this means that

N

⁢

(

i

)

≠

∅

. Since

i

↔

j

, there are

r

,

s

≥

0

such that

P

j

⁢

i

r

>

0

and

P

i

⁢

j

s

>

0

, and so

P

j

⁢

j

r

+

s

>

0

, showing

r

+

s

∈

N

⁢

(

j

)

. If we pick any

n

∈

N

, we also have

P

j

⁢

j

r

+

n

+

s

≥

P

j

⁢

i

r

⁢

P

i

⁢

i

n

⁢

P

i

⁢

j

s

>

0

, or

r

+

s

+

n

∈

N

⁢

(

j

)

. But this means

d

⁢

(

j

)

divides both

r

+

s

and

r

+

s

+

n

, and so

d

⁢

(

j

)

divides their

difference

, which is

n

. Since

n

is arbitrarily picked,

d

(

j

)

∣

d

(

i

)

. Similarly,

d

(

i

)

∣

d

(

j

)

. Hence

d

⁢

(

i

)

=

d

⁢

(

j

)

.
∎

Title

periodicity of a Markov chain

Canonical

name

PeriodicityOfAMarkovChain

Date of creation

2013-03-22 16:24:28

Last modified on

2013-03-22 16:24:28

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 60J10

Defines

period of a state

Defines

aperiodic state

Defines

aperiodic Markov chain

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recurrence in a Markov chain

recurrence in a Markov chain

Let

{

X

n

}

be a

stationary

(

http://planetmath.org/StationaryProcess

)

Markov chain

and

I

the

state space

. Given

i

,

j

∈

I

and any non-negative integer

n

, define a number

F

i

⁢

j

n

as follows:

F

i

⁢

j

n

:=

{

0

if

⁢

n

=

0

,

P

(

X

n

=

j

and

X

m

≠

j

for

0

<

m

<

n

∣

X

0

=

i

)

otherwise

.

In other words,

F

i

⁢

j

n

is the

probability

that the process

first

reaches state

j

at time

n

from state

i

at time

0

.

From the

definition

of

F

i

⁢

j

n

, we see that the probability of the process reaching state

j

within and including

time

n

from state

i

at time

0

is given by

∑

m

=

0

n

F

i

⁢

j

m

.

As

n

→

∞

, we have the limiting probability of the process reaching

j

eventually

from the initial state of

i

at

0

, which we denote by

F

i

⁢

j

:

F

i

⁢

j

:=

∑

m

=

0

∞

F

i

⁢

j

m

.

Definitions

. A state

i

∈

I

is said to be

recurrent

or

persistent

if

F

i

⁢

i

=

1

, and

transient

otherwise.

Given a

recurrent state

i

, we can further classify it according to “how soon” the state

i

returns after its initial appearance. Given

F

i

⁢

i

n

, we can calculate the expected number of steps or transitions required to

return

to state

i

by time

n

. This

expectation

is given by

∑

m

=

0

n

m

⁢

F

i

⁢

i

m

.

When

n

→

∞

, the above

expression

may or may not approach a limit. It is the expected number of transitions needed to return to state

i

at all

from the beginning. We denote this figure by

μ

i

:

μ

i

:=

∑

m

=

0

∞

m

⁢

F

i

⁢

i

m

.

Definitions

. A recurrent state

i

∈

I

is said to be

or

strongly ergodic

if

μ

i

<

∞

, otherwise it is called

null

or

weakly ergodic

. If a stronly ergodic state is in

addition

aperiodic (

http://planetmath.org/PeriodicityOfAMarkovChain

), then it is said to be an

ergodic state

.

Title

recurrence in a Markov chain

Canonical

name

RecurrenceInAMarkovChain

Date of creation

2013-03-22 16:24:43

Last modified on

2013-03-22 16:24:43

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

5

Author

CWoo (3771)

Entry type

Definition

Classification

msc 60J10

Synonym

null recurrent

Synonym

positive recurrent

Synonym

strongly ergodic

Synonym

weakly ergodic

Defines

recurrent state

Defines

persistent state

Defines

transient state

Defines

null state

Defines

positive state

Defines

ergodic state

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Example of stochastic matrix of mapping

Example of stochastic matrix of mapping

In order to understand the notion of

stochastic matrix

associated
to a mapping and its dual, we will work through a simple example.
Let

X

=

{

a

,

b

,

c

}

and let

Y

=

{

d

,

e

}

, and define the mapping

f

:

X

→

Y

as follows:

f

⁢

(

a

)

=

d

f

⁢

(

b

)

=

d

f

⁢

(

c

)

=

e

Then

𝒱

⁢

X

is a 3-dimensional

real vector space

with basis

δ

a

=

(

1

0

0

)

,

δ

b

=

(

0

1

0

)

,

δ

c

=

(

0

0

1

)

and

𝒱

⁢

Y

is a 3-dimensional real vector space with basis

δ

c

=

(

1

0

)

,

δ

d

=

(

0

1

)

and

𝒱

⁢

f

=

(

1

1

0

0

0

1

)

.

To form the dual, we first renormalize the rows to sum to unity,
then

transpose

:

(

1

1

0

0

0

1

)

→

r

⁢

e

⁢

n

(

1

2

1

2

0

0

0

1

)

→

*

(

1

2

0

1

2

0

0

1

)

Next, to illustrate inclusions, we shall examine the map

i

:

Y

↪

X

defined as follows:

f

⁢

(

d

)

=

a

f

⁢

(

e

)

=

b

Following the same procedures as above, for this map we find that

𝒱

⁢

i

=

(

1

0

0

1

0

0

)

and

(

𝒱

⁢

i

)

♮

=

(

1

0

0

0

1

0

)

Title

Example of stochastic matrix of mapping

Canonical name

ExampleOfStochasticMatrixOfMapping

Date of creation

2014-04-28 3:33:09

Last modified on

2014-04-28 3:33:09

Owner

rspuzio (6075)

Last modified by

PMBookProject (1000683)

Numerical id

21

Author

rspuzio (1000683)

Entry type

Example

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Feller process

Feller process

Let

E

be a LCCB space (

locally compact

with a countable base; usually a subset of

ℝ

n

for some

n

∈

ℕ

) and

C

0

⁢

(

E

)

=

C

0

⁢

(

E

,

ℝ

)

be the space of

continuous functions

on

E

that

vanish at infinity

. (We may write

C

0

as shorthand.) A

Feller semigroup

on

C

0

⁢

(

E

)

is a family of

positive

linear operators

T

t

,

t

≥

0

, on

C

0

⁢

(

E

)

such that

•

T

0

=

I

⁢

d

and

||

T

t

||

≤

1

for every

t

∈

T

,

i.e.

{

T

t

}

t

∈

T

is a family of contracting maps;

•

T

t

+

s

=

T

t

∘

T

s

(the semigroup

property

);

•

lim

t

↓

0

⁡

||

T

t

⁢

f

-

f

||

=

0

for every

f

∈

C

0

⁢

(

E

)

.

A

probability transition function

associated with a Feller semigroup is called a

Feller transition function

. A

Markov process

having a Feller transition function is called a

Feller process

.

References

1

D. Revuz & M. Yor,

Continuous

Martingales

and

Brownian Motion

, Third Edition Corrected. Volume 293, Grundlehren der mathematischen Wissenschaften. Springer, Berlin, 2005.

Title

Feller process

Canonical

name

FellerProcess

Date of creation

2013-03-22 16:12:40

Last modified on

2013-03-22 16:12:40

Owner

mcarlisle (7591)

Last modified by

mcarlisle (7591)

Numerical id

6

Author

mcarlisle (7591)

Entry type

Definition

Classification

msc 60J35

Defines

Feller semigroup

Defines

Feller transition function

Defines

Feller process

Defines

LCCB

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probability transition function

probability transition function

A

probability transition function

(p.t.f., or just t.f. in context) on a

measurable space

(

Ω

,

ℱ

)

is a family

P

s

,

t

,

0

≤

s

<

t

of

transition probabilities

on

(

Ω

,

ℱ

)

such that for every three

real numbers

s

<

t

<

v

, the family the

Chapman-Kolmogorov equation

∫

P

s

,

t

⁢

(

x

,

d

⁢

y

)

⁢

P

t

,

v

⁢

(

y

,

A

)

=

P

s

,

v

⁢

(

x

,

A

)

for every

x

∈

Ω

and

A

∈

ℱ

. The t.f. is said to be if

P

s

,

t

depends on

s

and

t

only through their

t

-

s

. In this case, we write

P

t

,

0

=

P

t

and the family

{

P

t

,

t

≥

0

}

is a semigroup, and the Chapman-Kolmogorov equation reads

P

t

+

s

⁢

(

x

,

A

)

=

∫

P

s

⁢

(

x

,

d

⁢

y

)

⁢

P

t

⁢

(

y

,

A

)

.

References

1

D. Revuz & M. Yor,

Continuous

Martingales

and

Brownian Motion

, Third Edition Corrected. Volume 293, Grundlehren der mathematischen Wissenschaften. Springer, Berlin, 2005.

Title

probability transition function

Canonical name

ProbabilityTransitionFunction

Date of creation

2013-03-22 16:12:37

Last modified on

2013-03-22 16:12:37

Owner

mcarlisle (7591)

Last modified by

mcarlisle (7591)

Numerical id

8

Author

mcarlisle (7591)

Entry type

Definition

Classification

msc 60J35

Defines

probability transition function

Defines

homogeneous probability transition function

Defines

Chapman-Kolmogorov equation

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Brownian motion

Brownian motion

Definition

.

One-dimensional

Brownian motion

is a

stochastic process

W

⁢

(

t

)

, defined for

t

∈

[

0

,

∞

)

such that

1.

W

⁢

(

0

)

=

0

almost surely

2.

The

sample paths

t

↦

W

⁢

(

t

)

are almost surely

continuous

.

3.

For any

finite sequence

of times

t

0

<

t

1

<

⋯

<

t

n

,
the increments

W

⁢

(

t

1

)

-

W

⁢

(

t

0

)

,

W

⁢

(

t

2

)

-

W

⁢

(

t

1

)

,

…

,

W

⁢

(

t

n

)

-

W

⁢

(

t

n

-

1

)

are

independent

.

4.

For any times

s

<

t

,

W

⁢

(

t

)

-

W

⁢

(

s

)

is normally distributed with mean zero
and

variance

t

-

s

.

Definition

.

A

d

-dimensional Brownian motion is a stochastic process

W

⁢

(

t

)

=

(

W

1

⁢

(

t

)

,

…

,

W

d

⁢

(

t

)

)

in

ℝ

d

whose coordinate processes

W

i

⁢

(

t

)

are
independent one-dimensional Brownian motions.

Figure 1:

Sample paths of a standard Brownian motion

Title

Brownian motion

Canonical

name

BrownianMotion

Date of creation

2013-03-22 15:12:46

Last modified on

2013-03-22 15:12:46

Owner

skubeedooo (5401)

Last modified by

skubeedooo (5401)

Numerical id

16

Author

skubeedooo (5401)

Entry type

Definition

Classification

msc 60J65

Synonym

Wiener process

Related topic

WienerMeasure

Related topic

StochasticCalculusAndSDE

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Levy martingale characterization

Levy martingale characterization

Theorem

(Levy’s martingale characterisation).

Let

W

⁢

(

t

)

,

t

≥

0

, be a

stochastic process

and let

F

t

=

σ

(

W

s

,

s

≤

t

)

be the

filtration

generated by it. Then

W

⁢

(

t

)

is a

Wiener process

if and only if the following conditions hold:

1.

W

⁢

(

0

)

=

0

almost surely;

2.

The

sample paths

t

↦

W

⁢

(

t

)

are

continuous

almost surely;

3.

W

⁢

(

t

)

is a

martingale

with respect to the filtration

ℱ

t

;

4.

|

W

⁢

(

t

)

|

2

-

t

is a martingale with respect to

ℱ

t

.

Title

Levy martingale characterization

Canonical name

LevyMartingaleCharacterization

Date of creation

2013-03-22 15:12:48

Last modified on

2013-03-22 15:12:48

Owner

skubeedooo (5401)

Last modified by

skubeedooo (5401)

Numerical id

5

Author

skubeedooo (5401)

Entry type

Theorem

Classification

msc 60J65

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