aarjaneiro/planetmath huggingface dataset:
predictable stopping time

predictable stopping time

A

predictable

, or

previsible

stopping time

is a random time which is possible to predict just before the event.
Letting

(

ℱ

t

)

t

∈

ℝ

+

be a

filtration

(

http://planetmath.org/FiltrationOfSigmaAlgebras

) on a

measurable space

(

Ω

,

ℱ

)

, then, a stopping time

τ

is predictable if there exists an increasing sequence of stopping times

τ

n

satisfying the following.

•

τ

n

<

τ

whenever

τ

>

0

.

•

τ

n

→

τ

as

n

→

∞

.

The

sequence

τ

n

is said to announce or foretell

τ

.

For example, if

X

is a

continuous

adapted process

with

X

0

=

0

, such as

Brownian motion

, then the first time

τ

at which it hits a given level

K

≠

0

is a

predictable stopping time

.
In this case, if

τ

n

is the first time at which

X

hits the level

K

⁢

(

1

-

1

/

n

)

, then the sequence

τ

n

announces

τ

.

On the other hand, if

X

is a

Poisson process

then the first time

τ

at which it is nonzero is not predictable. To show this, suppose that

τ

n

<

τ

are stopping times. The fact that

X

t

-

λ

⁢

t

is a

martingale

means that Doob’s

optional

sampling theorem

can be applied, giving

𝔼

⁢

[

X

τ

n

-

λ

⁢

τ

n

]

=

0

.
Then,

X

t

=

0

for

t

<

τ

gives

𝔼

⁢

[

τ

n

]

=

0

. So,

τ

n

=

0

with

probability

one, and the sequence

τ

n

cannot announce

τ

.

In discrete time, where the filtration

(

ℱ

t

)

has time

t

running over the

index set

ℤ

+

, then a stopping time is said to be predictable if

{

τ

≤

t

}

is

ℱ

t

-

1

-

measurable

for every time

t

=

1

,

2

,

…

.

This can be generalized to an arbitrary index set

𝕋

, where a stopping time

τ

:

Ω

→

𝕋

∪

{

∞

}

is predictable if there exists an increasing sequence of stopping times

τ

n

≤

τ

such that

τ

n

<

τ

whenever

τ

is not equal to a

minimal element

of

𝕋

, and

⋂

n

(

τ

n

,

τ

)

contains no elements of

𝕋

.

Title

predictable stopping time

Canonical name

PredictableStoppingTime

Date of creation

2013-03-22 18:37:19

Last modified on

2013-03-22 18:37:19

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

6

Author

gel (22282)

Entry type

Definition

Classification

msc 60G40

Classification

msc 60G05

Synonym

predictable time

Synonym

previsible time

Synonym

previsible stopping time

Related topic

StoppingTime

Related topic

PredictableProcess

Generated on Sat Feb 10 11:59:04 2018 by

LaTeXML
proof of the début theorem

proof of the début theorem

Let

(

ℱ

)

t

∈

𝕋

be a right-continuous filtration (

http://planetmath.org/FiltrationOfSigmaAlgebras

) on the

measurable space

(

Ω

,

ℱ

)

, It is assumed that

𝕋

is a

closed subset

of

ℝ

and that

ℱ

t

is

universally complete

for each

t

∈

𝕋

.

If

A

⊆

𝕋

×

Ω

is a

progressively measurable

set, then we show that its début

D

⁢

(

A

)

=

inf

⁡

{

t

∈

𝕋

:

(

t

,

ω

)

∈

A

}

is a

stopping time

.

As

A

is progressively measurable, the set

A

∩

(

(

-

∞

,

t

)

×

Ω

)

is

ℬ

⁢

(

𝕋

)

×

ℱ

t

-measurable. By the

measurable projection theorem

it follows that

{

D

(

A

)

<

t

}

=

{

ω

∈

Ω

:

(

s

,

ω

)

∈

A

∩

(

(

-

∞

,

t

)

×

Ω

)

for some

s

∈

𝕋

}

is in

ℱ

t

. If there exists a sequence

t

n

∈

𝕋

with

t

n

>

t

and

t

n

→

t

, then

{

D

(

A

)

≤

t

}

=

⋂

n

{

D

(

A

)

<

t

n

}

∈

⋂

n

ℱ

t

n

=

ℱ

t

+

=

ℱ

t

.

On the other hand, if

t

is not a right limit point of

𝕋

then

{

D

(

A

)

≤

t

}

=

{

D

(

A

)

<

t

}

∪

{

ω

∈

Ω

:

(

t

,

ω

)

∈

A

}

∈

ℱ

t

.

In either case,

{

D

⁢

(

A

)

≤

t

}

is in

ℱ

t

, so

D

⁢

(

A

)

is a stopping time.

Title

proof of the début theorem

Canonical name

ProofOfTheDebutTheorem

Date of creation

2013-03-22 18:39:15

Last modified on

2013-03-22 18:39:15

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

6

Author

gel (22282)

Entry type

Proof

Classification

msc 60G40

Classification

msc 60G05

Generated on Sat Feb 10 11:59:06 2018 by

LaTeXML
stopped process

stopped process

A

stochastic process

(

X

t

)

t

∈

𝕋

defined on a

measurable space

(

Ω

,

ℱ

)

can be stopped at a random time

τ

:

Ω

→

𝕋

∪

{

∞

}

. The resulting

stopped process

is denoted by

X

τ

,

X

t

τ

≡

X

min

⁡

(

t

,

τ

)

.

The random time

τ

used is typically a

stopping time

.

If the process

X

t

has left limits (

http://planetmath.org/CadlagProcess

) for every

t

∈

𝕋

, then it can alternatively be stopped just before the time

τ

, resulting in the pre-stopped process

X

τ

-

≡

{

X

t

,

if

⁢

t

<

τ

,

X

τ

-

,

if

⁢

t

≥

τ

.

Stopping is often used to enforce boundedness or integrability constraints on a process.
For example, if

B

is a

Brownian motion

and

τ

is the first time at which

|

B

τ

|

hits some given

positive

value, then the stopped process

B

τ

will be a

continuous

and

bounded

martingale

.
It can be shown that many

properties

of stochastic processes, such as the martingale property, are stable under stopping at any stopping time

τ

. On the other hand, a pre-stopped martingale need not be a martingale.

For continuous processes, stopping and pre-stopping are

equivalent

procedures.
If

τ

is the first time at which

|

X

τ

|

≥

K

, for any given

real number

K

, then the pre-stopped process

X

τ

-

will be uniformly bounded.
However, for some noncontinuous processes it is not possible to find a stopping time

τ

>

0

making

X

τ

into a uniformly bounded process. For example, this is the case for any

Levy process

(

http://planetmath.org/LevyProcess

) with

unbounded

jump

distribution

.

Stopping is used to generalize properties of stochastic processes to obtain the related localized property. See, for example,

local martingales

.

Title

stopped process

Canonical

name

StoppedProcess

Date of creation

2013-03-22 18:37:38

Last modified on

2013-03-22 18:37:38

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

5

Author

gel (22282)

Entry type

Definition

Classification

msc 60G40

Classification

msc 60G05

Synonym

optional stopping

Defines

pre-stopped process

Defines

prestopped process

Generated on Sat Feb 10 11:59:08 2018 by

LaTeXML
Doob’s optional sampling theorem

Doob’s optional sampling theorem

Given a

filtered probability space

(

Ω

,

ℱ

,

(

ℱ

t

)

t

∈

𝕋

,

ℙ

)

, a process

(

X

t

)

t

∈

𝕋

is a

martingale

if it

satisfies

the equality

𝔼

[

X

t

∣

ℱ

s

]

=

X

s

for all

s

<

t

in the

index set

𝕋

. Doob’s

optional

sampling theorem

says that this equality still holds if the times

s

,

t

are replaced by bounded

stopping times

S

,

T

. In this case, the

σ

-

algebra

ℱ

s

is replaced by the

collection

of events observable at the random time

S

(

http://planetmath.org/SigmaAlgebraAtAStoppingTime

),

ℱ

S

=

{

A

∈

ℱ

:

A

∩

{

S

≤

t

}

∈

ℱ

t

for all

t

∈

𝕋

}

.

In discrete-time, when the index set

𝕋

is

countable

, the result is as follows.

Doob’s Optional Sampling Theorem

.

Suppose that the index set

T

is countable and that

S

≤

T

are stopping times bounded above by some

constant

c

∈

T

.
If

(

X

t

)

is a martingale then

X

T

is an integrable

random variable

and

𝔼

[

X

T

|

ℱ

S

]

=

X

S

,

ℙ

almost surely

.

(1)

Similarly, if

X

is a submartingale then

X

T

is integrable and

𝔼

[

X

T

|

ℱ

S

]

≥

X

S

,

ℙ

almost surely

.

(2)

If

X

is a supermartingale then

X

T

is integrable and

𝔼

[

X

T

|

ℱ

S

]

≤

X

S

,

ℙ

almost surely

.

(3)

This

theorem

shows, amongst other things, that in the case of a fair casino, where your return is a martingale, betting strategies involving ‘knowing when to quit’ do not enhance your expected return.

In continuous-time, when the index set

𝕋

an interval of the real numbers, then the stopping times

S

,

T

can have a

continuous distribution

and

X

S

,

X

T

need not be measurable quantities. Then, it is necessary to place conditions on the

sample paths

of the process

X

. In particular, Doob’s optional sampling theorem holds in continuous-time if

X

is assumed to be right-continuous.

Title

Doob’s optional sampling theorem

Canonical name

DoobsOptionalSamplingTheorem

Date of creation

2013-03-22 16:43:41

Last modified on

2013-03-22 16:43:41

Owner

skubeedooo (5401)

Last modified by

skubeedooo (5401)

Numerical id

8

Author

skubeedooo (5401)

Entry type

Theorem

Classification

msc 60G44

Classification

msc 60G46

Classification

msc 60G42

Related topic

Martingale

Related topic

StoppingTime

Related topic

SigmaAlgebraAtAStoppingTime

Generated on Sat Feb 10 11:59:19 2018 by

LaTeXML
Kolmogorov’s martingale inequality

Kolmogorov’s martingale inequality

Theorem

(Kolmogorov’s martingale inequality).

Let

X

⁢

(

t

)

, for

0

≤

t

≤

T

, be a

submartingale

with

continuous

sample paths

.
Then for any

constant

α

>

0

,

ℙ

(

max

0

≤

t

≤

T

X

(

t

)

≥

α

)

≤

𝔼

⁢

[

X

⁢

(

T

)

+

]

α

.

(The notation

X

⁢

(

T

)

+

means

max

⁡

(

X

⁢

(

T

)

,

0

)

, the positive part of

X

⁢

(

T

)

.)

Notice the

analogy

with Markov’s

inequality

.
Of course, the

conclusion

is much

stronger

than Markov’s inequality,
as the probabilistic bound applies to an

uncountable

number
of

random variables

. The

continuity

and submartingale hypotheses
are used to establish the stronger bound.

Proof.

Let

{

t

i

}

i

=

1

n

be a

partition

of the

interval

[

0

,

T

]

.
Let

B

=

{

max

1

≤

i

≤

n

X

(

t

i

)

≥

α

}

and split

B

into

disjoint

parts

B

i

,
defined by

B

i

=

{

X

(

t

j

)

<

α

for all

j

<

i

but

X

(

t

i

)

≥

α

}

.

Also let

{

ℱ

t

}

be the

filtration

under which

X

⁢

(

t

)

is a submartingale.

Then

ℙ

⁢

(

B

)

=

∑

i

=

1

n

𝔼

⁢

[

1

⁢

(

B

i

)

]

≤

∑

i

=

1

n

𝔼

⁢

[

X

⁢

(

t

i

)

α

⁢

1

⁢

(

B

i

)

]

definition

of

B

i

≤

1

α

∑

i

=

1

n

𝔼

[

𝔼

[

X

(

T

)

∣

ℱ

t

i

]

1

(

B

i

)

]

X

⁢

(

t

)

is submartingale

=

1

α

∑

i

=

1

n

𝔼

[

𝔼

[

X

(

T

)

1

(

B

i

)

∣

ℱ

t

i

]

]

B

i

is

ℱ

t

i

-

measurable

=

1

α

⁢

∑

i

=

1

n

𝔼

⁢

[

X

⁢

(

T

)

⁢

1

⁢

(

B

i

)

]

iterated

expectation

=

1

α

⁢

𝔼

⁢

[

X

⁢

(

T

)

⁢

1

⁢

(

B

)

]

≤

1

α

⁢

𝔼

⁢

[

X

⁢

(

T

)

+

⁢

1

⁢

(

B

)

]

≤

1

α

⁢

𝔼

⁢

[

X

⁢

(

T

)

+

]

monotonicity.

Since the sample paths are continuous by

hypothesis

,
the event

A

=

{

max

0

≤

t

≤

T

X

(

t

)

≥

α

}

can be expressed as an

countably infinite

intersection

of events of the form

B

with finer and finer partitions

{

t

i

}

of the time interval

[

0

,

T

]

.
By taking limits, it follows

ℙ

⁢

(

A

)

has the same bound as the

probabilities

ℙ

⁢

(

B

)

.
∎

Corollary

.

Let

X

⁢

(

t

)

, for

0

≤

t

≤

T

, be a square-integrable

martingale

possessing continuous sample paths, whose
unconditional mean is

m

=

E

⁢

[

X

⁢

(

0

)

]

.
For any constant

α

>

0

,

ℙ

(

max

0

≤

t

≤

T

|

X

(

t

)

-

m

|

≥

α

)

≤

Var

⁡

[

X

⁢

(

T

)

]

α

2

.

Proof.

Apply Kolmogorov’s martingale inequality to

(

X

⁢

(

t

)

-

m

)

2

,
which is a submartingale by Jensen’s inequality.
∎

Title

Kolmogorov’s martingale inequality

Canonical

name

KolmogorovsMartingaleInequality

Date of creation

2013-03-22 17:20:22

Last modified on

2013-03-22 17:20:22

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

8

Author

stevecheng (10074)

Entry type

Theorem

Classification

msc 60G44

Classification

msc 60G07

Synonym

Kolmogorov’s submartingale inequality

Related topic

MarkovsInequality

Related topic

DoobsInequalities

Generated on Sat Feb 10 11:59:23 2018 by

LaTeXML
Doob’s inequalities

Doob’s inequalities

Doob’s

inequalities

place bounds on the

maximum

value attained by a

martingale

in terms of the terminal value.
We consider a process

(

X

t

)

t

∈

𝕋

defined on the

filtered probability space

(

Ω

,

ℱ

,

(

ℱ

)

t

∈

𝕋

,

ℙ

)

. The associated maximum process

(

X

t

*

)

is

X

t

*

≡

sup

s

≤

t

⁡

|

X

s

|

.

The notation

∥

⋅

∥

p

for the

L

p

-norm (

http://planetmath.org/LpSpace

) of a

random variable

will be used.
In discrete-time or, more generally whenever the

index set

𝕋

is

countable

, then Doob’s inequalities are as follows.

Theorem 1

(Doob).

Let

(

X

t

)

t

∈

T

be a submartingale with countable index set

T

. Then,

ℙ

(

sup

s

≤

t

X

s

≥

K

)

≤

K

-

1

𝔼

[

(

X

t

)

+

]

(1)

If

X

is either a martingale or nonnegative submartingale then,

ℙ

(

X

t

*

≥

K

)

≤

K

-

1

𝔼

[

|

X

t

|

]

,

(2)

∥

X

t

*

∥

p

≤

p

p

-

1

⁢

∥

X

t

∥

p

.

(3)

for every

K

>

0

and

p

>

1

.

In particular, (

3

) shows that the maximum of any

L

p

-bounded martingale is itself

L

p

-bounded and, martingales

X

n

converge

to

X

in the

L

p

-norm if and only if

(

X

n

-

X

)

*

→

0

in the

L

p

-norm. The special case where

p

=

2

gives

𝔼

⁢

[

(

X

t

*

)

2

]

≤

4

⁢

𝔼

⁢

[

X

t

2

]

which is known as

Doob’s

maximal

quadratic inequality

.

Similarly, (

2

) shows that any

L

1

-bounded martingale is almost surely bounded and that convergence in the

L

1

-norm implies

ucp convergence

. Inequality (

1

) is also known as Kolmogorov’s submartingale inequality.

Doob’s inequalities are often applied to

continuous-time processes

, where

𝕋

=

ℝ

+

. In this case,

X

t

*

=

sup

s

≤

t

⁡

|

X

s

|

is a

supremum

of uncountably many random variables, and need not be

measurable

. Instead, it is typically assumed that the processes are right-continuous, in which case, for any

t

>

0

the supremum may be restricted to the

countable set

𝕋

′

=

{

s

∈

ℝ

+

:

s

/

t

∈

ℚ

}

.

Putting this into

Theorem

1

gives the following continuous-time version of the inequalities.

Theorem 2

(Doob).

Let

(

X

t

)

t

∈

R

+

be a right-continuous submartingale. Then,

ℙ

(

sup

s

≤

t

X

s

≥

K

)

≤

K

-

1

𝔼

[

(

X

t

)

]

for every

K

>

0

.
If

X

is right-continuous and either a martingale or nonnegative submartingale then,

ℙ

(

X

t

*

≥

K

)

≤

K

-

1

𝔼

[

|

X

t

|

]

,

∥

X

t

*

∥

p

≤

p

p

-

1

⁢

∥

X

t

∥

p

.

for every

K

>

0

and

p

>

1

.

Title

Doob’s inequalities

Canonical name

DoobsInequalities

Date of creation

2013-03-22 18:39:52

Last modified on

2013-03-22 18:39:52

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

7

Author

gel (22282)

Entry type

Theorem

Classification

msc 60G46

Classification

msc 60G44

Classification

msc 60G42

Synonym

Doob’s inequality

Related topic

KolmogorovsMartingaleInequality

Defines

Doob’s maximal quadratic inequality

Generated on Sat Feb 10 11:59:25 2018 by

LaTeXML
gale

gale

Let

ν

be a

probability measure

on Cantor space

𝐂

, and let

s

∈

[

0

,

∞

)

.

1.

A

ν

-

s

-

supergale

is a

function

d

:

{

0

,

1

}

*

→

[

0

,

∞

)

that

satisfies

the condition

d

⁢

(

w

)

⁢

ν

⁢

(

w

)

s

≥

d

⁢

(

w

⁢

0

)

⁢

ν

⁢

(

w

⁢

0

)

s

+

d

⁢

(

w

⁢

1

)

⁢

ν

⁢

(

w

⁢

1

)

s

(1)

for all

w

∈

{

0

,

1

}

*

, the set of all finite strings of

0

’s and

1

’s (including

e

, the

empty string

).

2.

A

ν

-

s

-

gale

is a

ν

-

s

-

supergale

that satisfies the condition with equality for all

w

∈

{

0

,

1

}

*

.

3.

A

ν

-

supermartingale

is a

ν

-1-supergale.

4.

A

ν

-

martingale

is a

ν

-1-gale.

5.

An

s

-

supergale

is a

μ

-

s

-supergale, where

μ

is the uniform probability measure.

6.

An

s

-

gale

is a

μ

-

s

-gale.

7.

A

supermartingale

is a 1-supergale.

8.

A

martingale

is a 1-gale.

Put in another way, a martingale is a function

d

:

{

0

,

1

}

*

→

[

0

,

∞

)

such that, for all

w

∈

{

0

,

1

}

*

,

d

⁢

(

w

)

=

(

d

⁢

(

w

⁢

0

)

+

d

⁢

(

w

⁢

1

)

)

/

2

.

Let

d

be a

ν

-

s

-

supergale

, where

ν

is a probability measure on

𝐂

and

s

∈

[

0

,

∞

)

. We say that

d

succeeds

on a

sequence

S

∈

𝐂

if

lim sup

n

→

∞

d

(

S

[

0

.

.

n

-

1

]

)

=

∞

.

The

success set

of

d

is

S

∞

[

d

]

=

{

S

∈

𝐂

|

d

succeeds on

S

}

.

d

succeeds on a

language

A

⊆

{

0

,

1

}

*

if

d

succeeds on the

characteristic sequence

χ

A

of

A

. We say that

d

succeeds strongly

on a sequence

S

∈

𝐂

if

lim inf

n

→

∞

d

(

S

[

0

.

.

n

-

1

]

)

=

∞

.

The

strong success set

of

d

is

S

str

∞

[

d

]

=

{

S

∈

𝐂

|

d

succeeds strongly on

S

}

.

Intuitively, a supergale

d

is a betting strategy that bets on the next bit of a sequence when the previous bits are known.

s

is the

parameter

that tunes the fairness of the betting. The smaller

s

is, the less fair the betting is. If

d

succeeds on a sequence, then the bonus we can get from applying

d

as the betting strategy on the sequence is

unbounded

. If

d

succeeds strongly on a sequence, then the bonus goes to

infinity

.

Title

gale

Canonical

name

Gale

Date of creation

2013-03-22 16:43:37

Last modified on

2013-03-22 16:43:37

Owner

skubeedooo (5401)

Last modified by

skubeedooo (5401)

Numerical id

5

Author

skubeedooo (5401)

Entry type

Definition

Classification

msc 60G46

Classification

msc 60G44

Classification

msc 60G42

Defines

supergale

Defines

gale

Defines

supermartingale

Defines

succeed

Defines

succeed strongly

Defines

success set

Defines

strong success set

Generated on Sat Feb 10 11:59:28 2018 by

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martingale

martingale

Martingales

definition

Definition

. Let

(

Ω

,

ℱ

,

(

ℱ

t

)

t

∈

𝕋

,

ℙ

)

be a

filtered probability space

and

(

X

t

)

be a

stochastic process

such that

X

t

is integrable (

http://planetmath.org/Integral2

) for all

t

∈

𝕋

. Then,

X

=

(

X

t

,

ℱ

t

)

is called a

submartingale

if

𝔼

ℙ

[

X

t

|

ℱ

s

]

≥

X

s

,

for every

s

<

t

, a.e.[

ℙ

],

and a

supermartigale

if

𝔼

ℙ

[

X

t

|

ℱ

s

]

≤

X

s

,

for every

s

<

t

, a.e.[

ℙ

].

A submartingale that is also a supermartingale is called a

martingale

, i.e., a martingale

satisfies

𝔼

ℙ

[

X

t

|

ℱ

s

]

=

X

s

,

for
every

s

<

t

, a.e.[

ℙ

].

Similarly, if the

{

ℱ

t

}

form a decreasing

collection

of

σ

-

subalgebras

of

ℱ

, then

X

is called a

reverse submartingale

if

𝔼

ℙ

[

X

s

|

ℱ

t

]

≥

X

t

,

for every

s

<

t

, a.e.[

ℙ

],

and a

reverse supermartingale

if

𝔼

ℙ

[

X

s

|

ℱ

t

]

≤

X

t

,

for every

s

<

t

, a.e.[

ℙ

].

Remarks

•

The martingale property captures the idea of a fair bet, where the
expected future value is equal to the current value.

•

The submartingale property is

equivalent

to

∫

A

X

t

⁢

𝑑

ℙ

≥

∫

A

X

s

⁢

𝑑

ℙ

⁢

for every

A

∈

ℱ

s

and

s

<

t

and similarly for the other definitions. This is immediate
from the definition of

conditional expectation

.

Title

martingale

Canonical name

Martingale

Date of creation

2013-03-22 13:33:09

Last modified on

2013-03-22 13:33:09

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

25

Author

CWoo (3771)

Entry type

Definition

Classification

msc 60G46

Classification

msc 60G44

Classification

msc 60G42

Related topic

LocalMartingale

Related topic

DoobsOptionalSamplingTheorem

Related topic

ConditionalExpectationUnderChangeOfMeasure

Related topic

MartingaleConvergenceTheorem

Defines

martingale

Defines

supermartingale

Defines

submartingale

Defines

reverse submartingale

Defines

reverse supermartingale

Generated on Sat Feb 10 11:59:30 2018 by

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martingale convergence theorem

martingale convergence theorem

There are several convergence

theorems

for

martingales

, which follow from Doob’s

upcrossing

lemma. The following says that any

L

1

-bounded martingale

X

n

in discrete time

converges

almost surely.
Note that almost-sure convergence (i.e. convergence with probability one) is quite strong, implying the weaker property of

convergence in probability

. Here, a martingale

(

X

n

)

n

∈

ℕ

is understood to be defined with respect to a

probability space

(

Ω

,

ℱ

,

ℙ

)

and

filtration

(

ℱ

n

)

n

∈

ℕ

.

Theorem

(Doob’s Forward Convergence Theorem).

Let

(

X

n

)

n

∈

N

be a martingale (or submartingale, or supermartingale) such that

E

⁢

[

|

X

n

|

]

is bounded over all

n

∈

N

. Then, with

probability

one, the limit

X

∞

=

lim

n

→

∞

⁡

X

n

exists and is finite.

The condition that

X

n

is

L

1

-bounded is automatically satisfied in many cases. In particular, if

X

is a non-negative supermartingale then

𝔼

⁢

[

|

X

n

|

]

=

𝔼

⁢

[

X

n

]

≤

𝔼

⁢

[

X

1

]

for all

n

≥

1

, so

𝔼

⁢

[

|

X

n

|

]

is bounded, giving the following corollary.

Corollary

.

Let

(

X

n

)

n

∈

N

be a non-negative martingale (or supermartingale). Then, with probability one, the limit

X

∞

=

lim

n

→

∞

⁡

X

n

exists and is finite.

As an example application of the

martingale convergence theorem

, it is easy to show that a standard

random walk

started started at

0

will visit every level with probability one.

Corollary

.

Let

(

X

n

)

n

∈

N

be a standard random walk. That is,

X

1

=

0

and

ℙ

(

X

n

+

1

=

X

n

+

1

∣

ℱ

n

)

=

ℙ

(

X

n

+

1

=

X

n

-

1

∣

ℱ

n

)

=

1

/

2

.

Then, for every integer

a

, with probability one

X

n

=

a

for some

n

.

Proof.

Without loss of generality, suppose that

a

≤

0

. Let

T

:

Ω

→

ℕ

∪

{

∞

}

be the first time

n

for which

X

n

=

a

. It is easy to see that the

stopped process

X

n

T

defined by

X

n

T

=

X

min

⁡

(

n

,

T

)

is a martingale and

X

T

-

a

is non-negative. Therefore, by the martingale convergence theorem, the limit

X

∞

T

=

lim

n

→

∞

⁡

X

n

T

exists and is finite (almost surely). In particular,

|

X

n

+

1

T

-

X

n

T

|

converges to

0

and must be less than

1

for large

n

. However,

|

X

n

+

1

T

-

X

n

T

|

=

1

whenever

n

<

T

, so we have

T

<

∞

and therefore

X

n

=

a

for some

n

.
∎

Title

martingale convergence theorem

Canonical name

MartingaleConvergenceTheorem

Date of creation

2013-03-22 18:33:47

Last modified on

2013-03-22 18:33:47

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

5

Author

gel (22282)

Entry type

Theorem

Classification

msc 60G46

Classification

msc 60G44

Classification

msc 60G42

Classification

msc 60F15

Related topic

Martingale

Related topic

MartingaleProofOfKolmogorovsStrongLawForSquareIntegrableVariables

Related topic

MartingaleProofOfTheRadonNikodymTheorem

Related topic

UpcrossingsAndDowncrossings

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