predictable process

predictable process

A

predictable process

is a real-valued

stochastic process

whose values are known, in a sense, just in advance of time. Predictable processes are also called

previsible

.

1

predictable processes in discrete time

Suppose we have a

filtration

(

http://planetmath.org/FiltrationOfSigmaAlgebras

)

(

ℱ

n

)

n

∈

ℤ

+

on a

measurable space

(

Ω

,

ℱ

)

. Then a stochastic process

X

n

is predictable if

X

n

is

ℱ

n

-

1

-measurable (

http://planetmath.org/MeasurableFunctions

) for every

n

≥

1

and

X

0

is

ℱ

0

-measurable. So, the value of

X

n

is known at the previous time step. Compare with the definition of

adapted processes

for which

X

n

is

ℱ

n

-measurable.

2

predictable processes in continuous time

In

continuous

time, the definition of predictable processes is a little more subtle. Given a filtration

(

ℱ

t

)

with time index

t

ranging over the non-negative

real numbers

, the class of predictable processes forms the smallest set of real valued stochastic processes containing all left-continuous

ℱ

t

-adapted processes and which is

closed under

taking limits of a

sequence

of processes.

Equivalently, a real-valued stochastic process

X

:

ℝ

+

×

Ω

→

ℝ

(

t

,

ω

)

↦

X

t

⁢

(

ω

)

is predictable if it is measurable with respect to the predictable

sigma algebra

℘

. This is defined as the smallest

σ

-algebra on

ℝ

+

×

Ω

making all left-continuous and adapted processes measurable.

Alternatively,

℘

is generated by either of the following

collections

of subsets of

ℝ

+

×

Ω

℘

=

σ

⁢

(

{

(

t

,

∞

)

×

A

:

t

≥

0

,

A

∈

ℱ

t

}

∪

{

{

0

}

×

A

:

A

∈

ℱ

0

}

)

=

σ

⁢

(

{

(

T

,

∞

)

:

T

⁢

is a stopping time

}

∪

{

{

0

}

×

A

:

A

∈

ℱ

0

}

)

=

σ

⁢

(

{

[

T

,

∞

)

:

T

⁢

is a predictable stopping time

}

)

Note that in these definitions, the sets

(

T

,

∞

)

and

[

T

,

∞

)

are stochastic intervals, and subsets of

ℝ

+

×

Ω

.

3

general predictable processes

The definition of predictable process given above can be extended to a filtration

(

ℱ

t

)

with time index

t

lying in an arbitrary subset

𝕋

of the

extended real numbers

. In this case, the predictable sets form a

σ

-algebra on

𝕋

×

Ω

. If

𝕋

has a minimum

element

t

0

then let

S

be the collection of sets of the form

{

t

0

}

×

A

for

A

∈

ℱ

t

0

, otherwise let

S

be the

empty set

.Then, the predictable

σ

-algebra is defined by

℘

=

σ

⁢

(

{

(

t

,

∞

]

×

A

:

t

∈

𝕋

,

A

∈

ℱ

t

}

∪

S

)

=

σ

⁢

(

{

(

T

,

∞

]

:

T

:

Ω

→

𝕋

⁢

is a stopping time

}

∪

S

)

.

Here,

(

t

,

∞

]

and

(

T

,

∞

]

are understood to be

intervals

containing only times in the

index set

𝕋

. If

𝕋

is an interval of the real numbers then

℘

can be equivalently defined as the

σ

-algebra generated by the class of left-continuous and adapted processes with time index ranging over

𝕋

.

A stochastic process

X

:

𝕋

×

Ω

→

ℝ

is predictable if it is

℘

-measurable. It can be verified that in the cases where

𝕋

=

ℤ

+

or

𝕋

=

ℝ

+

then this definition agrees with the ones given above.

Title

predictable process

Canonical name

PredictableProcess

Date of creation

2013-03-22 18:36:30

Last modified on

2013-03-22 18:36:30

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

12

Author

gel (22282)

Entry type

Definition

Classification

msc 60G07

Related topic

PredictableStoppingTime

Related topic

ProgressivelyMeasurableProcess

Related topic

OptionalProcess

Defines

predictable

Defines

previsible

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