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F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
Model of information available at a given point of a random process
In the theory of stochastic processes , a subdiscipline of probability theory , filtrations are totally ordered collections of subsets that are used to model the information that is available at a given point and therefore play an important role in the formalization of random (stochastic) processes.
Definition
[ edit ]
Let
(
Ω
,
A
,
P
)
{\displaystyle (\Omega ,{\mathcal {A}},P)}
be a probability space and let
I
{\displaystyle I}
be an index set with a total order
≤
{\displaystyle \leq }
(often
N
{\displaystyle \mathbb {N} }
,
R
+
{\displaystyle \mathbb {R} ^{+}}
, or a subset of
R
+
{\displaystyle \mathbb {R} ^{+}}
).
For every
i
∈
I
{\displaystyle i\in I}
let
F
i
{\displaystyle {\mathcal {F}}_{i}}
be a sub-σ -algebra of
A
{\displaystyle {\mathcal {A}}}
. Then
F
:=
(
F
i
)
i
∈
I
{\displaystyle \mathbb {F} :=({\mathcal {F}}_{i})_{i\in I}}
is called a filtration, if
F
k
⊆
F
ℓ
{\displaystyle {\mathcal {F}}_{k}\subseteq {\mathcal {F}}_{\ell }}
for all
k
≤
ℓ
{\displaystyle k\leq \ell }
. So filtrations are families of σ -algebras that are ordered non-decreasingly.[1] If
F
{\displaystyle \mathbb {F} }
is a filtration, then
(
Ω
,
A
,
F
,
P
)
{\displaystyle (\Omega ,{\mathcal {A}},\mathbb {F} ,P)}
is called a filtered probability space .
Example
[ edit ]
Let
(
X
n
)
n
∈
N
{\displaystyle (X_{n})_{n\in \mathbb {N} }}
be a stochastic process on the probability space
(
Ω
,
A
,
P
)
{\displaystyle (\Omega ,{\mathcal {A}},P)}
.
Let
σ
(
X
k
∣
k
≤
n
)
{\displaystyle \sigma (X_{k}\mid k\leq n)}
denote the σ -algebra generated by the random variables
X
1
,
X
2
,
…
,
X
n
{\displaystyle X_{1},X_{2},\dots ,X_{n}}
.
Then
F
n
:=
σ
(
X
k
∣
k
≤
n
)
{\displaystyle {\mathcal {F}}_{n}:=\sigma (X_{k}\mid k\leq n)}
is a σ -algebra and
F
=
(
F
n
)
n
∈
N
{\displaystyle \mathbb {F} =({\mathcal {F}}_{n})_{n\in \mathbb {N} }}
is a filtration.
F
{\displaystyle \mathbb {F} }
really is a filtration, since by definition all
F
n
{\displaystyle {\mathcal {F}}_{n}}
are σ -algebras and
σ
(
X
k
∣
k
≤
n
)
⊆
σ
(
X
k
∣
k
≤
n
+
1
)
.
{\displaystyle \sigma (X_{k}\mid k\leq n)\subseteq \sigma (X_{k}\mid k\leq n+1).}
This is known as the natural filtration of
A
{\displaystyle {\mathcal {A}}}
with respect to
(
X
n
)
n
∈
N
{\displaystyle (X_{n})_{n\in \mathbb {N} }}
.
Types of filtrations
[ edit ]
Right-continuous filtration
[ edit ]
If
F
=
(
F
i
)
i
∈
I
{\displaystyle \mathbb {F} =({\mathcal {F}}_{i})_{i\in I}}
is a filtration, then the corresponding right-continuous filtration is defined as[2]
F
+
:=
(
F
i
+
)
i
∈
I
,
{\displaystyle \mathbb {F} ^{+}:=({\mathcal {F}}_{i}^{+})_{i\in I},}
with
F
i
+
:=
⋂
i
<
z
F
z
.
{\displaystyle {\mathcal {F}}_{i}^{+}:=\bigcap _{i<z}{\mathcal {F}}_{z}.}
The filtration
F
{\displaystyle \mathbb {F} }
itself is called right-continuous if
F
+
=
F
{\displaystyle \mathbb {F} ^{+}=\mathbb {F} }
.[3]
Complete filtration
[ edit ]
Let
(
Ω
,
F
,
P
)
{\displaystyle (\Omega ,{\mathcal {F}},P)}
be a probability space and let,
N
P
:=
{
A
⊆
Ω
∣
A
⊆
B
for some
B
∈
F
with
P
(
B
)
=
0
}
{\displaystyle {\mathcal {N}}_{P}:=\{A\subseteq \Omega \mid A\subseteq B{\text{ for some }}B\in {\mathcal {F}}{\text{ with }}P(B )=0\}}
be the set of all sets that are contained within a
P
{\displaystyle P}
-null set .
A filtration
F
=
(
F
i
)
i
∈
I
{\displaystyle \mathbb {F} =({\mathcal {F}}_{i})_{i\in I}}
is called a complete filtration , if every
F
i
{\displaystyle {\mathcal {F}}_{i}}
contains
N
P
{\displaystyle {\mathcal {N}}_{P}}
. This implies
(
Ω
,
F
i
,
P
)
{\displaystyle (\Omega ,{\mathcal {F}}_{i},P)}
is a complete measure space for every
i
∈
I
.
{\displaystyle i\in I.}
(The converse is not necessarily true.)
Augmented filtration
[ edit ]
A filtration is called an augmented filtration if it is complete and right continuous. For every filtration
F
{\displaystyle \mathbb {F} }
there exists a smallest augmented filtration
F
~
{\displaystyle {\tilde {\mathbb {F} }}}
refining
F
{\displaystyle \mathbb {F} }
.
If a filtration is an augmented filtration, it is said to satisfy the usual hypotheses or the usual conditions .[3]
See also
[ edit ]
References
[ edit ]
^ Kallenberg, Olav (2017). Random Measures, Theory and Applications . Probability Theory and Stochastic Modelling. Vol. 77. Switzerland: Springer. p. 350-351. doi :10.1007/978-3-319-41598-7 . ISBN 978-3-319-41596-3 .
^ a b Klenke, Achim (2008). Probability Theory . Berlin: Springer. p. 462 . doi :10.1007/978-1-84800-048-3 . ISBN 978-1-84800-047-6 .
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Filtration_(probability_theory)&oldid=1227857598 "
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● T h i s p a g e w a s l a s t e d i t e d o n 8 J u n e 2 0 2 4 , a t 0 5 : 1 9 ( U T C ) .
● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
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