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Algebra of a branch of probability theory
The σ-algebra of τ-past , (also named stopped σ-algebra , stopped σ-field , or σ-field of τ-past ) is a
σ-algebra associated with a
stopping time in the theory of
stochastic processes , a branch of
probability theory .
[1]
[2]
Definition
Let
τ
{\displaystyle \tau }
be a
stopping time on the
filtered probability space
(
Ω
,
A
,
(
F
t
)
t
∈
T
,
P
)
{\displaystyle (\Omega ,{\mathcal {A}},({\mathcal {F}}_{t})_{t\in T},P)}
. Then the
σ-algebra
F
τ
:=
{
A
∈
A
∣
∀
t
∈
T
:
{
τ
≤
t
}
∩
A
∈
F
t
}
{\displaystyle {\mathcal {F}}_{\tau }:=\{A\in {\mathcal {A}}\mid \forall t\in T\colon \{\tau \leq t\}\cap A\in {\mathcal {F}}_{t}\}}
is called the σ-algebra of τ-past.
[1]
[2]
Properties
Monotonicity
Is
σ
,
τ
{\displaystyle \sigma ,\tau }
are two stopping times and
σ
≤
τ
{\displaystyle \sigma \leq \tau }
almost surely, then
F
σ
⊂
F
τ
.
{\displaystyle {\mathcal {F}}_{\sigma }\subset {\mathcal {F}}_{\tau }.}
Measurability
A stopping time
τ
{\displaystyle \tau }
is always
F
τ
{\displaystyle {\mathcal {F}}_{\tau }}
-
measurable .
Intuition
The same way
F
t
{\displaystyle {\mathcal {F}}_{t}}
is all the information up to time
t
{\displaystyle t}
,
F
τ
{\displaystyle {\mathcal {F}}_{\tau }}
is all the information up time
τ
{\displaystyle \tau }
. The only difference is that
τ
{\displaystyle \tau }
is random. For example, if you had a random walk, and you wanted to ask, “How many times did the random walk hit −5 before it first hit 10?”, then letting
τ
{\displaystyle \tau }
be the first time the random walk hit 10,
F
τ
{\displaystyle {\mathcal {F}}_{\tau }}
would give you the information to answer that question.
[3]
References