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In the mathematical theory of probability, Blumenthal's on
[
0
,
∞
)
{\displaystyle [0,\infty )}
starting from deterministic point has also deterministic initial movement.
Statement
Suppose that
X
=
(
X
t
:
t
≥
0
)
{\displaystyle X=(X_{t}:t\geq 0)}
is an adapted
(
Ω
,
F
,
{
F
t
}
t
≥
0
,
P
)
{\displaystyle (\Omega ,{\mathcal {F}},\{{\mathcal {F}}_{t}\}_{t\geq 0},\mathbb {P} )}
such that
X
0
{\displaystyle X_{0}}
is constant with probability one. Let
F
t
X
:=
σ
(
X
s
;
s
≤
t
)
,
F
t
+
X
:=
⋂
s
>
t
F
s
X
{\displaystyle {\mathcal {F}}_{t}^{X}:=\sigma (X_{s};s\leq t),{\mathcal {F}}_{t^{+}}^{X}:=\bigcap _{s>t}{\mathcal {F}}_{s}^{X}}
. Then any event in the
sigma algebra
Λ
∈
F
0
+
X
{\displaystyle \Lambda \in {\mathcal {F}}_{0+}^{X}}
has either
P
(
Λ
)
=
0
{\displaystyle \mathbb {P} (\Lambda )=0}
or
P
(
Λ
)
=
1.
{\displaystyle \mathbb {P} (\Lambda )=1.}
Generalization
Suppose that
X
=
(
X
t
:
t
≥
0
)
{\displaystyle X=(X_{t}:t\geq 0)}
is an adapted stochastic process on a probability space
(
Ω
,
F
,
{
F
t
}
t
≥
0
,
P
)
{\displaystyle (\Omega ,{\mathcal {F}},\{{\mathcal {F}}_{t}\}_{t\geq 0},\mathbb {P} )}
such that
X
0
{\displaystyle X_{0}}
is constant with probability one. If
X
{\displaystyle X}
has Markov property with respect to the filtration
{
F
t
+
}
t
≥
0
{\displaystyle \{{\mathcal {F}}_{t^{+}}\}_{t\geq 0}}
then any event
Λ
∈
F
0
+
X
{\displaystyle \Lambda \in {\mathcal {F}}_{0+}^{X}}
has either
P
(
Λ
)
=
0
{\displaystyle \mathbb {P} (\Lambda )=0}
or
P
(
Λ
)
=
1.
{\displaystyle \mathbb {P} (\Lambda )=1.}
Note that every
(
Ω
,
F
,
{
F
t
}
t
≥
0
,
P
)
{\displaystyle (\Omega ,{\mathcal {F}},\{{\mathcal {F}}_{t}\}_{t\geq 0},\mathbb {P} )}
has
strong Markov property
with respect to the filtration
{
F
t
+
}
t
≥
0
{\displaystyle \{{\mathcal {F}}_{t^{+}}\}_{t\geq 0}}
.
References