CHAPTER 10.
MARKOV CHARACTERIZATIONS
55
Lemma 108
If
X
is adapted to
G
t
, then
F
X
t
±
G
t
.
Proof:
For each
t
,
X
t
is
G
t
measurable.
But
F
X
t
is, by construction, the
smallest
σ
algebra with respect to which
X
t
is measurable,
so,
for every
t
,
F
X
t
⊆ G
t
, and the result follows.
±
Theorem 109
If
X
is Markovian with respect to
G
t
, then it is Markovian with
respect to any coarser ﬁltration to which it is adapted, and in particular with
respect to its natural ﬁltration.
Proof:
Use the smoothing property of conditional expectations: For any two
σ
ﬁelds
F ⊂ G
and random variable
Y
,
E
[
Y
F
] =
E
[
E
[
Y
G
]
F
] a.s. So, if
F
t
is
coarser than
G
t
, and
X
is Markovian with respect to the latter, for any function
f
∈
L
1
and time
s > t
,
E
[
f
(
X
s
)
F
t
]
=
E
[
E
[
f
(
X
s
)
G
t
]
F
t
]
a.s.
(10.1)
=
E
[
E
[
f
(
X
s
)

X
t
]
F
t
]
(10.2)
=
E
[
f
(
X
s
)

X
t
]
(10.3)
where the last line uses the facts that (i)
E
[
f
(
X
s
)

X
t
] is a function
X
t
, (ii)
X
is adapted to
F
t
, so
X
t
is
F
t
measurable, and (iii) if
Y
is
F
measurable, then
E
[
Y
F
] =
Y
. Since this holds for all
f
∈
L
1
, it holds in particular for
1
A
, where
A
is any measurable set, and this established the conditional independence which
constitutes the Markov property.
Since (Lemma 108) the natural ﬁltration is
the coarsest ﬁltration to which
X
is adapted, the remainder of the theorem
follows.
±
The converse is false, as the following example shows.