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−
0
(
Then
,
the process M
t
:=
A
f(X
t
)
f )(X
s
)
d
s is a martingale with respect to the
filtration of W
.
Proof
We apply the Itô formula (1.7)to
f(X
t
)
and obtain, in integral form,
t
t
1
2
∂
xx
f(X
s
)σ
2
(X
s
)
d
s.
f(X
t
)
=
f(X
0
)
+
∂
x
f(X
s
)
d
X
s
+
0
0
Using d
X
t
=
b(X
t
)
d
t
+
σ(X
t
)
d
W
t
, and the definition of
A
,wehave
t
t
σ(X
s
)f
(X
s
)
d
W
s
.
f(X
t
)
=
f(X
0
)
+
(
A
f )(X
s
)
d
s
+
0
0
The result follows if we can show that the stochastic integral
0
σ(X
s
)∂
x
f(X
s
)
d
W
s
is a martingale (with respect to the filtration of
W
). According to Proposition 1.2.7,
it is sufficient to show that
0
|
2
d
s
E[
σ(X
s
)∂
x
f(X
s
)
|
]
<
∞
. Since
f
has bounded
derivatives and
σ
satisfies (1.4), we obtain
t
0
|
2
d
s
2
)
d
s
t
2
C
2
sup
x
∈R
2
E
σ(X
s
)
|
|
∂
x
f(X
s
)
|
≤
|
∂
x
f(x)
|
E
(
1
+|
X
s
|
0
2
1
+ E
sup
0
2
(
1.5
)
≤
TC
2
sup
x
|
∂
x
f(x)
|
T
|
X
s
|
<
∞
.
∈R
≤
s
≤
Remark 4.1.2
For
t>
0 denote by
X
t
the solution of the SDE (1.2) starting from
−
0
(
x
at time 0. Then, since
M
t
=
f(X
t
)
A
f )(X
s
)
d
s
is a martingale by Proposi-
tion
4.1.1
, we know that
E[
M
0
]=
f(x)
= E[
M
t
]
. Therefore,
f )(X
s
)
d
s
.
t
f(X
t
)
E[
]=
f(x)
+ E
(
A
0
Since by assumption
f
has bounded derivatives and
b, σ
satisfy the global Lipschitz
and linear growth condition (1.3)-(1.4)
E
sup
0
|
≤
E
sup
0
|
f )(X
s
)
σ
2
(X
s
)
b(X
s
)
≤
s
≤
T
|
(
A
C
≤
s
≤
T
|
|+|
C
1
+ E
sup
0
2
<
X
s
|
≤
T
|
∞
.
≤
s
≤
f
and
X
t
Thus, since
A
are continuous, the dominated convergence theorem gives
1
t
f )(X
s
)
d
s
t
d
d
t
E[
f(X
t
)
]|
t
=
0
=
lim
t
0
E
(
A
=
(
A
f )(x).
→
0
is called the
infinitesimal generator
of the process
X
t
.
Therefore,
A
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