A Proof of Corollary 2.2
Since
QMall.tf
and
QJumptf
have the forms
QMall.tf(x)=E[f(ˉXEM0,t(x))Mxt(Bt)]
it holds
Q
t
/
2
Jump
∘
Q
t
Mall
.
∘
Q
t
/
2
Jump
f
(
x
)
=
E
[
f
(
X
¯
t
/
2
,
t
Jump
∘
X
¯
t
EM
∘
X
¯
0
,
t
/
2
Jump
(
x
)
)
M
t
X
¯
0
,
t
/
2
Jump
(
x
)
(
B
t
)
]
.
Then we have
Q
s
1
/
2
Jump
∘
Q
s
1
Mall
.
∘
Q
s
1
/
2
Jump
∘
Q
s
2
/
2
Jump
∘
Q
s
2
Mall
.
∘
Q
s
2
/
2
Jump
∘
⋯
∘
Q
s
n
/
2
Jump
∘
Q
s
n
Mall
.
∘
Q
s
n
/
2
Jump
f
(
x
)
=
E
[
f
(
X
¯
T
(
n
)
(
x
)
)
∏
i
=
1
n
M
s
i
X
¯
t
i
-
1
,
t
i
-
1
+
s
i
/
2
Jump
∘
X
¯
t
i
-
1
(
n
)
(
x
)
(
B
t
i
-
B
t
i
-
1
)
]
,
x
∈
ℝ
N
,
where
X
¯
t
i
(
n
)
(
x
)
=
X
¯
t
i
-
1
+
s
i
/
2
,
t
i
Jump
∘
X
¯
t
i
-
1
,
t
i
EM
∘
X
¯
t
i
-
1
,
t
i
-
1
+
s
i
/
2
Jump
∘
⋯
∘
X
¯
t
2
+
s
2
/
2
,
t
3
Jump
∘
X
¯
t
1
,
t
2
EM
∘
X
¯
t
2
,
t
2
+
s
2
/
2
Jump
∘
X
¯
t
1
/
2
,
t
2
Jump
∘
X
¯
0
,
t
1
EM
∘
X
¯
0
,
t
1
/
2
Jump
(
x
)
for
i
=
1
,
…
,
n
.
Note that we have
E
[
|
X
¯
t
,
s
EM
(
x
)
-
x
|
2
]
=
O
(
s
-
t
)
and
E
[
|
X
¯
t
,
s
Jump
(
x
)
-
x
|
2
]
=
O
(
s
-
t
)
.
Thus, using (3.1), it holds that there is
C
1
(
T
)
>
0
independent of n such that
E
[
|
X
¯
T
(
n
)
(
x
)
|
2
]
≤
C
1
(
T
)
.
Also, the estimate of the Malliavin weight:
sup
x
∈
ℝ
N
E
[
|
M
(
s
-
t
)
x
(
B
s
-
B
t
)
|
2
]
≤
(
1
+
O
(
s
-
t
)
)
as in [22] gives
E
[
|
∏
i
=
1
n
M
s
i
X
¯
t
i
-
1
,
t
i
-
1
+
s
i
/
2
Jump
∘
X
¯
t
i
-
1
(
n
)
(
x
)
(
B
t
i
-
B
t
i
-
1
)
|
2
]
≤
C
2
(
T
)
for some
C
2
(
T
)
>
0
independent of n, where (3.1) is again used. Therefore, there exists
C
(
T
)
>
0
independent of n such that
∥
f
(
X
¯
T
(
n
)
(
x
)
)
∏
i
=
1
n
M
s
i
X
¯
t
i
-
1
,
t
i
-
1
+
s
i
/
2
Jump
∘
X
¯
t
i
-
1
(
n
)
(
x
)
(
B
t
i
-
B
t
i
-
1
)
∥
2
≤
C
(
T
)
.
The proof is finished.
B Proof of Lemma 3.1
We first prepare an approximation for the continuous diffusion part using Malliavin calculus and show how the operator
Q
t
Mall
.
is constructed using polynomials of Brownian motion
via Malliavin calculus [11, 13]. It holds that there exists
C
>
0
such that
(B.1)
∥
e
t
ℒ
Conti
.
f
-
Q
t
Mall
.
f
∥
∞
≤
C
∑
j
=
1
3
∥
∇
j
f
∥
∞
t
3
for all
t
>
0
and
f
∈
C
b
∞
(
ℝ
N
)
.
We decompose the upper bound as follows:
∥
P
t
f
-
Q
t
/
2
Jump
∘
Q
t
Mall
.
∘
Q
t
/
2
Jump
f
∥
∞
≤
∥
P
t
f
-
e
1
2
t
ℒ
Jump
e
t
ℒ
Conti
.
e
1
2
t
ℒ
Jump
f
∥
∞
+
∥
e
1
2
t
ℒ
Jump
e
t
ℒ
Conti
.
e
1
2
t
ℒ
Jump
f
-
Q
t
/
2
Jump
∘
Q
t
Mall
.
∘
Q
t
/
2
Jump
f
∥
∞
.
By semigroup expansions, we have for
m
∈
ℕ
,
f
∈
C
b
∞
(
ℝ
N
)
and
x
∈
ℝ
N
,
e
t
ℒ
f
(
x
)
=
f
(
x
)
+
∑
i
=
1
m
t
i
i
!
ℒ
i
f
(
x
)
+
∫
0
t
(
t
-
s
)
m
m
!
e
s
ℒ
ℒ
m
+
1
f
(
x
)
𝑑
s
,
e
t
ℒ
Jump
f
(
x
)
=
f
(
x
)
+
∑
i
=
1
m
t
i
i
!
(
ℒ
Jump
)
i
f
(
x
)
+
∫
0
t
(
t
-
s
)
m
m
!
e
s
ℒ
Jump
(
ℒ
Jump
)
m
+
1
f
(
x
)
𝑑
s
,
e
t
ℒ
Conti
.
f
(
x
)
=
f
(
x
)
+
∑
i
=
1
m
t
i
i
!
(
ℒ
Conti
.
)
i
f
(
x
)
+
∫
0
t
(
t
-
s
)
m
m
!
e
s
ℒ
Conti
.
(
ℒ
Conti
.
)
m
+
1
f
(
x
)
𝑑
s
.
We note that
(B.2)
sup
x
|
P
t
f
(
x
)
-
{
f
(
x
)
+
t
ℒ
f
(
x
)
+
1
2
t
2
ℒ
2
f
(
x
)
}
|
≤
C
∑
k
=
1
6
∥
∇
k
f
∥
∞
t
3
.
Then we have
e
t
ℒ
Conti
.
∘
e
1
2
t
ℒ
Jump
f
(
x
)
=
f
(
x
)
+
1
2
t
ℒ
Jump
f
(
x
)
+
1
8
t
2
ℒ
Jump
2
f
(
x
)
+
∫
0
t
2
(
t
2
-
s
)
2
2
e
s
ℒ
Jump
ℒ
Jump
3
f
(
x
)
𝑑
s
+
t
ℒ
Conti
.
{
f
(
x
)
+
1
2
t
ℒ
Jump
f
(
x
)
+
∫
0
t
2
(
t
2
-
s
)
e
s
ℒ
Jump
ℒ
Jump
2
f
(
x
)
𝑑
s
}
+
1
2
t
2
ℒ
Conti
.
2
{
f
(
x
)
+
∫
0
t
2
e
s
ℒ
Jump
ℒ
Jump
f
(
x
)
𝑑
s
}
+
∫
0
t
(
t
-
s
)
2
2
e
s
ℒ
Conti
.
(
ℒ
Conti
.
)
3
e
1
2
t
ℒ
Jump
f
(
x
)
𝑑
s
=
f
(
x
)
+
1
2
t
ℒ
Jump
f
(
x
)
+
∫
0
t
2
(
t
2
-
s
)
e
s
ℒ
Jump
ℒ
Jump
2
f
(
x
)
𝑑
s
+
t
ℒ
Conti
.
{
f
(
x
)
+
∫
0
t
2
e
s
ℒ
Jump
ℒ
Jump
f
(
x
)
𝑑
s
}
+
∫
0
t
(
t
-
s
)
e
s
ℒ
Conti
.
(
ℒ
Conti
.
)
2
e
1
2
t
ℒ
Jump
f
(
x
)
𝑑
s
=
f
(
x
)
+
∫
0
t
2
e
s
ℒ
Jump
ℒ
Jump
f
(
x
)
𝑑
s
+
∫
0
t
e
s
ℒ
Conti
.
ℒ
Conti
.
e
1
2
t
ℒ
Jump
f
(
x
)
𝑑
s
.
Thus, an easy calculation gives
e
1
2
t
ℒ
Jump
e
t
ℒ
Conti
.
e
1
2
t
ℒ
Jump
f
(
x
)
=
f
(
x
)
+
t
ℒ
f
(
x
)
+
1
2
t
2
ℒ
2
f
(
x
)
+
R
f
(
t
,
x
)
,
with
R
f
(
t
,
x
)
=
∫
0
t
2
(
t
2
-
s
)
2
2
e
s
ℒ
Jump
ℒ
Jump
3
f
(
x
)
𝑑
s
+
t
ℒ
Conti
.
∫
0
t
2
(
t
2
-
s
)
e
s
ℒ
Jump
ℒ
Jump
2
f
(
x
)
𝑑
s
+
1
2
t
2
ℒ
Conti
.
2
∫
0
t
2
e
s
ℒ
Jump
ℒ
Jump
f
(
x
)
𝑑
s
+
∫
0
t
(
t
-
s
)
2
2
e
s
ℒ
Conti
.
(
ℒ
Conti
.
)
3
e
1
2
t
ℒ
Jump
f
(
x
)
𝑑
s
+
1
2
t
ℒ
Jump
∫
0
t
2
(
t
2
-
s
)
e
s
ℒ
Jump
ℒ
Jump
2
f
(
x
)
𝑑
s
+
1
2
t
2
ℒ
Jump
ℒ
Conti
.
∫
0
t
2
e
s
ℒ
Jump
ℒ
Jump
f
(
x
)
𝑑
s
+
1
2
t
ℒ
Jump
∫
0
t
(
t
-
s
)
e
s
ℒ
Conti
.
(
ℒ
Conti
.
)
2
e
1
2
t
ℒ
Jump
f
(
x
)
𝑑
s
+
1
8
t
2
ℒ
Jump
2
∫
0
t
2
e
s
ℒ
Jump
ℒ
Jump
f
(
x
)
𝑑
s
+
1
8
t
2
ℒ
Jump
2
∫
0
t
e
s
ℒ
Conti
.
ℒ
Conti
.
e
1
2
t
ℒ
Jump
f
(
x
)
𝑑
s
+
∫
0
t
2
(
t
2
-
s
)
2
2
e
s
ℒ
Jump
ℒ
Jump
3
e
t
ℒ
Conti
.
e
1
2
t
ℒ
Jump
f
(
x
)
𝑑
s
.
Since, by [19, Proposition A.4], we have
∥
ℒ
Jump
f
∥
∞
≤
C
∑
k
=
1
2
∥
∇
k
f
∥
∞
and
∥
ℒ
Conti
.
f
∥
∞
≤
C
∑
k
=
1
2
∥
∇
k
f
∥
∞
,
it holds that
(B.3)
sup
x
|
e
1
2
t
ℒ
Jump
e
t
ℒ
Conti
.
e
1
2
t
ℒ
Jump
f
(
x
)
-
{
f
(
x
)
+
t
ℒ
f
(
x
)
+
1
2
t
2
ℒ
2
f
(
x
)
}
|
≤
C
∑
k
=
1
6
∥
∇
k
f
∥
∞
.
Then, by (B.2) and (B.3), we have
∥
P
t
f
-
e
1
2
t
ℒ
Jump
e
t
ℒ
Conti
.
e
1
2
t
ℒ
Jump
f
∥
∞
≤
C
∑
k
=
1
6
∥
∇
k
f
∥
∞
.
From
∥
e
1
2
t
ℒ
Jump
f
∥
∞
≤
∥
f
∥
∞
,
∥
∇
k
e
1
2
t
ℒ
Jump
f
∥
∞
≤
C
∑
ℓ
=
1
k
∥
∇
ℓ
f
∥
∞
,
and (B.1), we have
∥
e
1
2
t
ℒ
Jump
e
t
ℒ
Conti
.
e
1
2
t
ℒ
Jump
f
-
Q
t
/
2
Jump
∘
Q
t
Mall
.
∘
Q
t
/
2
Jump
f
∥
∞
=
∥
e
1
2
t
ℒ
Jump
(
e
t
ℒ
Conti
.
-
Q
t
Mall
.
)
e
1
2
t
ℒ
Jump
f
∥
∞
≤
C
∑
k
=
1
3
∥
∇
k
e
1
2
t
ℒ
Jump
f
∥
∞
t
3
≤
C
∑
k
=
1
3
∥
∇
k
f
∥
∞
t
3
.
Therefore, we obtain that
∥
P
t
f
-
Q
t
/
2
Jump
∘
Q
t
Mall
.
∘
Q
t
/
2
Jump
f
∥
∞
≤
C
∑
j
=
1
6
∥
∇
j
f
∥
∞
t
3
.
The proof is finished.
C Proof of Lemma 3.2
The bound of
P
t
f
-
Q
t
/
2
Jump
∘
Q
t
Mall
.
∘
Q
t
/
2
Jump
f
is decomposed as
∥
P
t
f
-
Q
t
/
2
Jump
∘
Q
t
Mall
.
∘
Q
t
/
2
Jump
f
∥
∞
≤
∥
P
t
f
-
f
∥
∞
+
∥
Q
t
/
2
Jump
∘
Q
t
Mall
.
∘
Q
t
/
2
Jump
f
-
f
∥
∞
.
We immediately have
(C.1)
∥
P
t
f
-
f
∥
∞
≤
C
C
Lip
[
f
]
t
1
/
2
.
The bound of
Q
t
/
2
Jump
∘
Q
t
Mall
.
∘
Q
t
/
2
Jump
f
-
f
is split into
∥
Q
t
/
2
Jump
∘
Q
t
Mall
.
∘
Q
t
/
2
Jump
f
-
f
∥
∞
≤
∥
Q
t
/
2
Jump
∘
Q
t
Mall
.
∘
Q
t
/
2
Jump
f
-
Q
t
Mall
.
∘
Q
t
/
2
Jump
f
∥
∞
+
∥
Q
t
Mall
.
∘
Q
t
/
2
Jump
f
-
Q
t
/
2
Jump
f
∥
∞
+
∥
Q
t
/
2
Jump
f
-
f
∥
∞
.
First, we have
(C.2)
∥
Q
s
Jump
f
-
f
∥
∞
≤
C
Lip
[
f
]
E
[
|
X
s
d
+
1
-
x
|
]
≤
C
Lip
[
f
]
∥
X
s
d
+
1
-
x
∥
2
≤
C
C
Lip
[
f
]
s
1
/
2
by [19, Corollary A.7]. Next we give the bound of
Q
t
Mall
.
∘
Q
t
/
2
Jump
f
-
Q
t
/
2
Jump
f
. We note that,
for
g
∈
C
b
1
(
ℝ
N
)
,
Q
t
Mall
.
g
(
x
)
-
g
(
x
)
can be expressed by
E
[
g
(
X
¯
t
EM
(
x
)
)
M
t
x
(
B
t
)
]
-
g
(
x
)
=
E
[
g
(
X
¯
t
EM
(
x
)
)
M
t
x
(
B
t
)
-
g
(
x
)
M
t
x
(
B
t
)
]
=
E
[
{
g
(
X
¯
t
EM
(
x
)
)
-
g
(
x
)
}
M
t
x
(
B
t
)
]
,
which gives
|
E
[
Q
t
/
2
Jump
f
(
X
¯
t
EM
(
x
)
)
M
t
x
(
B
t
)
]
-
Q
t
/
2
Jump
f
(
x
)
|
=
∥
Q
t
/
2
Jump
f
(
X
¯
t
EM
(
x
)
)
-
Q
t
/
2
Jump
f
(
x
)
∥
2
∥
M
t
x
(
B
t
)
∥
2
.
Since
∥
Q
t
/
2
Jump
f
(
X
¯
t
EM
(
x
)
)
-
Q
t
/
2
Jump
f
(
x
)
∥
2
=
E
[
|
Q
t
/
2
Jump
f
(
X
¯
t
EM
(
x
)
)
-
Q
t
/
2
Jump
f
(
x
)
|
2
]
1
/
2
≤
C
Lip
[
Q
t
/
2
Jump
f
]
t
1
/
2
,
∥
M
t
x
(
B
t
)
∥
2
≤
1
+
C
t
,
with
C
Lip
[
Q
t
Jump
f
]
=
∥
∇
Q
t
Jump
f
∥
∞
≤
C
C
Lip
[
f
]
,
we have
(C.3)
∥
Q
t
Mall
.
∘
Q
t
/
2
Jump
f
-
Q
t
/
2
Jump
f
∥
∞
≤
C
C
Lip
[
f
]
t
1
/
2
.
Finally, we check the bound
Q
t
/
2
Jump
∘
Q
t
Mall
.
∘
Q
t
/
2
Jump
f
-
Q
t
Mall
.
∘
Q
t
/
2
Jump
f
.
By (C.2), we have
∥
Q
t
/
2
Jump
∘
Q
t
Mall
.
∘
Q
t
/
2
Jump
f
-
Q
t
Mall
.
∘
Q
t
/
2
Jump
f
∥
∞
≤
C
C
Lip
[
Q
t
Mall
.
∘
Q
t
/
2
Jump
f
]
t
1
/
2
.
We need the bound
C
Lip
[
Q
t
Mall
.
g
]
=
∥
∇
Q
t
Mall
.
g
∥
∞
for
g
∈
C
b
1
(
ℝ
N
)
.
Hereafter, we use notation of the Malliavin integration by parts in [13, (2.31)]
and the Skorohod integral of Brownian motion as in [22, Definition 3.6], i.e.
𝔹
t
Skor
,
α
=
δ
α
p
(
⋯
δ
α
2
(
B
t
α
1
)
)
for
α
∈
{
0
,
1
,
…
,
d
}
p
, where
δ
k
is the Skorohod integral operator given by
δ
k
(
F
⋅
h
)
=
F
∫
0
t
h
(
s
)
𝑑
B
s
k
-
∫
0
t
D
s
k
F
h
(
s
)
𝑑
s
with k-th Malliavin derivative
D
k
for
k
=
1
,
…
,
d
,
δ
0
(
F
⋅
h
)
=
F
∫
0
t
h
(
s
)
𝑑
s
and
B
t
0
=
t
.
Note that
Q
t
Mall
.
g
has the form
Q
t
Mall
.
g
(
x
)
=
E
[
g
(
X
¯
t
EM
(
x
)
)
]
+
∑
i
∑
α
E
[
∂
i
g
(
X
¯
t
EM
(
x
)
)
L
α
1
V
α
2
i
(
x
)
1
2
𝔹
t
Skor
,
α
]
+
∑
i
1
,
i
2
=
1
N
∑
α
E
[
∂
i
1
∂
i
2
g
(
X
¯
t
EM
(
x
)
)
[
J
¯
t
x
]
ℓ
j
L
α
1
V
α
2
i
1
(
x
)
L
α
1
V
α
2
i
2
(
x
)
1
2
t
2
]
.
Then
∂
∂
x
j
Q
t
Mall
.
g
(
x
)
=
∑
ℓ
=
1
N
E
[
∂
ℓ
g
(
X
¯
t
EM
(
x
)
)
[
J
¯
t
x
]
ℓ
j
]
+
∑
ℓ
=
1
N
∑
i
∑
α
E
[
∂
ℓ
∂
i
g
(
X
¯
t
EM
(
x
)
)
[
J
¯
t
x
]
ℓ
j
L
α
1
V
α
2
i
(
x
)
1
2
𝔹
t
Skor
,
α
]
+
∑
i
∑
α
E
[
∂
i
g
(
X
¯
t
EM
(
x
)
)
∂
j
L
α
1
V
α
2
i
(
x
)
1
2
𝔹
t
Skor
,
α
]
+
∑
ℓ
=
1
N
∑
i
1
,
i
2
=
1
N
∑
α
E
[
∂
ℓ
∂
i
1
∂
i
2
g
(
X
¯
t
EM
(
x
)
)
[
J
¯
t
x
]
ℓ
j
L
α
1
V
α
2
i
1
(
x
)
L
α
1
V
α
2
i
2
(
x
)
1
2
t
2
]
+
∑
i
1
,
i
2
=
1
N
∑
α
E
[
∂
i
1
∂
i
2
g
(
X
¯
t
EM
(
x
)
)
∂
j
(
L
α
1
V
α
2
i
1
(
x
)
L
α
1
V
α
2
i
2
(
x
)
)
1
2
t
2
]
=
∑
ℓ
=
1
N
E
[
∂
ℓ
g
(
X
¯
t
EM
(
x
)
)
[
J
¯
t
x
]
ℓ
j
]
+
∑
ℓ
=
1
N
∑
i
∑
α
E
[
∂
i
f
(
X
¯
t
EM
(
x
)
)
H
(
ℓ
)
(
X
¯
t
EM
(
x
)
,
[
J
¯
t
x
]
ℓ
j
L
α
1
V
α
2
i
(
x
)
1
2
𝔹
t
Skor
,
α
)
]
+
∑
i
∑
α
E
[
∂
i
g
(
X
¯
t
EM
(
x
)
)
∂
j
L
α
1
V
α
2
i
(
x
)
1
2
𝔹
t
Skor
,
α
]
+
∑
ℓ
=
1
N
∑
i
1
,
i
2
=
1
N
∑
α
E
[
∂
i
2
g
(
X
¯
t
EM
(
x
)
)
H
(
ℓ
,
i
1
)
(
X
¯
t
EM
(
x
)
,
[
J
¯
t
x
]
ℓ
j
L
α
1
V
α
2
i
1
(
x
)
L
α
1
V
α
2
i
2
(
x
)
1
2
t
2
)
]
+
∑
i
1
,
i
2
=
1
N
∑
α
E
[
∂
i
1
g
(
X
¯
t
EM
(
x
)
)
H
(
i
1
)
(
X
¯
t
EM
(
x
)
,
∂
j
(
L
α
1
V
α
2
i
1
(
x
)
L
α
1
V
α
2
i
2
(
x
)
)
1
2
t
2
)
]
.
Therefore, we have
C
Lip
[
Q
t
Mall
.
f
]
=
∥
∇
Q
t
Mall
.
f
∥
∞
≤
C
C
Lip
[
f
]
and
∥
Q
t
/
2
Jump
∘
Q
t
Mall
.
∘
Q
t
/
2
Jump
f
-
Q
t
Mall
.
∘
Q
t
/
2
Jump
f
∥
∞
≤
C
C
Lip
[
Q
t
Mall
.
∘
Q
t
/
2
Jump
f
]
t
1
/
2
≤
C
′
C
Lip
[
Q
t
/
2
Jump
f
]
t
1
/
2
(C.4)
≤
C
′′
C
Lip
[
f
]
t
1
/
2
.
By (C.1) and (C.2)–(C.4), we obtain
∥
P
t
f
-
Q
t
/
2
Jump
∘
Q
t
Mall
.
∘
Q
t
/
2
Jump
f
∥
∞
≤
∥
P
t
f
-
f
∥
∞
+
∥
Q
t
/
2
Jump
∘
Q
t
Mall
.
∘
Q
t
/
2
Jump
f
-
f
∥
∞
≤
C
C
Lip
[
f
]
t
1
/
2
.
The proof is finished.
D Proof of Lemma 3.3
We prove
k
=
1
,
2
. Let
{
f
n
}
n
∈
ℕ
be smooth functions satisfying
f
n
→
f
uniformly as
n
→
∞
and
∥
∇
f
n
∥
∞
≤
C
Lip
[
f
]
for all
n
∈
ℕ
. Using the formulas in [3, Theorems 2 and 3] and the proof of [4, Theorem 2.3]
on Bismut’s representation [2], we have
∂
∂
x
E
[
f
n
(
X
T
-
t
x
)
]
=
E
[
(
∇
f
n
)
(
X
T
-
t
x
)
∂
∂
x
X
T
-
t
x
]
,
∥
∇
P
T
-
t
f
∥
∞
≤
C
C
Lip
[
f
]
,
∂
2
∂
x
2
E
[
f
n
(
X
T
-
t
x
)
]
=
1
T
-
t
E
[
(
∇
f
n
)
(
X
T
-
t
x
)
∂
∂
x
X
T
-
t
x
∫
0
T
-
t
(
V
-
1
(
X
s
-
(
x
)
)
∂
∂
x
X
s
-
(
x
)
)
𝑑
B
s
]
-
1
T
-
t
E
[
∫
0
T
-
t
(
∇
P
T
-
t
-
s
f
n
)
(
X
s
-
(
x
)
)
(
∇
V
(
X
s
-
(
x
)
)
∂
∂
x
X
s
-
(
x
)
V
-
1
(
X
s
-
(
x
)
)
∂
∂
x
X
s
-
(
x
)
)
𝑑
s
]
+
1
T
-
t
E
[
∫
0
T
-
t
(
∇
P
T
-
t
-
s
f
n
)
(
X
s
-
(
x
)
)
∂
2
∂
x
2
X
s
-
(
x
)
𝑑
s
]
,
∥
∇
2
P
T
-
t
f
∥
∞
≤
C
C
Lip
[
f
]
∑
ℓ
=
1
2
1
(
T
-
t
)
(
ℓ
-
1
)
/
2
.
Iterating a similar procedure with an integration by parts argument for
k
≥
2
, we obtain the assertion.
The proof is finished.
References
[1]
N. Akiyama and T. Yamada,
A weak approximation method for irregular functionals of hypoelliptic diffusions,
Appl. Numer. Math. 172 (2022), 27–49.
10.1016/j.apnum.2021.09.020Search in Google Scholar
[2]
J.-M. Bismut,
Large Deviations and the Malliavin Calculus,
Progr. Math. 45,
Birkhäuser, Boston, 1984.
Search in Google Scholar
[3]
T. Cass and P. Friz,
The Bismut–Elworthy–Li formula for jump-diffusions and applications to Monte Carlo methods in finance,
preprint (2007), https://arxiv.org/abs/math/0604311.
Search in Google Scholar
[4]
K. D. Elworthy and X.-M. Li,
Formulae for the derivatives of heat semigroups,
J. Funct. Anal. 125 (1994), no. 1, 252–286.
10.1006/jfan.1994.1124Search in Google Scholar
[5]
Y. Iguchi, R. Naito, Y. Okano, A. Takahashi and T. Yamada,
Deep asymptotic expansion: Application to financial mathematics,
Proceedings of IEEE CSDE 2021,
to appear.
10.1109/CSDE53843.2021.9718463Search in Google Scholar
[6]
Y. Iguchi and T. Yamada,
Operator splitting around Euler–Maruyama scheme and high order discretization of heat kernels,
ESAIM Math. Model. Numer. Anal. 55 (2021), S323–S367.
10.1051/m2an/2020043Search in Google Scholar
[7]
J. Jacod and P. Protter,
Asymptotic error distributions for the Euler method for stochastic differential equations,
Ann. Probab. 26 (1998), no. 1, 267–307.
10.1214/aop/1022855419Search in Google Scholar
[8]
P. E. Kloeden and E. Platen,
Numerical Solution of Stochastic Differential Equations,
Appl. Math. (New York) 23,
Springer, Berlin, 1992.
10.1007/978-3-662-12616-5Search in Google Scholar
[9]
S. Kusuoka,
Approximation of expectation of diffusion process and mathematical finance,
Taniguchi Conference on Mathematics Nara ’98,
Adv. Stud. Pure Math. 31,
Mathematical Society of Japan, Tokyo (2001), 147–165.
10.2969/aspm/03110147Search in Google Scholar
[10]
S. Kusuoka,
Approximation of expectation of diffusion processes based on Lie algebra and Malliavin calculus,
Advances in Mathematical Economics. Vol. 6,
Adv. Math. Econ. 6,
Springer, Tokyo (2004), 69–83.
10.1007/978-4-431-68450-3_4Search in Google Scholar
[11]
P. Malliavin and A. Thalmaier,
Stochastic Calculus of Variations in Mathematical Finance,
Springer, Berlin, 2006.
Search in Google Scholar
[12]
R. Naito and T. Yamada,
A third-order weak approximation of multidimensional Itô stochastic differential equations,
Monte Carlo Methods Appl. 25 (2019), no. 2, 97–120.
10.1515/mcma-2019-2036Search in Google Scholar
[13]
D. Nualart,
The Malliavin Calculus and Related Topics,
Springer, Berlin, 2006.
Search in Google Scholar
[14]
Y. Okano and T. Yamada,
A control variate method for weak approximation of SDEs via discretization of numerical error of asymptotic expansion,
Monte Carlo Methods Appl. 25 (2019), no. 3, 239–252.
10.1515/mcma-2019-2044Search in Google Scholar
[15]
E. Platen and N. Bruti-Liberati,
Numerical Solution of Stochastic Differential Equations with Jumps in Finance,
Springer, Berlin, 2010.
10.1007/978-3-642-13694-8Search in Google Scholar
[16]
A. Takahashi and T. Yamada,
An asymptotic expansion with push-down of Malliavin weights,
SIAM J. Financial Math. 3 (2012), no. 1, 95–136.
10.1137/100807624Search in Google Scholar
[17]
A. Takahashi and T. Yamada,
On error estimates for asymptotic expansions with Malliavin weights: Application to stochastic volatility model,
Math. Oper. Res. 40 (2015), no. 3, 513–541.
10.1287/moor.2014.0683Search in Google Scholar
[18]
A. Takahashi and T. Yamada,
A weak approximation with asymptotic expansion and multidimensional Malliavin weights,
Ann. Appl. Probab. 26 (2016), no. 2, 818–856.
10.1214/15-AAP1105Search in Google Scholar
[19]
H. Tanaka and A. Kohatsu-Higa,
An operator approach for Markov chain weak approximations with an application to infinite activity Lévy driven SDEs,
Ann. Appl. Probab. 19 (2009), no. 3, 1026–1062.
10.1214/08-AAP568Search in Google Scholar
[20]
H. Tanaka and T. Yamada,
Strong convergence for Euler–Maruyama and Milstein schemes with asymptotic method,
Int. J. Theor. Appl. Finance 17 (2014), no. 2, Article ID 1450014.
10.1142/S0219024914500149Search in Google Scholar
[21]
T. Yamada,
An arbitrary high order weak approximation of SDE and Malliavin Monte Carlo: Analysis of probability distribution functions,
SIAM J. Numer. Anal. 57 (2019), no. 2, 563–591.
10.1137/17M114412XSearch in Google Scholar
[22]
T. Yamada,
High order weak approximation for irregular functionals of time-inhomogeneous SDEs,
Monte Carlo Methods Appl. 27 (2021), no. 2, 117–136.
10.1515/mcma-2021-2085Search in Google Scholar
