section is devoted to showing that this particular choice of h
ξ
is “good” in the sense that it allows to satisfy 49. We are going to assume throughout this section that Assumptions A.1 and B.1-B.4 hold,
so that we are in the setting of Theorem 5.5, and that h
ξ
is defined as in 51.
5.4 Preliminary bounds and definitions
We start by a stating a few straightforward consequences of Assumption B.2:
Proposition 5.11 For any
α ≤ 1, one has the bound
E exp αV u
1
≤ exp αη
′
V u + αC
L
. Furthermore, for
η 0 and p 0 such that ηp ≤ 1, one has
E exp ηpV u
n
≤ exppηη
′ n
V u + pκC
L
. Finally, setting
κ = η1 − η
′
as before, one has the bound
E exp ηp
n
X
k=0
V u
k
≤ exppκV u + pκC
L
n , provided that
κp ≤ 1. Proof.
The first bound follows immediately from Jensen’s inequality. The second and third inequalities are shown by rewriting the estimate from Assumption B.2 as
E exp ηpV u
n
|F
n −1
≤ exp ηpη
′
V u
n −1
+ ηpC
L
, and iterating it.
Similarly, we obtain a bound on the Jacobian and on the Malliavin derivative A
n
of the solution flow between times n and n + 1:
Proposition 5.12 For any p ∈ [0, ¯p], one has
sup
n ≤st≤n+1
E kJ
s,t
k
p
≤ exp pη
′ n
ηV u + pC
J
+ pκC
L
52
E kA
n
k
p
≤ kGk
p
expp ηη
′ n
V u + pκC
L
+ pC
J
. 53
Furthermore, 52 also holds for J
2 s,t
with C
J
replaced by C
2 J
. Proof.
We only need to show the bound for p = ¯ p
, since lower values follow again from Jensen’s inequality. The bound 52 is an immediate consequence of Assumption B.2 and Proposition 5.11.
The second bound follows by writing
kA
n
h k
p
= Z
n+1 n
J
r,n+1
Gh
r
d r
p
691
≤ kGk
p
Z
n+1 n
kJ
r,n+1
k
2
d r
p 2
Z
n+1 n
|h
r
|
2
d r
p 2
≤ kGk
p
Z
n+1 n
kJ
r,n+1
k
p
d r |||h|||
p n
, and then applying the first bound.
In addition to these first Malliavin derivatives, we will need the control of the derivative of various objects involving the Malliavin derivative. The following lemma gives control over two objects related
to the second Malliavin derivative:
Lemma 5.13 For all p ∈ [0, ¯p2], one has the bounds
sup
s,r ∈[n,n+1]
E kD
i s
J
r,n+1
k
p
≤ exp2pηη
′ n
V u + 2pκC
L
+ pC
J
+ pC
2 J
, sup
s ∈[n,n+1]
E kD
i s
A
n
k
p
≤ |||G|||
p
exp2p ηη
′ n
V u + 2pκC
L
+ pC
J
+ pC
2 J
. Proof. For this, we note that by 35 one has the identities
D
i s
J
r,n+1
ξ =
J
2 s,n+1
J
r,s
ξ, g
i
for r ≤ s,
J
2 r,n+1
J
s,r
g
i
, ξ for s ≤ r,
D
i s
A
n
v = Z
n+1 n
D
i s
J
r,n+1
G v
r
d r . Hence if p
∈ [0, ¯p2] which by the way also ensures that 2pκ 1 it follows from Proposition 5.12 that
E kD
i s
J
r,n+1
k
p
≤ EkJ
2 s,n+1
k
2p
E kJ
r,s
k
2p
1 2
≤ E exp2pηV u
n
+ pC
J
+ pC
2 J
≤ exp2pηη
′ n
V u + 2pκC
L
+ pC
J
+ pC
2 J
for r ≤ s and similarly for s ≤ r. Since, for p ≥ 1, we can write
E kD
i s
A
n
k
p
≤ kGk
p
Z
n+1 n
E kD
i s
J
r,n+1
k
p
d r , the second estimate then follows from the first one.
5.5 Controlling the error term
