Proposition 3.11 If β
⋆
a − 1 then there exists γ γ
⋆
+ 1 such that, for every n 0, there exist exponents N
n
and M
n
such that kJ
n s,t
ϕk ≤ C sup
r ∈[0,t]
1 + ku
r
k
γ N
n
sup
≤uv≤t
1 + kJ
u,v
k
M
n
, uniformly over all n-uples
ϕ with kϕ
k
k ≤ 1 for every k. Proof. We proceed by induction. As a shorthand, we set
E M, N = sup
r ∈[0,t]
1 + ku
r
k
γ N
sup
≤uv≤t
1 + kJ
u,v
k
M
. The result is trivially true for n = 1 with M
1
= 1 and N
1
= 0. For n 1, we combine 26 and 25, and we use Assumption A.1, part 2., to obtain
kJ
n s,t
ϕk ≤ C Z
t
kJ
r,t
k
−a→0
1 + ku
r
k
n
+ X
I
kJ
|I| s
I
,r
ϕ
I
k
n
d r ≤ CE nM
n −1
, nN
n −1
+ 1 Z
t
kK
r,t
k
→a
d r . To go from the first to the second line, we used the induction hypothesis, the fact that K
r,t
= J
∗ r,t
, and the duality between
H
a
and H
−a
. It remains to apply Proposition 3.9 with
β = a to obtain the required bound.
4 Malliavin calculus
In this section, we show that the solution to the SPDE 8 has a Malliavin derivative and we give an expression for it. Actually, since we are dealing with additive noise, we show the stronger result that
the solution is Fréchet differentiable with respect to the driving noise. In this section, we will make the standing assumption that the explosion time
τ from Proposition 3.4 is infinite.
4.1 Malliavin derivative
In light of Proposition 3.4, for fixed initial condition u ∈ H there exists an “Itô map” Φ
u t
:
C [0, t], R
d
→ H with u
t
= Φ
u t
W . We have:
Proposition 4.1 For every t 0 and every u ∈ H , the map Φ
u t
is Fréchet differentiable and its Fréchet derivative DΦ
u t
v in the direction v
∈ C R
+
, R
d
satisfies the equation d DΦ
u t
v = −LDΦ
u t
v d t + DN u
t
DΦ
u t
v d t + Gd vt 27
in the mild sense.
Remark 4.2 Note that 27 has a unique H -valued mild solution for every continuous function
v because it follows from our assumptions that G v ∈ C R
+
, H
γ
for some γ 0 and therefore R
t
e
−Lt−s
G d vs = G vt − e
−Lt
G v0 −
R
t
Le
−Lt−s
G vs ds is a continuous H -valued process.
680
Proof of Proposition 4.1. The proof works in exactly the same way as the arguments presented in Section 3.3: it follows from Remark 4.2 that for any given u
∈ H and t 0, the map W, u 7→ e
−Lt
u +
Z
t
e
−Lt−s
N us ds + Z
t
e
−Lt−s
G dW s is Fréchet differentiable in
C [0, t], R
d
× C [0, t], H . Furthermore, for t sufficiently small de- pending on u and W , it satisfies the assumptions of the implicit functions theorem, so that the claim
follows in this case. The claim for arbitrary values of t follows by iterating the statement.
As a consequence, it follows from Duhamel’s formula and the fact that J
s,t
is the unique solution to 17 that
Corollary 4.3 If v is absolutely continuous and of bounded variation, then
DΦ
u t
v = Z
t
J
s,t
Gd vs , 28
where the integral is to be understood as a Riemann-Stieltjes integral and the Jacobian J
s,t
is as in 17. In particular, 28 holds for every v in the Cameron-Martin space
CM = v :
∂
t
v ∈ L
2
[0, ∞, R
d
, v0 = 0
, which is a Hilbert space endowed with the norm
kvk
2 CM
= R
∞
|∂
t
vt |
2 R
d
d t
def
= |||∂
t
v |||
2
. Obviously, CM is isometric to CM
′
= L
2
[0, ∞, R
d
, so we will in the sequel use the notation D
h
Φ
u t
def
= DΦ
u t
v = Z
t
J
s,t
Gd vs = Z
t
J
s,t
Ghs ds , if
∂
t
v = h . 29
The representation 28 is still valid for arbitrary stochastic processes h such that h ∈ CM
′
almost surely.
Since G : R
d
→ H
γ
∗
+1
is a bounded operator whose norm we denote kGk, we obtain the bound
kD
h
Φ
u t
k ≤ kGk Z
t
kJ
s,t
k |hs| ds ≤ CkJ
·,t
k
L
2
0,t,H
|||h||| , valid for every h
∈ CM
′
. In particular, by Riesz’s representation theorem, this shows that there exists a random element
DΦ
u t
of CM
′
⊗ H such that D
h
Φ
u t
= 〈DΦ
u t
, h 〉
CM
′
= Z
∞
D
s
Φ
u t
hs ds , 30
for every h ∈ CM
′
. This abuse of notation is partially justified by the fact that, at least formally, D
s
Φ
u t
= D
h
Φ
u t
with hr = δs − r. In our particular case, it follows from 28 that one has
D
s
Φ
u t
= J
s,t
G ∈ R
d
⊗ H , t s , 681
and D
s
Φ
u t
= 0 for s t. With this notation, the identity 28 can be rewritten as D
h
u
t
= R
t
D
s
u
t
hsds. It follows from the theory of Malliavin calculus, see for example [Mal97, Nua95] that, for any Hilbert
space H , there exists a closed unbounded linear operator D : L
2
Ω, R ⊗ H → L
2 ad
Ω, F
t
, CM
′
⊗ H such that
DΦ
t
coincides with the object described above whenever Φ
t
is the solution map to 8. Here,
F
t
is the σ-algebra generated by the increments of W up to time t and L
2 ad
denotes the space of L
2
functions adapted to the filtration {F
t
}. The operator
D simply acts as the identity on the factor H , so that we really interpret it as an operator from L
2
Ω, R to L
2
Ω, CM
′
. The operator D is called the “Malliavin derivative.” We define a family of random linear operators
A
t
: CM
′
→ H depending also on the initial condition u
∈ H for 8 by h 7→ 〈DΦ
u t
, h 〉. It follows from 29 that their adjoints A
∗ t
: H → CM
′
are given for ξ ∈ H by
A
∗ t
ξs = G
∗
J
∗ s,t
ξ = G
∗
K
s,t
ξ for s
≤ t , for s
t . 31
Similarly, we define A
s,t
: CM
′
→ H by A
t,s
h
def
= A
t
h1
[t,s]
= 〈Du
t
, h 1
[t,s]
〉 = R
t s
J
r,t
Gh
r
d r . Ob-
serve that A
∗ s,t
: H → CM
′
is given for ξ ∈ H by A
∗ s,t
ξr = G
∗
J
∗ r,t
ξ = G
∗
K
r,t
ξ for r ∈ [s, t] and zero otherwise.
Recall that the Skorokhod integral h 7→
R
t
hs · dW s
def
= D
∗
h is defined as the adjoint of the Malliavin
derivative operator or rather of the part acting on L
2
Ω, F
t
, R and not on
H . In other words, one has the following identity between elements of
H : E
D
h
Φ
u t
= E〈DΦ
u t
, h
〉 = E Φ
u t
Z
t
hs · dW s
, 32
for every h ∈ L
2
Ω, CM
′
belonging to the domain of D
∗
. It is well-established [Nua95, Ch. 1.3] that the Skorokhod integral has the following two important
properties: 1. Every adapted process h with
E |||h|||
2
∞ belongs to the domain of D
∗
and the Skorokhod integral then coincides with the usual Itô integral.
2. For non-adapted processes h, if hs belongs to the domain of D for almost every s and is such
that
E Z
t
Z
t
|D
s
hr |
2 R
d
ds = E
Z
t
|||D
s
h |||
2
ds ∞ ,
then one has the following modification of the Itô isometry:
E Z
t
hs · dW s
2
= E Z
t
|hs|
2 R
d
ds
+ E Z
t
Z
t
tr D
s
hr D
r
hs ds d r . 33 Note here that since hs
∈ R
d
, we interpret D
r
hs as a d × d matrix.
682
4.2 Malliavin derivative of the Jacobian
