#### Infinite-Dimensional Calculus Under Weak Spatial Regularity of the Processes

Infinite-Dimensional Calculus Under Weak Spatial Regularity of the Processes
Franco Flandoli 0 1 2
Francesco Russo 0 1 2
Giovanni Zanco 0 1 2
0 Institute of Science and Technology Austria (IST Austria) , Am Campus 1, 3400 Klosterneuburg , Austria
1 ENSTA ParisTech, Université Paris-Saclay, Unité de Mathématiques appliquées , 828, boulevard des Maréchaux, 91120 Palaiseau , France
2 Dipartimento di Matematica, Università di Pisa , Largo Bruno Pontecorvo 5, 56127 Pisa , Italy
Two generalizations of Itô formula to infinite-dimensional spaces are given. The first one, in Hilbert spaces, extends the classical one by taking advantage of cancellations when they occur in examples and it is applied to the case of a group generator. The second one, based on the previous one and a limit procedure, is an Itô formula in a special class of Banach spaces having a product structure with the noise in a Hilbert component; again the key point is the extension due to a cancellation. This extension to Banach spaces and in particular the specific cancellation are motivated by path-dependent Itô calculus.
Stochastic calculus in Hilbert (Banach) spaces; Itô formula
1 Introduction
Stochastic calculus and in particular Itô formula have been extended since long time
ago from the finite to the infinite-dimensional case. The final result in infinite
dimensions is rather similar to the finite-dimensional one except for some important details.
One of them is the fact that unbounded (often linear) operators usually appear in
infi
B Giovanni Zanco
nite dimensions and Itô formula has to cope with them. If the Itô process X (t ), taking
values in a Hilbert space H , satisfies an identity of the form
d X (t ) = A X (t ) dt + B (t ) dt + C (t ) dW (t ) ,
where A : D ( A) ⊂ H → H is an unbounded linear operator, then in the Itô formula
for a functional F : [0, T ] × H → R we have the term
A X (t ) , Dx F (t, X (t )) ,
which requires X (t ) ∈ D ( A) to be defined. The fact that A X (t ) also appears in (1)
is not equally dramatic: Eq. (1) could be interpreted in weak or mild form,
depending on the case. But the term (2) is less flexible. Sometimes it helps to interpret it as
X (t ) , A∗ Dx F (t, X (t )) or similar reformulations, but this requires strong
assumptions on F . Thus, in general, direct application of Itô formula is restricted to the case
when X (t ) ∈ D ( A). Among the ways to escape this difficulty let us mention the
general trick to replace (1) by a regularized relation of the form d Xn (t ) = A Xn (t ) dt +· · ·
where Xn (t ) ∈ D ( A) (the proof of Theorem 2.1 below is an example) and the
socalled mild Itô formula proved in [6]. Another example of mild Itô formula under the
assumptions that D F ∈ D ( A∗) was the object of [12], Theorem 4.8.
Less common but important for some classes of infinite-dimensional problems,
for example the so-called path-dependent problems, is the phenomenon that ∂∂Ft (t, x )
exists only when x lives in a smaller space than H , for instance D ( A) (we shall clarify
this issue in the examples of Sect. 5). And in Itô formula we have the term
(t, X (t )) ,
so again we need the condition X (t ) ∈ D ( A).
The purpose of this paper is to give a generalization of Itô formula in
infinitedimensional Hilbert and Banach spaces which solves the difficulties described above
when the two terms
A X (t ) , Dx F (t, X (t ))
(t, X (t ))
compensate each other when they sum, although they are not well defined separately.
This happens in a number of cases related to hyperbolic equations and path-dependent
problems. This gives rise to a new Itô formula in which the term
A X (s), Dx F (s, X (s)) ,
G(s, X (s)),
where G (t, x ) is an extension of ∂∂Ft (t, x ) +
Ax , Dx F (t, x ) .
In this introduction, for simplicity of exposition, we have insisted on the formula
in a Hilbert space H, but one of the main purposes of our work is the further extension
to a suitable class of Banach spaces, motivated by precise applications. Since the
notations in the Banach case require more preliminaries, we address to Sect. 3 for this
generalization.
Itô formulae for this kind of problems, both at the abstract level and in applications
to path-dependent functionals, have been investigated by many authors; see [3,4,9–
11,15]; however, the idea to exploit the compensation explained above appears to be
new and could be relevant for several applications.
Related to these problems is also the study of Kolmogorov equations in Banach
spaces; see, for instance, [2,13,14,16].
The paper is organized as follows. In Sect. 2 we give a first generalization of Itô
formula in Hilbert spaces. It serves as a first step to prove a generalization to Banach
spaces, described in Sect. 3. But this also applies directly to examples of hyperbolic
SPDEs, as described in Sect. 4. Finally, in Sects. 5 and 6, we apply the extension
to Banach spaces to several path-dependent problems: in Sect. 5 we consider typical
path-dependent functionals; in Sect. 6 we deal with the important case when F (t, x )
satisfies an infinite-dimensional Kolmogorov equation.
2 An Itô Formula in Hilbert Spaces
Let H, U be two separable Hilbert spaces (which will be always identified with their
dual spaces) and A : D ( A) ⊂ H → H be the infinitesimal generator of a strongly
continuous semigroup et A, t ≥ 0, in H . Let (Ω, F , P) be a complete probability
space, F = (Ft )t≥0 be a complete filtration and (W (t ))t≥0 be a Wiener process in
U with nuclear covariance operator Q; we address to [7] for a detailed description of
these notions of stochastic calculus in Hilbert spaces.
Let B : Ω × [0, T ] → H be a progressively measurable process with
0T |B (s)| ds < ∞ a.s., C : Ω × [0, T ] → L (U, H ) be progressively
measurable with 0T C (s) 2L(U,H) ds < ∞ a.s. and X 0 : Ω → H be a random vector,
measurable w.r.t. F0; here L (U, H ) denotes the space of bounded linear operators
from U to H , with the corresponding norm · L(U,H).
Let X = (X (t ))t∈[0,T ] be the stochastic process in H defined by
X (t ) = et A X 0 +
e(t−s)A B (s) ds +
e(t−s)AC (s) dW (s) ,
formally solution to the equation
d X (t ) = A X (t ) dt + B (t ) dt + C (t ) dW (t ) ,
X (0) = X 0.
We are interested here in examples where X (s) ∈/ D ( A) and also, for a given
function F : [0, T ] × H → R, the derivative ∂∂Fs (s, x ) exists only for a.e. s and for
x ∈ D ( A). In these cases the two terms ∂∂Fs (s, X (s)) and A X (s) , D F (s, X (s))
have no meaning, in general.
However, there are examples where the sum ∂∂Fs (s, X (s))+ A X (s) , D F (s, X (s))
has a meaning even if the two addends do not, separately. This paper is devoted to this
class of examples.
We assume that there exists a Banach space E continuously embedded in H such
(I) D( A) ⊂ E ;
(II) et A is strongly continuous in E ;
(III) X (t ) ∈ E ;
(IV) almost surely the set {X (t )}t∈[0,T ] is relatively compact in E .
The space E can possibly coincide with the whole space H , but in general it is a smaller
space endowed with a finer topology and it is not required to be a inner product space.
In the setting described above, our abstract result is the following one.
Theorem 1 Let F ∈ C ([0, T ] × H ; R) be twice differentiable with respect to its
second variable, with D F ∈ C ([0, T ] × H ; H ) and D2 F ∈ C ([0, T ] × H ; L (H, H )),
and assume the time derivative ∂∂Ft (t, x ) exists for (t, x ) ∈ T × D( A) where
T ⊂ [0, T ] has Lebesgue measure λ (T ) = T and does not depend on x . Assume,
moreover, that there exists a continuous function G : [0, T ] × E → R such that
Ax , D F (s, x )
for all (t, x ) ∈ T × D ( A) .
Let X be the process defined in (3). Then
F (t, X (t )) = F 0, X 0 +
G (s, X (s)) ds
0
t
D F (s, X (s)) , C (s) dW (s) ,
where D F and D2 F denote the first and second Fréchet differentials of F with respect
to its second variable (the same notation will be used everywhere in this article).
Proposition 1 Let β : Ω × [0, T ] → H and θ : Ω × [0, T ] → L(U, H ) be two
progressively measurable processes such that |β(s)| and θ (s) 2L(U,H) are integrable
on [0, T ] a.s.; consider the Itô process Z in H given by
Z (t ) = Z 0 +
t ∂ F
(s, X (s)) ds
0
t
By Section 3.3 in [8]
[X, X ] dz (t ) =
C (s)Q1/2 C (s)Q1/2 ∗ ds,
where [X, X ] dz is the Da Prato–Zabczyk quadratic variation; hence, proposition 6.12
of [8] implies that
Proof According to [10] we have that
F (t, X (t )) = F (0, X (0)) +
D F (s, X (s)) , d− X (s)
t ∂ F
1 t
(s, X (s)) ds + 2 0 D2 F (s, X (s)) d[X, X ](s),
where d− X denotes the integral via regularization introduced in [10]. We remark that
[X, X ] is here the global quadratic variation of the process in (3).
By Theorem 3.6 and Proposition 3.8 of [12] we get
D F (s, X (s)) , d− X (s) =
D F (s, X (s)) , C (s) dW (s)
D F (s, X (s)) , A X (s) + B(s) ds.
D2 F (s, X (s)) d[X, X ](s) =
This concludes the proof.
Proof of Theorem 1 Let {ρε}ε∈(0,1], ρε : R → R, be a family of mollifiers with
supp(ρε) ⊆ [0, 1] for every ε. For x ∈ H set F (t, x ) = F (0, x ) if t ∈ [−1, 0) and
F (t, x ) = F (T , x ) if t ∈ (T , T + 1].
Denote by Jn the Yosida approximations Jn = n (n − A)−1 : H → D ( A), defined
for every n ∈ N, which satisfy limn→∞ Jn x = x for every x ∈ H . One also has
limn→∞ Jn∗x = x , limn→∞ Jn2x = x and limn→∞ Jn2 ∗ x = x for every x ∈ H , used
several times below, along with the fact that the operators Jn and Jn∗ are equibounded.
Tr D2 F (s, X (s)) C (s)Q1/2 C (s)Q1/2 ∗
All these facts are well known and can be found also in [7]. Moreover, it is easy to
show that the family Jn2 converges uniformly on compact sets to the identity (in the
strong operator topology). Since A generates a strongly continuous semigroup in E
as well, all the properties of Jn and Jn2 just listed hold also in E (with respect to its
topology).
Define now Fε,n : [0, T ] × H → R as
Fε,n(t, x ) = (ρε ∗ F (·, Jn x )) (t ).
∂ Fε,n (t, x ) = (ρ˙ε ∗ F (·, Jn x )) (t ),
∂t
D Fε,n (t, x ) , h = (ρε ∗ D F (·, Jn x ) , Jn h ) (t )
ρε ∗ D2 F (t, Jn x ) ( Jn h, Jnk) (t ).
1
= ali→m0 a R ρε(r ) [F (t + a − r, Jn x ) − F (t − r, Jn x )] dr
ρε(r ) [F (t + a − r, Jn x ) − F (t − r, Jn x )] dr
1
= ali→m0 a Rt ρε(r ) [F (t + a − r, Jn x ) − F (t − r, Jn x )] dr
ε
∂ F
= ρε(r ) ∂t (t − r, Jn x ) dr
Now set Xn (t ) = Jn X (t ), X 0
n = Jn X 0, Bn (t ) = Jn B (t ), Cn (t ) = JnC (t ). Since
Jn commutes with et A, we have
e(t−s)A Bn (s) ds +
e(t−s)ACn (s) dW (s) .
Moreover, Xn (t ), Bn(t ), Cn(t ) belong to D ( A) for a.e. t ∈ [0, T ], with | A Xn (·)|
integrable P-a.s.; hence,
Xn(t ) = Xn0 +
[ A Xn(s) + Bn(s)] ds +
Cn(s) dW (s)
and by the Itô formula in Hilbert spaces given in Proposition 1 we have
Fε,n (t, Xn (t )) = Fε,n 0, Xn0
0
t
A Xn (s) , D Fε,n (s, Xn (s)) + ∂ Fε,n (s, Xn (s)) ds
T r Cn (s) QCn (s)∗ D2 Fε,n (s, Xn (s)) ds.
Let us prove the convergence (as n → ∞ and ε → 0) of each term to the
corresponding one of the formula stated by the theorem. We fix t and prove the a.s. (hence in
probability) convergence of each term, except for the convergence in probability of
the Itô term; this yields the conclusion.
Given (ω, t ), we have Fε,n (t, Xn (ω, t )) = ρε ∗ F ·, Jn2 X (ω, t ) (t ) and thus
Fε,n (t, Xn(ω, t )) − F (t, X (ω, t ))
ρε(r )F t − r, Jn2 X (ω, t ) dr − F (t, X (ω, t ))
ρε(r ) F t − r, Jn2 X (ω, t ) − F (t, X (ω, t )) dr,
which is arbitrarily small for ε small enough and n big enough, because Jn2 converges
strongly to the identity and F is continuous; similarly
From now on we work in the set Ω1 where X has relatively compact paths in E (hence
in H ). Fix δ > 0. Since for ω ∈ Ω1 the set {X (ω, s)}s∈[0,t] is relatively compact, we
it admits a finite subcover Bδ/2 (X (si )) i=1,...,M for some finite set {s1, . . . , sM } ⊂
[0, t ]; therefore, for any s there exists i ∈ {1, . . . , N } such that |X (s) − X (si )| < δ/2
and
for n > N where N does not depend on s since the convergence is uniform. This
shows that the set Jn2 X (s) n,s is totally bounded both in E and in H .
Therefore, we can study the convergence of the other terms as follows. First we
consider the difference
The second term in this last sum is bounded by
B(s), D F (s, X (s)) ds
Jn2 B(s), ρε ∗ D F ·, Jn2 X (s)
Jn2 B(s), D F (s, X (s)) ds
Jn2 B(s) − B(s), D F (s, X (s)) ds .
Jn2 B(s) − B(s) |D F (s, X (s))| ds
and {X (s)}s is compact, hence |D F (s, X (s))| is bounded uniformly in s and, since
the Jn2 are equibounded and converge strongly to the identity and B is integrable,
Lebesgue dominated convergence theorem applies. The first term in the previous sum
instead is bounded by
ρε(r ) D F s − r, Jn2 X (s) − D F (s, X (s)) dr ds;
by the discussion above the set [0, t ] × Jn2 X (s) n,s ∪ {X (s)}s is contained in a
compact subset of [0, T ] × H , hence |D F | is bounded on that set uniformly in s and
r . Thanks again to the equicontinuity of the operators Jn2 and the integrability of B,
(7) is shown to go to 0 by the dominated convergence theorem and the continuity of
D F .
About the critical term involving G we have
ds −
G (s, X (s)) ds
·, Jn2 X (s) + A Jn2 X (s), D F ·, Jn2 X (s)
Jn∗ 2 ρε ∗ D F ·, Jn2 X (s) (s) − D F (s, X (s))
Jn∗ 2 ρε ∗ D F ·, Jn2 X (s) (s) − Jn∗ 2 D F (s, X (s))
(s) − G (s, X (s)) ds
ρε(r ) G s − r, Jn2 X (s) − G (s, X (s)) dr ds
C ∗(s) Jn2 ∗
(s) − C ∗(s)D F (s, X (s))
Jn∗ 2 ρε ∗ D f ·, Jn2 X (s) (s) − D F (s, X (s))
[0,t]∩T
[0,t]∩T
it is immediate to see that the right-hand side of (8) converges to 0 almost surely,
hence
D Fε,n (s, Xn(s)) , Cn(s) dW (s) →
D F (s, X (s)) , C (s) dW (s)
in probability.
It remains to treat the trace term. Let h j be an orthonormal complete system in
H ; then
Tr Cn(s)QCn(s)∗ D2 Fε,n (s, Xn(s)) ds
Tr C (s)QC (s)∗ D F (s, X (s)) ds
JnC (s)QC (s)∗ Jn∗ 2 ρε ∗ D2 F ·, Jn2 X (s) (s) Jn
−C (s)QC (s)∗ D2 F (s, X (s)) h j , h j
Now for any j
JnC (s)QC (s)∗ Jn∗ 2 ρε ∗ D2 F ·, Jn2 X (s) (s) Jn h j
− C (s)QC (s)∗ D2 F (s, X (s)) h j
≤ JnC (s)QC (s)∗ Jn∗ 2 ρε ∗ D2 F ·, Jn2 X (s) (s) Jn h j
− C (s)QC (s)∗ D2 F (s, X (s)) Jn h j
The second term in the sum converges to 0 thanks to the properties of Jn; the first one
is bounded by the sum
JnC (s)QC (s)∗ Jn∗ 2 ρε ∗ D2 F ·, Jn2 X (s) (s) Jn h j
− JnC (s)QC (s)∗ J ∗ 2 D2 F (s, X (s)) Jn h j
n
whose first addend is less or equal to
ρε D2 F s − r, Jn2 X (s) − D2 F (s, X (s))
which is shown to go to zero as before. For the second addend of (10) notice that for
any k ∈ H
JnC (s)QC (s)∗ Jn∗ 2 − C (s)QC (s)∗ k
JnC (s)QC (s)∗ Jn∗ 2 − JnC (s)QC (s)∗ k
JnC (s)QC (s)∗ − C (s)QC (s)∗ k
≤ JnC (s)QC (s)∗
+ JnC (s)QC (s)∗k − C (s)QC (s)∗k ,
which tends to 0 as n tends to ∞.
The same compactness arguments used in the previous steps, the continuity of
D2 F and the equiboundedness of the family { Jn} allow to apply Lebesgue dominated
convergence theorem both to the series and to the integral with respect to s in (9). This
concludes the proof.
3 Extension to Particular Banach Spaces
In this section we consider the following framework. Let H1 be a separable Hilbert
space with scalar product · 1 and norm · 1 and let E2 be a Banach space, with norm
· E2 and duality pairing denoted by ·, · , densely and continuously embedded in
another separable Hilbert space H2 with scalar product and norm denoted, respectively,
by · 2 and · 2. Then set H := H1 × H2 so that
E := H1 × E2 ⊂ H
with continuous and dense embedding. We adopt here the standard identification of
H with H ∗ so that
E ⊂ H ∼= H ∗ ⊂ E ∗.
Our aim here is to extend the results exposed so far to situations in which the process
X lives in a subset of E but the noise only acts on H1.
Example 1 In the application of this abstract framework to path-dependent functionals
(see Sect. 5), we will choose the spaces
H = Rd × L2
−T , 0; Rd
Similarly to the setup we introduced in Sect. 2, consider a complete probability space
(Ω, F , P) with a complete filtration F = (Ft )t≥0 and a Wiener process (W (t ))t≥0
in another separable Hilbert space U with nuclear covariance operator Q.
Consider a linear operator A on H with domain D( A) ⊂ E and assume that it
generates a strongly continuous semigroup et A in H . Let B : Ω × [0, T ] → E
be a progressively measurable process s.t. 0t |B(t )| dt < ∞ as in Sect. 2; let then
C : Ω ×[0, T ] → L(U, H1) be another progressively measurable process that satisfies
0T C (t ) 2L(U,H1) dt < ∞ and define C : Ω × [0, T ] → L(U, E ) as
C (t )u =
, u ∈ U ;
let X 0 be a F0-measurable random vector with values in H and set
X (t ) = et A X 0 +
e(t−s)A B(s) ds +
e(t−s)AC (s) dW (s).
E = D ( A)E ,
D = A−1(E ).
Notice that D ⊂ D( A) ⊂ E . In most examples the set D is not dense in E .
As in Sect. 2 we assume here that et A is strongly continuous in E (and this in turn
implies that D is dense in E ), X (t ) actually belongs to E and that almost surely the
set {X (t )}t∈[0,T ] is relatively compact in E .
Example 2 In the path-dependent case, we will have
D ( A) =
−T , 0; Rd , x1 = lim x2(s) ,
s→0−
A =
E =
D =
Finally consider a sequence Jn of linear continuous operators, Jn : H → E with
the properties:
(i) Jn x ∈ D for every x ∈ E ;
(ii) Jn x → x in the topology of E for every x ∈ E ;
(iii) Jn commutes with A on D( A).
By Banach–Steinhaus and Ascoli–Arzelà theorems it follows that the operators Jn
are equibounded and converge to the identity uniformly on compact sets of E .
Theorem 2 Assume there exists a sequence Jn as above and let F ∈ C ([0, T ]× E ; R)
be twice differentiable with respect to its second variable with D F ∈ C ([0, T ]× E ; E ∗)
and D2 F ∈ C ([0, T ] × E ; L (E ; E ∗)). Assume the time derivative ∂∂Ft (t, x ) exists
for (t, x ) ∈ T × D where T ⊂ [0, T ] has Lebesgue measure λ (T ) = T and does
not depend on x . If there exists a continuous function G : [0, T ] × E → R such that
Ax , D F (t, x )
∀ x ∈ D, ∀ t ∈ T ,
then, in probability,
F (t, X (t )) = F 0, X 0 +
G (s, X (s)) ds
0
t
D F (s, X (s)) , C(s) dW (s) ,
where Tr H1 is defined for T ∈ L (E , E ) as
Tr H1 T =
h j being an orthonormal complete system in H1.
Proof Set Fn : [0, T ] × H → R, Fn(t, x ) := F t, Jn x . Thanks to the assumptions
on F we have that Fn is twice differentiable with respect to the variable x and
D Fn(t, x ) = Jn∗ D F t, Jn x ∈ L (H ; R) ∼= H
Furthermore for any t ∈ T the derivative of Fn with respect to t is defined for all
x ∈ H and equals
Set Gn : [0, T ] × E → R, Gn (t, x ) := G t, Jn x . Gn is obviously continuous; we
check now that for any t ∈ T Gn(t, ·) extends ∂∂Ftn (t, ·) + A·, D Fn (t, ·) from D ( A)
to E . Since Jn maps E into D ⊂ D ( A) ⊂ H we have
= ∂∂Ftn (t, x ) +
Jn Ax , D F t, Jn x
Ax , D Fn (t, x ) .
Notice that here only the term Ax , D Fn (t, x ) has to be extended (since it is not well
defined outside D ( A)) while the time derivative of Fn makes sense on the whole space
H by definition.
We can now apply Theorem 1 to Fn and Gn, obtaining that for each n
Fn (t, X (t )) = Fn 0, X 0 +
Gn (s, X (s)) ds
0
t
D Fn (s, X (s)) , C(s) dW (s) .
Here C (s)QC (s)∗ maps E ∗ into E , therefore C (s)QC (s)∗ D2 Fn (s, X (s)) maps H
into E ⊂ H and the trace term can be interpreted as in H . Also, since C (s) belongs
to L (U ; H1 × {0}), we have that the stochastic integral above is well defined as a
stochastic integral in a Hilbert space.
Substituting the definition of Fn and identities (13), (14) in the previous equation
we get
F t, Jn X (t) = F 0, Jn X 0 +
G s, Jn X (s) ds
0
t
D F s, Jn X (s) , JnC(s) dW (s) .
Now we fix (ω, t ) and study the convergence of each of the terms above. Since
X (ω, t ) ∈ E , Jn X (ω, t ) → X (ω, t ) almost surely as n → ∞ and therefore by
continuity of F we have that F t, Jn X (ω, t ) converges to F (t, X (ω, t )) almost surely.
For the same reasons F 0, Jn X 0(ω) converges to F 0, X 0(ω) almost surely.
Denote by Ω1 the set of full probability where each of the trajectories {X (ω, t )}t
is relatively compact. Arguing as in Proof of Theorem 1 it can be shown that, thanks
to the uniform convergence on compact sets of the Jn, the set Jn X (ω, t ) n,t is
totally bounded in E for any ω ∈ Ω1. Therefore, the a.s. convergence of the terms
0t G s, Jn X (ω, s) ds and 0t Jn B(ω, s), D F s, Jn X (ω, s) ds follows from
the dominated convergence theorem since G and D F are continuous, B is integrable,
and the family Jn is equibounded.
To show the convergence of the stochastic integral term consider
Jn∗ D F s, Jn X (s) − D F (s, X (s)) E∗
e, Jn∗ D F s, Jn X (s) − D F (s, X (s))
Jne, D F s, Jn X (s)
− e, D F (s, X (s))
Jne, D F s, Jn X (s)
− Jne, D F (s, X (s))
Jne − e, D F (s, X (s))
Jne − e E
D F (s, X (s)) E∗
D F s, Jn X (s) − D F (s, X (s)) E∗
and this last quantity converges to zero as before, since Jn is equibounded, D F
is continuous (hence uniformly continuous on Jn X (s) n,s ∪ {X (s)}s ), and Jn
converges to the identity on E . Since C (s) 2 is integrable, we can apply again the
dominated convergence theorem in (16) to get that the left-hand side converges to 0
almost surely, hence
D F s, Jn X (s) , JnC (s) dW (s) →
D F (s, X (s)) , C (s) dW (s)
E C (s)u, f E∗ = E
hence C (s)∗ f = C (s)∗ f1 for any f ∈ E ∗.
Now let H1 and H2 be complete orthonormal systems of H1 and H2, respectively,
and set H1 := H1 × {0}, H2 := H2 × {0}, so that H := H1 ∪ H2 is a complete
orthonormal system for H . H is countable since H1 and H2 are separable. For h ∈ H
we have that
y := Jn∗ D2 F s, Jn X (s) Jn h ∈ H ⊂ E ∗ = H1 × E ∗,
2
so that, writing y = (y1, y2), we have
C (s)QC (s)∗ y = C (s)QC (s)∗ y1 =
Tr C (s)QC (s)∗Jn∗ D2 F s, Jn X (s) Jn
h∈H
C (s)QC (s)∗Jn∗ D2 F s, Jn X (s) Jn h, h
h∈H1
C (s)QC (s)∗Jn∗ D2 F s, Jn X (s) Jn h, h
Now, setting K := supn
Jn we have that for h ∈ H1
h∈H1
h∈H1
h∈H1
h∈H1
≤ K
D2 f t, Jn X (s) Jn h, JnC(s)QC(s)∗h
D2 F (t, X (s)) h, C(s)QC(s)∗h
JnC(s)QC(s)∗h − C(s)QC(s)∗h
D2 F t, Jn X (s) − D2 F (t, X (s))
h∈H1
h∈H1
Jn h − h ;
therefore thanks to the equiboundedness of Jn and the uniform continuity of D2 F
on the set Jn X (s) n,s ∪ {X (s)}s we can apply the dominated convergence theorem
to the sum over h ∈ H1 to obtain that
Tr H1 C (s)QC (s)∗Jn∗ D2 F s, Jn X (s) Jn
n−→→∞ Tr H1 C (s)QC (s)∗ D2 F (s, X (s)) .
Since D2 F is bounded also in s ∈ [0, T ] and C (s) 2L(U ;E) is integrable by
assumption, a second application of the dominated convergence theorem yields that for every
t ∈ [0, T ]
Tr H1 C (s)QC (s)∗ D2 F (s, X (s)) ds,
thus concluding the proof.
Remark 1 The use of both spaces E and E in the statement of the theorem can seem
unjustified at first sight: since the process X is supposed to live in E and the result is
a Itô formula valid on E (because the extension G is defined on E only), everything
could apparently be formulated in E . However, in most examples the space E is not
a product space (see Sect. 5) hence neither is its dual space, and the product structure
of the dual is needed to show that the second order term is concentrated only on the
H1-component. Since asking F to be defined on [0, T ] × H will leave out many
interesting examples (we typically want to endow E with a topology stronger that the
one of H ), the choice to use the intermediate space E seems to be the more adequate.
Corollary 1 Consider n + 1 points 0 = t0 ≤ t1 ≤ · · · ≤ tn = T and assume that
F ∈ C [t j , t j+1) × E ; R for j = 0, 1, . . . , n − 1. Suppose, moreover, that
3. D2 F ∈ C t j , t j+1 × E ; L (E , E ∗) for j = 0, 1, . . . , n − 1;
Ax , D F (t, x )
∀x ∈ D, ∀t ∈ T ∩ t j , t j+1 .
Then the formula
F (T , X (T )) = F 0, X 0 +
− F t j −, X t j
j=1
G (s, X (s)) ds +
D F (s, X (s)) , C(s) dW (s)
Proof Thanks to the assumptions, Theorem 2 can be applied to obtain n identities for
the increments F t j+1 − ε, X t j+1 − ε − F t j , X t j , j = 0, . . . , n − 1, with
0 < ε < min j t j+1 − t j . Summing up these identities and taking the limit as ε goes
to 0 yield the result.
4 Application to Generators of Groups
In a Hilbert space H , given a Wiener process (W (t ))t≥0 with covariance Q, defined on
a filtered probability space Ω, F , (Ft )t≥0 , P , given x 0 ∈ H , B : Ω × [0, T ] → H
progressively measurable and integrable in t , P-a.s., C : Ω × [0, T ] → L (H, H )
progressively measurable and square integrable in t , P-a.s., let X (t ) be the stochastic
process given by the mild formula
(D H0) s, e−(t−s)A x , e−(t−s)A Ax ds.
D F (t, x ) , h = (D F0) e−t A x , e−t Ah
D H0 s, e−(t−s)A x , e−(t−s)Ah ds.
X (t ) = et A x 0 +
e(t−s)A B (s) ds +
e(t−s)AC (s) dW (s) ,
where et A is a strongly continuous group. In this particular case we can also write
X (t ) = et A
e−s A B (s) ds +
e−s AC (s) dW (s) ,
from which we may deduce, for instance, that X is a continuous process in H . Formally
d X (t ) = A X (t ) dt + B (t ) dt + C (t ) dW (t ) ,
but A X (t ) is generally not well defined: typically the solution has the same spatial
regularity of the initial condition and the forcing terms. Thus in general, one cannot
apply the classical Itô formula to F (t, X (t )), due to this fact. A possibility is given
by the mild Itô formula [6]. We show here an alternative, which applies when suitable
cancellations in F (t, x ) occur.
As a first example, let F (t, x ) be given by
F (t, x ) = F0 e−t A x +
H0 s, e−(t−s)A x ds,
where F0 ∈ C 2 (H ; R), H0 ∈ C ([0, T ] × H ; R), with continuous derivatives D H0,
D2 H0. Then ∂∂Ft (t, x ) exists for all x ∈ D ( A), t ∈ [0, T ] and it is given by
Ax , D F (t, x ) = H0 (t, x ) .
Consider the function G (t, x ) := ∂∂Ft (t, x ) +
defined only on x ∈ D ( A). However, being
Ax , D F (t, x ) . It is a priori well
G (t, x ) = H0 (t, x ) ,
the function G extends to a continuous function on [0, T ] × H . Then Theorem 1
applies and Itô formula reads
F (t, X (t )) = F 0, x 0 +
H0 (s, X (s)) ds +
B (s) , D F (s, X (s)) ds
The previous example concerns a very particular class of functionals F . As a more
useful (but very related) example, assume we have a solution F (t, x ) of the following
Kolmogorov equation
Ax + B (t, x ) , D F (t, x )
1
+ 2 T r C (t, x ) QC ∗ (t, x ) D2 F (t, X (t )) = 0,
F ∈ C ([0, T ] × H ; R) ,
D2 F ∈ C ([0, T ] × H ; L (H, H )) ,
Here we assume that B : [0, T ] × H → H and C : [0, T ] × H → L (H, H ) are
continuous (we assume continuity of B and ∂∂Ft for simplicity of exposition, but this
detail can be generalized). Since
F (t, X (t )) = F t0, x 0 +
D F (s, X (s)) , C (s, X (s)) dW (s) .
then it has a continuous extension on [0, T ] × H and Theorem 1 is applicable, if
(X (t ))t∈[t0,T ] (for some t0 ∈ [0, T )) is a continuous process in H satisfying
X (t ) = e(t−t0)A x 0 +
e(t−s)A B (s, X (s)) ds +
e(t−s)AC (s, X (s)) dW (s) ,
thus we get
We can now easily prove the following uniqueness result. We do not repeat the
assumptions on H , W , et A, B.
Theorem 3 Assume that for every t0, x 0 ∈ [0, T ] × H , there exists at least one
continuous process X in H satisfying Eq. (20). Then the following holds.
∞ < ∞.
5 Application to Path-Dependent Functionals
We will now apply the abstract results of Sect. 3 to obtain an Itô formula for
pathdependent functionals of continuous processes, stated in Theorem 4. In the first
sections we will introduce the necessary spaces and operators and we will show that
the infinite-dimensional reformulation of path-dependent problems appears naturally
when dealing with path-dependent SDEs (see again [13] for a more detailed
discussion).
In this and the following sections we will denote by Ct the space of Rd -valued functions
on [0, t ] that can have a jump only at t , that is
Ct =
ϕ : [0, t ] → Rd : ϕ ∈ C [0, t ); Rd , ∃ lim ϕ(s) ∈ Rd
s→t−
y(t ) = y0 +
c(s) dW (s),
where b and c are progressively measurable processes, with values in Rd and Rk×d ,
respectively, such that
|b(s)| ds < ∞,
c(s) 2 ds < ∞
endowed with the supremum norm. The space Ct is clearly isomorphic to the product
space
Let y(t ) be the continuous process in Rd given by
and y0 is a F0-measurable random vector.
Let us introduce the following infinite-dimensional reformulation. We will work in
the space
whose elements we shall usually denote by x =
endowed with the norm xx21 2 = |x1|2 + x2 2∞; the notation ·, · will denote
the duality pairing between E and its dual space E ∗. The space E is densely and
continuously embedded in the product space
xx21 . E is a Banach space when
H = Rd × L2
−T , 0; Rd .
D ( A) =
−T , 0; Rd , x1 = lim x2(s) ,
s→0−
A =
where we identify an element in W 1,2 with the restriction of its continuous version to
[−T , 0). Therefore, we identify also the space
E = D ( A)E =
The operator A generates a strongly continuous semigroup et A in H . This semigroup
turns out to be not strongly continuous in E ; nevertheless et A maps E in itself and is
strongly continuous in E . This follows from the fact that the semigroup et A has the
explicit form
et A x =
for every γ ∈ Ct .
Using (28), it is easy to show (see also Proposition 3) that
γ (0)1[−T,−t) + γ (t + ·)1[−t,0)
as a H -valued process, is given by
X (t ) = Lt yt ,
X (t ) = et A X 0 +
e(t−s)A B(s) ds +
e(t−s)AC (s) dW (s),
and the processes B : [0, T ] →
B(t ) =
C (s)u =
for u ∈ Rk .
The validity of (30) corresponds to saying that X is the unique mild solution to the
linear equation
d X (t ) = A X (t ) dt + B(t ) dt + C (t ) dW (t );
hence, we see that our infinite-dimensional reformulation forces us to deal with
equations even if we start from finite-dimensional processes: the operator A appears as a
consequence of the introduction of the second component that represents the “past
trajectory” of the process (see Remark 3).
Proposition 2 The process X is such that X (t ) ∈ E for every t and the trajectories
t → X (t ) are almost surely continuous as maps from [0, T ] to E .
Proof The random variable X 0 takes values in E by definition. Since the process y has
almost surely continuous trajectories, Lt yt 2 ∈ C [−T , 0); Rd and Lt yt belongs to
E . To check the almost sure continuity of the trajectories of X as a E -valued process
denote again by Ω0 ⊂ Ω a null set such that t → y(ω, t ) is continuous for every
ω ∈ Ω \ Ω0, fix ω ∈ Ω \ Ω0, fix t, s ∈ [0, T ] and ε > 0; we can suppose t > s
without loss of generality. Since y (ω, ·) is uniformly continuous on [0, T ] we can find
δ such that |y(t ) − y(s)| < ε/2 if t − s < δ. Then for t − s < δ
X (t ) − X (s) E ≤ |y(t ) − y(s)|
|y(0) − y(r )| , sup |y(t − s + r ) − y(r )|
r∈[0,s]
Among the examples of path-dependent functional, let us mention the integral ones
f (t, γ ) = q (γ (t ), γ (t0)) 1t>t0 .
and those involving pointwise evaluations, like for instance
Here we assume that g : Rd × Rd → R is a measurable function in the first example,
with |g (a)| ≤ C 1 + |a|2 and that q : Rd × Rd → R is a measurable function in the
second example and t0 ∈ [0, T ] is a given point. In order to apply Itô calculus let us
simplify and assume that g and q are twice continuously differentiable with bounded
derivatives of all orders.
Given a path-dependent functional f (t, ·), t ∈ [0, T ] we may associate to it a map
F : [0, T ] × E → R setting
F (t, x ) = f (t, Mt x )
is defined as
Mt x (s) = x2 (s − t ) 1[0,t)(s) + x11{t}(s),
s ∈ [0, t ] .
Remark 2 Notice that Mt Lt is the identity on Ct .
Here we see that if f were defined only on C [0, t ]; Rd , then F would be defined on
[0, T ] × E , because of the definition of the operators Mt ; our abstract results require
instead F to be defined on [0, T ] × E ; see Remark 1.
Path-dependent functionals are often studied in spaces of càdlàg paths. The
framework presented here can be easily modified to do so, similarly to what is done in
[13]; however, this would require the introduction of further spaces, thus complicating
notations, but would not lead to generalizations of the result proved here.
The aim of this section is to show that Examples (33) and (34) fulfill the assumptions
of Theorem 2.
The abstract reformulation of the functional given in (33) is the map F : [0, T ] ×
E → R defined as
F (t, x ) =
g (Mt x (t ), Mt x (s)) ds =
g (x1, x2 (s − t )) ds
−t
g (x1, x2 (r )) dr.
−t
(t, x ) = g (x1, x1) −
Dg (x1, x2 (s − t )) · x2 (s − t ) ds
= g (x1, x1) −
g (x1, x2 (r )) · x2 (r ) dr,
D = A−1(E ) = A−1 ({0} × E2) ,
E2 =
D =
xx21 ∈ D( A) : x2 ∈ C 1 [−T , 0); Rd ! .
Moreover, the time derivative of F is defined for every t ∈ [0, T ]. Therefore, we see
that
is a natural assumption, while ∂∂Ft : [0, T ] × E → R would not be. Since g is
continuous we also have that ∂t F belongs to C [0, T ] × D; R .
Let us then investigate the function
For h ∈ E we have
Ax , D F (t, x )
x ∈ D, t ∈ [0, T ] .
−t
Dg (x1, x2 (s − t )) · h2 (s − t ) ds
Dg (x1, x2 (r )) · h2 (r ) dr.
G (s, x ) = g (x1, x1) ;
Hence, writing ∂1q and ∂2q for the derivatives of q with respect to its first and second
variable, respectively, for t = t0,
· ( Ax )2 (t0 − t )
= 0,
because ( Ax )1 = 0 and ( Ax )2 (t0 − t ) = x2 (t0 − t ). Again, G extends continuously
to E .
In Sect. 5.1 we have formulated a classical Itô process as an infinite-dimensional
process given by a mild formula; this apparently not natural formulation is suggested
by the case when the process is the solution of a path-dependent SDE. For these
equations, the mild formulation is natural, due to the similarity with delay equations,
where the infinite-dimensional approach is classical. Let us give here some details
about the case of a path-dependent SDE.
Let (Ω, F , P) be a complete probability space, F = (Ft )t≥0 a complete filtration,
(W (t ))t≥0 a Brownian motion in Rk (we shall write W i (t ) for its coordinates), y0 an
F0-measurable random vector of Rd . Consider the path-dependent SDE in Rd
d y (t ) = b (t, yt ) dt + σ (t, yt ) dW (t ) ,
y (0) = y0.
yt := {y (s)}s∈[0,t] .
About b and σ , initially we assume that, for each t ∈ [0, T ], the function b (t, ·)
maps C [0, t ]; Rd into Rd and the function σ (t, ·) maps C [0, t ]; Rd into Rk×d ;
moreover, we assume that b ad σ are locally bounded measurable functions, for each
t ∈ [0, T ], with bounds uniform in t and that the processes b (t, yt ) and σ (t, yt ) are
progressively measurable. These are relatively weak requirements to give a meaning
to the integral equation
y (t ) = y0 +
B (t, x ) =
Ci (t, x ) =
Finally, we set U = Rk , take Q equal to the identity in U and consider, for every
(t, x ) ∈ [0, T ]× E , the bounded linear operator C (t, x ) : U → E having components
Ci (t, x ).
Given a F0-measurable random variable X 0 with values in E we may now formulate
the path-dependent SDE in the Banach space E , i.e.,
X (0) = X 0.
The natural concept of solution here would be that of mild solution, but since under our
assumptions the stochastic convolution is a priori well defined only in H , we consider
is a solution to Eq. (41). We also have
Proof By (28) the first component of Eq. (41) reads
Lt yt 1 = y (t ) = X1 (t ) = X10 +
B (s, X (s))1 ds +
C (s, X (s))1 dW (s)
X (t ) = Lt yt
yt = Mt X (t ).
b (s, Ms X (s)) ds +
Eq. (40) in its mild form in the space H , that is
X (t ) = et A X 0 +
e(t−s)A B (s, X (s)) ds +
e(t−s)AC (s, X (s)) dW (s) . (41)
Proposition 3 Given an F0-measurable random vector y0 of Rd , set X 0 =
y0, y01[−T,0) T . Then, if {y (t )}t∈[0,T ] is a solution to Eq. (37), the process
= y0 +
which holds true because it is Eq. (37). About the second component, we have
b (s, Ms X (s)) 1[−t+s,0] (r ) ds
σ (s, Ms X (s)) 1[−t+s,0] (r ) dW (s)
b (s, ys ) 1[−t+s,0] (r ) ds +
σ (s, ys ) 1[−t+s,0] (r ) dW (s) .
y (t + r ) = yt (t + r ) = y0 +
because 1[−t+s,0] (r ) = 0 for s ∈ [t + r, t ]. This is again a copy of Eq. (37). The proof
is complete.
Remark 3 We have seen that, at the level of the mild formulation, the equation in
Hilbert space is just given by two copies of the original SDE. On the contrary, at the
level of the differential formulation, we formally have
d X1 (t ) = B (t, Mt X (t )) dt + Ci (t, Mt X (t )) dW (t )
d
d X2 (t ) = dr X2 (t ) dt.
The first equation, again, is a rewriting of the path-dependent SDE. But the second
equation is just a consistency equation, necessary since we need to introduce the
component X2 (t ). Here we see one of the technical problems which motivate this
paper: X2 (t ) = Lt yt 2 is “never” differentiable (being a trajectory of solution of the
SDE, it has the level of regularity of Brownian motion). In other words, X2 (t ) “never”
belongs to D ( A).
Having introduced the previous infinite-dimensional reformulations, we can apply
our abstract result of Sect. 3 to obtain a Itô formula for path-dependent functionals
of continuous paths. To this end we recall that we intend to apply Theorem 2 to the
spaces
H1 = Rd
E2 =
E =
D =
U = Rk
H2 = L2
−T , 0; Rd ,
E = H1 × E2,
H = H1 × H2,
and to the operator A on H given by
on the domain
As before y is a continuous process in Rd given by
R, be a path-dependent functional
y(t ) = y0 +
c(s) dW (s),
X (t ) = Lt yt .
Ax , D F (t, x )
for (t, x ) ∈ T × D.
Then the identity
f (t, yt ) = f (0, y0) +
G (s, X (s)) ds
0
t
D F (s, X (s)) , C (s) dW (s)
holds in probability.
Proof First notice that by Proposition 2 and the discussion in Sect. 5.1 the process X
has continuous paths in E , therefore the set {X (t )}t∈[0,T ] is a compact set in E . With
this choice of E and H , a sequence Jn : H → E satisfying the requirements of
Theorem 2 can be constructed (following [13]) in this way: for any ε ∈ (0, T/2) define
the function τε : [−T , 0] → [−T , 0] as
⎧⎪ −T + ε if x ∈ [−T , −T + ε]
τε(x ) = ⎨ x if x ∈ [−T + ε, −ε]
⎪⎩ −ε if x ∈ [−ε, 0].
Then choose any C ∞(R; R) function ρ such that ρ 1 = 1, 0 ≤ ρ ≤ 1 and supp(ρ) ⊆
[−1, 1] and define a sequence {ρn} of mollifiers by ρn(x ) := nρ(nx ). Set, for any
ϕ ∈ L2(−T , 0; Rd )
Jnϕ(x ) :=
ρn ρ2n ∗ τ n1 (x ) − y ϕ(y) dy
−T
and finally define Jn as
The proof is then completed applying Theorem 2 to the function F and its extension
G.
Remark 4 The choice of the spaces H and E and of the operators A and Jn does
not depend on F and is the same for all path-dependent functionals of continuous
processes. The only assumptions that need to be checked on each functional are the
regularity conditions and existence of the extension G.
The path-dependent functional given in (34) is not covered by the previous result
since it is not jointly continuous on [0, T ] × E . However, it satisfies the assumptions
of Corollary 1, which we now state in its path-dependent formulation.
Corollary 2 Let f and F be as in Theorem 4. If F satisfies the assumptions of
Corollary 1, then the formula
j=1
f (T , yT ) = F (0, y0) +
f t j , yt j − f t j −, yt j
G (s, X (s)) ds +
D F (s, X (s)) , C(s) dW (s)
holds in probability.
6 Application to Kolmogorov Equations
6.1 Uniqueness of Solutions
We begin investigating Kolmogorov equation in the abstract setting of Sect. 3,
discussing the particular case of path-dependent Kolmogorov equations afterward. Let
the spaces H , E , E , D and U , the Wiener process W and the operator A be as in
Sect. 3; given B : [0, T ] × E → E and C : [0, T ] × E → L (U ; H1 × {0}) we can
consider the partial differential equation
∂∂Vt (t, x ) + DV (t, x ), Ax + B (t, x ) + 21 Tr H1 C(s, x )QC(s, x )∗ D2V (t, x ) = 0,
V (T , ·) = Φ,
Definition 1 We say that a function V : [0, T ] × E →
Eq. (45) in E if
R is a classical solution to
V ∈ L∞ 0, T ; Cb2,α (E ; R) ∩ C ([0, T ] × E ; R) ,
V is differentiable with respect to t on T × D, T ⊂ [0, T ] being a set of full measure,
and satisfies identity (45) for every t ∈ T and x ∈ D.
Assume that B and C are continuous and such that the stochastic SDE
Theorem 5 Under the above assumptions any classical solution to Eq. (45) is uniquely
determined on the space E
Proof Suppose there exists a solution V . Since DV , D2V , B and C are defined on
[0, T ] × E and are continuous, the function
1
G(t, x ) = − B(t, x ), DV (t, x ) − 2 TrRd C (t, x )C (t, x )∗ D2V (t, x )
is a continuous extension of
Ax , DV (t, x )
from T × D to [0, T ] × E , because V satisfies Kolmogorov equation.
Therefore, we can apply Theorem 2 to obtain
Φ X t,x (T ) = V t, X t,x (t ) +
B s, X t,x (s) , DV s, X t,x (s) ds
DV s, X t,x (s) , C s, X t,x (s) dW (s)
= V t, X t,x (t ) +
DV s, X t,x (s) , C s, X t,x (s) dW (s) .
The integral in the last line is actually a stochastic integral in a Hilbert space, since for
every u ∈ U C (s, x )u belongs to H1 × {0}; taking expectations in the previous identity
we obtain that
V (t, x ) = E Φ X t,x (T ) .
In the path-dependent case the above discussion can be rephrased as follows. Choose
the spaces H , E , E , D, U and the operator A as in Sect. 5. Let, moreover, D [a, b]; Rd
denote the space of Rd -valued càdlàg functions on the interval [a, b], equipped with
the supremum norm and set
If the operators B, C are defined from b and σ as in (38), (39), we can consider the
infinite-dimensional Kolmogorov backward equation
∂∂Vt (t, x ) +
v (t, γt ) = V t, Lt γt .
Theorem 6 Let b and σ such that Eq. (47) has a continuous solution for every t ∈
[0, T ] and every γt ∈ C [0, t ]; Rd . Then, for any f ∈ Cb2,α D [0, T ]; Rd , any
path-dependent functional v such that the function V (t, x ) = v (t, Mt x ) is a classical
solution to the path-dependent Kolmogorov backward equation associated to (b, σ, f )
is uniquely determined on C [0, t ]; Rd .
= E f MT X t,x (T )
= E f
This is what one would expect to be the solution to a Kolmogorov equation with
terminal condition f associated (in some sense) to the SDE (47).
Notice that the extension of γt introduced by the operator Lt is arbitrary; nevertheless
it does not play any role in the path-dependent Kolmogorov equation since B and C
are defined using Mt ; compare Remark 2.
We try to identify a class of functions solving virtually a Kolmogorov type equations.
The inspiration comes from [8], Section 9.9, see also [5], Theorem 3.5 for a variant.
Let N ∈ N, g1, . . . , gN ∈ B V ([0, T ]). We set g0 = 1. We define by Σ (t ) the
(N + 1) × (N + 1) matrix
gi (s) g j (s) ds.
pt (x ) =
the Gaussian density with covariance Σ (t ), for t ∈ [0, T ), x ∈ RN +1. Let f :
RN +1 → R be a continuous function with polynomial growth. We set
[−T,0]
To simplify, let us assume gi continuous.
We define U : [0, T ] × R × C ([−T , 0]) → R by
[−T,0]
[−T,0]
where U : [0, T ] × R × RN → R is motivated by the following lines.
We consider the martingale
where (with W's = Ws+T , s ∈ [−T , 0]),
Mt = E [h|Ft ]
h = H W' = f
g1 (s) dWs , . . . ,
gN (s) dWs .
U (t, x , x1, . . . , xN )
= E f x + WT − Wt , x1 +
1 N
i, j=0
∂2U
∂ xi ∂ x j
= 0
U (T , x ) = f (x ) ,
where x = (x0, x1, . . . , xN ). This can be done via the property of the density
kernel (t, ξ ) → pt (ξ ) and classical integration theorems. We set U : [0, T ] × R ×
C ([−T , 0]) → R as in (51).
extends continuously on [0, T ] × R × C ([−T , 0]) to an operator still denoted
by A (U ) (t, x , ψ )
[−t,0]
[−t,0]
∂ j U(
× dt [−t,0]
[−t,0]
[−t,0]
Now we observe that
d d
dt [−t,0] g j (· + t ) dψ = dt [−t,0] g j (ξ ) ψ (ξ − t ) dξ
∂t U
+ A (U ) + 21 ∂x2x U
= 0.
Proof (i) Obvious.
(ii) We evaluate the different derivatives for (t , x , ψ ) ∈ [0, T ] × R × C 2. We get
from (51)
j=1
Now the application
[−t,0]
= −
= −
[−t,0]
[−t,0]
[−t,0]
(remark that, without restriction of generality, we can take g j (0) = 0). Now we
calculate
∂ j U
[−t,0]
[−t,0]
[−t,0]
[−t,0]
dg j (l) = −
[l−t,0)
has to be differentiated in the direction ψ . Taking into account (53), (54), (55),
it follows that
[−t,0]
[−t,0]
∂t U (t , x , ψ ) + Dψ U (t , x , ψ ) , ψ
= ∂t U
[−t,0]
∈C 1,2,1([0, T ] ×
A (U ) = ∂t U
[−t,0]
(iv) This claim follows by inspection, taking into account (52).
Acknowledgements Open access funding provided by Institute of Science and Technology (IST Austria).
The second named author benefited partially from the support of the “FMJH Program Gaspard Monge in
Optimization and Operations Research” (Project 2014-1607H). He is also grateful for the invitation to the
Department of Mathematics of the University of Pisa. The third named author is grateful for the invitation
to ENSTA.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0
International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit to the original author(s) and the
source, provide a link to the Creative Commons license, and indicate if changes were made.
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