Elements Of Stochastic Calculus Via Regularisation A La MÃ©moire De ...

Elements of stochastic calculus via
regularisation
A la m´emoire de Paul Andr´e Meyer
A la m´
emoire de Paul-Andr´
e Meyer
Francesco Russo (1) and Pierre Vallois (2)
(1) Universit´e Paris 13
Institut Galil´ee, Math´ematiques
99 avenue J.B. Cl´ement
F-93430 Villetaneuse, France
e-mail: russo@math.univ-paris13.fr
(2) Universit´e Henri Poincar´e
Institut de Math´ematiques Elie Cartan
B.P. 239
F-54506 Vandœuvre-l`es-Nancy Cedex, France
e-mail: vallois@iecn.u-nancy.fr
Summary. This paper ﬁrst summarizes the foundations of stochastic calculus via
regularization and constructs through this procedure Itˆo and Stratonovich integrals.
In the second part, a survey and new results are presented in relation with ﬁnite
quadratic variation processes, Dirichlet and weak Dirichlet processes.
Keywords: Integration via regularization, weak Dirichlet processes, covariation, Itˆo
formulae.
MSC 2000: 60H05, 60G44, 60G48
1 Introduction
Stochastic integration via regularization is a technique of integration devel-
oped in a series of papers by the authors starting from [46], continued in
[47, 48, 49, 50, 45] and later carried out by other authors, among them
[51, 12, 13, 55, 54, 56, 58, 17, 16, 18, 19, 24]. Among some recent applica-
tions to ﬁnance, we refer for instance to [32, 4].

2
Francesco Russo and Pierre Vallois
This approach constitutes a counterpart of a discretization approach ini-
tiated by F¨ollmer ([20]) and continued by many authors, see for instance
[2, 22, 15, 14, 11, 23].
The two theories run parallel and, at the axiomatic level, almost all the
results we obtained via regularization can essentially be translated in the
language of discretization.
The advantage of using regularization lies in the fact that this approach
is natural and relatively simple, and easily connects to other approaches. We
now list some typical features of stochastic calculus via regularization.
• Two fundamental notions are the quadratic variation of a process, see Def-
inition 2 and the forward integral, see Deﬁnition 1. Calculus via regulariza-
tion is ﬁrst of all a calculus related to ﬁnite quadratic variation processes,
see section 4. Itˆo integrals with respect to continuous semimartingales can
be deﬁned through forward integrals, see Section 3; this makes classical
stochastic calculus appear as a particular instance of calculus via regu-
larization. Let the integrator be a classical Brownian motion W and the
integrand a measurable adapted process H such that T H2
0
t dt < ∞ a.s.,
where a.s. means almost surely. We will show in section 3.5 that the forward
integral · Hd−W coincides with the Itˆo integral · HdW . On the other
0
0
hand, the discretization approach constitues a sort of Riemann-Stieltjes
type integral and only allows integration of processes that are not too
irregular, see Remark 14.
• Calculus via regularization constitutes a bridge between non causal and
causal calculus operating through substitution formulae, see subsection 3.6.
A precise link between forward integration and the theory of enlargement
of ﬁltrations may be given, see [47]. Our integrals can be connected to the
well-known Skorohod type integrals, see again [47].
• With the help of symmetric integrals a calculus with respect to processes
with a variation higher than 2 may be developed. For instance fractional
Brownian motion is the prototype of such processes.
• This stochastic calculus constitutes somehow a barrier separating the pure
pathwise calculus in the sense of T. Lyons and coauthors, see e.g. [36, 35,
31, 28], and any stochastic calculus taking into account an underlying
probability, see Section 6.
This paper will essentially focuse on the ﬁrst item.
The paper is organized as follows. First, in Section 2, we recall the ba-
sic deﬁnitions and properties of forward, backward, symmetric integrals and
covariations. Justifying the related deﬁnitions and properties needs no partic-
ular eﬀort. A signiﬁcant example is the Young integral, see [57]. In Section 3
we redeﬁne Itˆo integrals in the spirit of integrals via regularization and we
prove some typical properties. We essentially deﬁne Itˆo integrals as forward
integrals in a subclass and we then extend this deﬁnitionthrough functional
analysis methods. Section 4 is devoted to ﬁnite quadratic variation processes.

Stochastic calculus via regularisation
3
In particular we establish C1-stability properties and an Itˆo formula of C2-
type. Section 5 provides some survey material with new results related to the
class of weak Dirichlet processes introduced by [12] with later developments
discussed by [24, 7]. Considerations about Itˆo formulae under C1-conditions
are discussed as well.
2 Stochastic integration via regularization
2.1 Deﬁnitions and fundamental properties
In this paper T will be a ﬁxed positive real number. By convention, any real
continuous function f deﬁned either on [0, T ] or R+ will be prolongated (with
the same name) to the real line, setting
f (0) if t ≤ 0
f (t) =
(1)
f (T ) if t > T.
Let (Xt)
be a continuous process and (Y
be a process with paths
t≥0
t)t≥0
a
in L1 (R
|Y
loc
+), i.e. for any a > 0,
t|dt < ∞ a.s.
0
Our generalized stochastic integrals and covariations will be deﬁned through
a regularization procedure. More precisely, let I−(ε, Y, dX) (resp. I+(ε, Y, dX),
I0(ε, Y, dX) and C(ε, Y, X)) be the ε-forward integral (resp. ε-backward inte-
gral, ε-symmetric integral and ε-covariation):
t
X(s + ε) − X(s)
I−(ε, Y, dX)(t) =
Y (s)
ds;
t ≥ 0,
(2)
0
ε
t
X(s) − X(s − ε
I+(ε, Y, dX)(t) =
Y (s)
ds;
t ≥ 0,
(3)
0
ε
t
X(s + ε) − X(s − ε)
I0(ε, Y, dX)(t) =
Y (s)
ds;
t ≥ 0,
(4)
0
2ε
t
X(s + ε) − X(s) Y (s + ε) − Y (s)
C(ε, X, Y )(t) =
ds;
t ≥ 0.
(5)
0
ε
Observe that these four processes are continuous.
Deﬁnition 1. 1) A family of processes (H(ε)
t
)t∈[0,T] is said to converge to
(Ht)t∈[0,T] in the ucp sense, if sup |H(ε)
t
− Ht| goes to 0 in probability,
0≤t≤T
as ε → 0.
2) Provided the corresponding limits exist in the ucp sense, we deﬁne the
following integrals and covariations by the following formulae
t
a) Forward integral:
Y d−X = lim I−(ε, Y, dX)(t).
0
ε→0+

4
Francesco Russo and Pierre Vallois
t
b) Backward integral:
Y d+X = lim I+(ε, Y, dX)(t).
0
ε→0+
t
c) Symmetric integral:
Y d◦X = lim I◦(ε, Y, dX)(t).
0
ε→0+
d) Covariation: [X, Y ]t = lim C(ε, X, Y )(t). When X = Y we often
ε→0+
put [X] = [X, X].
Remark 1. Let X, X , Y, Y be four processes with X, X continuous and Y, Y
having paths in L1 (R
loc
+).
will stand for one of the three symbols −, + or ◦.
1. (X, Y ) → · Y d X and (X, Y ) → [X, Y ] are bilinear operations.
0
2. The covariation of continuous processes is a symmetric operation.
3. When it exists, [X] is an increasing process.
4. If τ is a random time, [Xτ , Xτ ] = [X, X]
and
t
t∧τ
t
t
t
t∧τ
Y 1[0,τ]d X =
Y d Xτ =
Y τ d Xτ =
Y d X,
0
0
0
0
where Xτ is the process X stopped at time τ , deﬁned by Xτt = Xt∧τ .
·
·
5. If ξ and η are two ﬁxed r.v.,
(ξYs)d (ηXs) = ξη
Ysd Xs.
0
0
6. Integrals via regularization also have the following localization property.
Suppose that Xt = Xt, Yt = Yt , ∀t ∈ [0, T ] on some subset Ω0 of Ω. Then
t
t
1Ω
Y
Y
0
sd Xs = 1Ω0
s d Xs,
t ∈ [0, T ].
0
0
N
7. If Y is an elementary process of the type Yt =
Ai1I , where A
i
i are
i=1
random variables and (Ii) a family of real intervals with end-points ai < bi,
then
t
N
Ysd Xs =
Ai(Xbi∧t − Xai∧t).
0
i=1
Deﬁnition 2. 1) If [X] exists, X is said to be a ﬁnite quadratic variation
process and [X] is called the quadratic variation of X.
2) If [X] = 0, X is called a zero quadratic variation process.
3) A vector (X1, . . . , Xn) of continuous processes is said to have all its mu-
tual covariations if [Xi, Xj] exists for all 1 ≤ i, j ≤ n.
We will also use the terminology bracket instead of covariation.
Remark 2. 1) If (X1, . . . , Xn) has all its mutual covariations, then
[Xi + Xj, Xi + Xj] = [Xi, Xi] + 2[Xi, Xj] + [Xj, Xj].
(6)
From the previous equality, it follows that [Xi, Xj] is the diﬀerence of two
increasing processes, having therefore bounded variation; consequently the
bracket is a classical integrator in the Lebesgue-Stieltjes sense.

Stochastic calculus via regularisation
5
2) Relation (6) holds as soon as three brackets among the four exist. More
generally, by convention, an identity of the type I1 + · · · + In = 0 has the
following meaning: if n − 1 terms among the Ij exist, the remaining one
also makes sense and the identity holds true.
3) We will see later, in Remark 23, that there exist processes X and Y such
that [X, Y ] exists but does not have ﬁnite variation; in particular (X, Y )
does not have all its mutual brackets.
The properties below follow elementarily from the deﬁnition of integrals
via regularization.
Proposition 1. Let X = (Xt)
be a continuous process and Y = (Y
t≥0
t)t≥0
be a process with paths in L1 (R
loc
+). Then
t
t
1) [X, Y ]t =
Y d+X −
Y d−X.
0
0
t
1
t
t
2)
Y d◦X =
Y d+X +
Y d−X .
0
2
0
0
3) Time reversal. Set ˆ
Xt = XT−t, t ∈ [0, T ]. Then
t
T
1.
Y d±X = −
ˆ
Y d ˆ
X,
0 ≤ t ≤ T ;
0
T −t
t
T
2.
Y d◦X = −
ˆ
Y d◦ ˆ
X,
0 ≤ t ≤ T ;
0
T −t
3. [ ˆ
X, ˆ
Y ]t = [X, Y ]T − [X, Y ]T−t,
0 ≤ t ≤ T .
4) Integration by parts. If Y is continuous,
t
t
XtYt = X0Y0 +
Xd−Y +
Y d+X
0
0
t
t
= X0Y0 +
Xd−Y +
Y d−X + [X, Y ]t.
0
0
5) Kunita-Watanabe inequality. If X and Y are ﬁnite quadratic variation
processes,
[X, Y ] ≤ [X] [Y ] 1/2.
6) If X is a ﬁnite quadratic variation process and Y is a zero quadratic
variation process then (X, Y ) has all its mutual brackets and [X, Y ] = 0.
7) Let X be a bounded variation process and Y be a process with locally
bounded paths, and at most countably many discontinuities. Then
t
t
t
t
a)
Y d+X =
Y d−X =
Y dX, where
Y dX is a Lebesgue-
0
0
0
0
Stieltjes integral.
b) [X, Y ] = 0. In particular a bounded variation and continuous process
is a zero quadratic variation process.

6
Francesco Russo and Pierre Vallois
8) Let X be an absolutely continuous process and Y be a process with locally
bounded paths. Then
t
t
t
Y d+X =
Y d−X =
Y X ds.
0
0
0
Remark 3. If Y has uncountably many discontinuities, 7) may fail. Take for
instance Y = 1supp dV , where V is an increasing continuous function such that
V (t) = 0 a.e. (almost everywhere) with respect to Lebesgue measure. Then
Y = 0 Lebesgue a.e., and Y = 1, dV a.e. Consequently
t
t
Y dV = V (t) − V (0),
I−(ε, Y, dV )(t) = 0
Y d−V = 0.
0
0
Remark 4. Point 2) of Proposition 1 states that the symmetric integral is the
average of the forward and backward integrals.
Proof of Proposition 1. Points 1), 2), 3), 4) follow immediately from the
deﬁnition. For illustration, we only prove 3); operating a change of variable
u = T − s, we obtain
t
X
T
ˆ
X
Y
s − Xs−ε
ˆ
u+ε − ˆ
Xu
s
ds = −
Yu
du,
0 ≤ t ≤ T.
0
ε
T −t
ε
Since X is continuous, one can take the limit of both members and the result
follows.
5) follows by Cauchy-Schwarz inequality which says that
1
t
(X
ε
s+ε − Xs) (Ys+ε − Ys)ds
0
1
1
t
1
t
2
≤
(X
(Y
.
ε
s+ε − Xs)2ds
s+ε − Ys)2ds
0
ε 0
6) is a consequence of 5).
7) Using Fubini, one has
1
t
1
t
s+ε
Y
ds Y
dX
ε
s(Xs+ε − Xs)ds =
s
u
0
ε 0
s
t+ε
1
u∧t
=
dXu
Ysds.
0
ε u−ε
1
u
Since the jumps of Y are at most countable,
Y
ε
sds → Yu, d|X | a.e.
u−ε
where |X| denotes the total variation of X. Since t → Yt is locally bounded,
t
t
Lebesgue’s convergence theorem implies that
Y d−X =
Y dX.
0
0
t
t
The fact that
Y d+X =
Y dX follows similarly.
0
0
b) is a consequence of point 1).
8) can be reached using similar elementary integration properties.

Stochastic calculus via regularisation
7
2.2 Young integral in a simpliﬁed framework
We will consider the integral deﬁned by Young ([57]) in 1936, and implemented
in the stochastic framework by Bertoin, see [3]. Here we will restrict ourselves
to the case when integrand and integrator are H¨older continuous processes.
As a result, that integral will be shown to coincide with the forward integral,
but also with backward and symmetric ones.
Deﬁnition 3. 1. Let Cα be the set of H¨older continuous functions deﬁned
on [0, T ], with index α > 0. Recall that f : [0, T ] → R belongs to Cα if
|f (t) − f (s)|
Nα(f) := sup
< ∞.
0≤s,t≤T
|t − s|α
2. If X, Y : [0, T ] → R are two functions of class C1, the Young integral of
Y with respect to X on [a, b] ⊂ [0, T ] is deﬁned as :
b
b
Y d(y)X :=
Y (t)X (t)dt,
0 ≤ a ≤ b ≤ T.
a
a
To extend the Young integral to H¨older functions we need some estimate
T
of
Y d(y)X in terms of the H¨older norms of X and Y . More precisely, let
0
X and Y be as in Deﬁnition 3 above; then in [15], it is proved:
T
(Y − Y (a))d(y)X ≤ CρT 1+ρNα(X)Nβ(Y ),
0 ≤ a ≤ T,
(7)
a
where α, β > 0, α + β > 1, ρ ∈]0, α + β − 1[, and Cρ is a universal constant.
·
Proposition 2. 1. The map (X, Y ) ∈ C1([0, T ]) × C1([0, T ]) →
Y d(y)X
0
with values in Cα, extends to a continuous bilinear map from Cα × Cβ
to Cα. The value of this extension at point (X, Y ) ∈ Cα × Cβ will still be
·
denoted by
Y d(y)X and called the Young integral of Y with respect
0
to X.
2. Inequality (7) is still valid for any X ∈ Cα and Y ∈ Cβ.
Proof. 1. Let X, Y be of class C1([0, T ]) and
t
t
F (t) =
Y d(y)X =
Y (s)X (s)ds,
t ∈ [0, T ].
0
0
For any a, b ∈ [0, T ], a < b, we have
b
F (b) − F (a) =
Y (t) − Y (a) d(y)X + Y (a) X(b) − X(a) .
a

8
Francesco Russo and Pierre Vallois
Then (7) implies
|F (b)−F (a)| ≤ Cρ(b−a)1+ρNα(X)Nβ(Y )+ sup |Y (t)| Nα(X)(b−a)α; (8)
0≤t≤T
consequently F ∈ Cα.
·
Then the map (X, Y ) ∈ C1([0, T ]) × C1([0, T ]) →
Y d(y)X, which is
0
bilinear, extends to a continuous bilinear map from Cα × Cβ to Cα.
2. is a consequence of point 1.
Before discussing the relation between Young integrals and integrals via
regularization, here is useful technical result.
Lemma 1. Let 0 < γ < γ ≤ 1, ε > 0. With Z ∈ Cγ we associate
1
t
Zε(t) =
Z(u + ε) − Z(u) du, t ∈ [0, T ].
ε 0
Then Zε converges to Z in Cγ , as ε → 0.
Proof. For any 0 ≤ t ≤ T ,
1
t
1
t+ε
1
ε
Zε(t) =
Z(u + ε) − Z(u) du =
Z(u)du −
Z(u)du.
ε 0
ε t
ε 0
Setting ∆ε(t) = Zε(t) − Z(t), we get
1
t+ε
1
s+ε
∆ε(t) − ∆ε(s) =
Z(u)du − Z(t) −
Z(u)du + Z(s)
ε t
ε s
1
t+ε
1
s+ε
=
Z(u) − Z(t) du −
Z(u) − Z(s) du,
ε t
ε s
where 0 ≤ s ≤ t ≤ T .
a) Suppose 0 ≤ s < s + ε < t. The above inequality implies
1
t+ε
1
s+ε
|∆ε(t) − ∆ε(s)| ≤
Z(u) − Z(t) du +
Z(u) − Z(s) du.
ε t
ε s
Since Z ∈ Cγ, then
N
t+ε
s+ε
|∆
γ (Z)
ε(t) − ∆ε(s)| ≤
(u − t)γdu +
(u − s)γdu
ε
t
s
2N
≤
γ (Z) εγ.
γ + 1
But ε < t − s, consequently
2N
|∆
γ (Z)
ε(t) − ∆ε(s)| ≤
εγ−γ |t − s|γ .
(9)
γ + 1

Stochastic calculus via regularisation
9
b) We now investigate the case 0 ≤ s < t < s+ε. The diﬀerence ∆ε(t)−∆ε(s)
may be decomposed as follows :
1
t+ε
1
t
∆ε(t) − ∆ε(s) =
Z(u) − Z(s + ε) du −
Z(u) − Z(s) du
ε s+ε
ε s
t − s
+
Z(s + ε) − Z(s) + Z(s) − Z(t).
ε
Proceeding as in the previous step and using the inequality 0 < t − s < ε, we
obtain
2
(t − s)γ+1
t − s
|∆ε(t) − ∆ε(s)| ≤ Nγ(Z)
+
+ (t − s)γ
γ + 1
ε
ε1−γ
γ + 2
≤ 2Nγ(Z)
εγ−γ |t − s|γ .
γ + 1
At this point, the above inequality and (9) directly imply that Nγ (Zε − Z) ≤
Cεγ−γ and the claim is ﬁnally established.
In the sequel of this section X and Y will denote stochastic processes.
Remark 5. If X and Y have a.s. H¨older continuous paths respectively of order
α and β with α > 0, β > 0 and α + β > 1. Then one can easily prove that
[X, Y ] = 0.
Proposition 3. Let X, Y be two real processes indexed by [0, T ] whose paths
are respectively a.s. in Cα and Cβ, with α > 0, β > 0 and α + β > 1. Then
·
·
·
the three integrals
Y d+X,
Y d−X and
Y d◦X exist and coincide with
0
0
0
·
the Young integral
Y d(y)X.
0
Proof. We establish that the forward integral coincides with the Young in-
tegral. The equality concerning the two other integrals is a consequence of
Proposition 1 1., 2. and Remark 5.
By additivity we can suppose, without lost generality, that Y (0) = 0.
Set
t
t
∆ε(t) :=
Y d(y)X −
Y dXε,
0
0
where
1
t
Xε(t) =
X(u + ε) − X(u) du, t ∈ [0, T ].
ε 0
t
Since t → Xε(t) is of class C1([0, T ]), then
Y dXε is equal to the Young
0
t
integral
Y d(y)Xε and therefore
0

10
Francesco Russo and Pierre Vallois
t
∆ε(t) =
Y d(y) X − Xε .
0
Let α be such that : 0 < α < α and α + β > 1. Applying inequality (7) we
obtain
sup |∆ε(t)| ≤ CρT 1+ρNα (X − Xε)Nβ(Y ),
ρ ∈]0, α +β−1[.
0≤t≤T
Lemma 1 with Z = X and γ = α directly implies that ∆ε(t) goes to 0,
uniformly a.s. on [0, T ], as ε → 0, concluding the proof of the Proposition.
3 Itˆ
o integrals and related topics
The section presents the construction of Itˆo integrals with respect to contin-
uous local martingales; it is based on McKean’s idea (see section 2.1 of [37]),
which ﬁts the spirit of calculus via regularization.
3.1 Some reminders on martingales theory
In this subsection, we recall basic notions related to martingale theory, essen-
tially without proofs, except when they help the reader. For detailed comple-
ments, see [30], chap. 1., in particular for deﬁnition of adapted and progres-
sively measurable processes.
Let (Ft)t≥0 be a ﬁltration on the probability space (Ω, F, P ) satisfying
the usual conditions, see Deﬁnition 2.25, chap. 1 in [30].
An adapted process (Mt) of integrable random variables, i.e. verifying
E(|Mt|) < ∞, ∀t ≥ 0 is:
• an (Ft)-martingale if E(Mt|Fs) = Ms,
∀t ≥ s;
• a (Ft)- submartingale if E(Mt|Fs) ≥ Ms,
∀t ≥ s
In this paper, all submartingales (and therefore all martingales) will be
supposed to be continuous.
Remark 6. It follows from the deﬁnition that if (Mt)t≥0 is a martingale, then
E(Mt) = E(M0), ∀t ≥ 0. If (Mt)t≥0 is a supermartingale (resp. submartin-
gale) then t −→ E(Mt) is decreasing (resp. increasing).
Deﬁnition 4. A process X is said to be square integrable if E(X2t) < ∞
for each t ≥ 0.
When we speak of a martingale without specifying the σ-ﬁelds, we refer
to the canonical ﬁltration generated by the process and satisfying the usual
conditions.

Stochastic calculus via regularisation
11
Deﬁnition 5. 1. A (continuous) process (Xt)
, is called a (F
t≥0
t)-local
martingale (resp. (Ft)-local submartingale) if there exists an in-
creasing sequence (τn) of stopping times such that Xτn1τn>0 is an (Ft)-
martingale (resp. submartingale) and lim τn = +∞ a.s.
n→∞
Remark 7. • An (Ft)-martingale is an (Ft)-local martingale. A bounded
(Ft)-local martingale is an (Ft)-martingale.
• The set of (Ft)-local martingales is a linear space.
• If M is an (Ft)-local martingale and τ a stopping time, then Mτ is again
an (Ft)-local martingale.
• If M0 is bounded, in the deﬁnition of a local martingale one can choose a
localizing sequence (τn) such that each Mτn is bounded.
• A convex function of an (Ft)-local submartingale is an (Ft)-local sub-
martingale.
Deﬁnition 6. A process S is called a (continuous) (Ft)-semimartingale if it
is the sum of an (Ft)-local martingale and an (Ft)-adapted continuous bounded
variation process.
A basic decomposition in stochastic analysis is the following.
Theorem 1. (Doob decomposition of a submartingale)
Let X be a (Ft)-local submartingale. Then, there is an (Ft)-local martin-
gale M and an adapted, continuous, and ﬁnite variation process V (such that
V0 = 0) with X = M + V . The decomposition is unique.
Deﬁnition 7. Let M be an (Ft)-local martingale. We denote by < M > the
bounded variation process featuring in the Doob decomposition of the local
submartingale M 2. In particular M 2− < M > is an (Ft)-local martingale.
In Corollary 2, we will prove that < M > coincides with [M, M ], so that the
skew bracket < M > does not depend on the underlying ﬁltration.
The following result will be needed in section 3.2.
Lemma 2. Let (M n
) be a sequence of (F
t∈[0,T ]
t) local martingales such that
M n
0 = 0 and < M n >T converges to 0 in probability as n → ∞. Then M n → 0
ucp, when n → ∞.
Proof. It suﬃces to apply to N = M n the following inequality stated in [30],
Problem 5.25 Chap. 1, which holds for any (Ft)-local martingale (Nt) such
that N0 = 0:
1
P
sup |Nu| ≥ λ ≤ P < N >t≥ δ +
E δ∧ < N >t ,
(10)
0≤u≤t
λ2
for any t ≥ 0, λ, δ > 0.
An immediate consequence of the previous lemma is the following.
Corollary 1. Let M be an (Ft)-local martingale vanishing at zero, with
< M > = 0. Then M is identically zero.

12
Francesco Russo and Pierre Vallois
3.2 The Itˆ
o integral
Let M be an (Ft)-local martingale. We construct here the Itˆo integral with
respect to M using stochastic calculus via regularization. We will proceed
in two steps. First we deﬁne the Itˆo integral · HdM for a smooth integrand
0
process H as the forward integral · Hd−M . Second, we extend H → · HdM
0
0
via functional analytical arguments. We remark that the classical theory of
Itˆo integrals ﬁrst deﬁnes the integral of simple step processes H, see Remark
9, for details.
Observe ﬁrst that the forward integral of a continuous process H of
bounded variation is well deﬁned because Proposition 1 4), 7) imply that
t
t
t
Hd−M = HtMt−H0M0−
M d+H = HtMt−H0M0−
MsdHs. (11)
0
0
0
Call C the vector algebra of adapted processes whose paths are of class C0.
This linear space, equipped with the metrizable topology which governs the
ucp convergence, is an F -space. For the deﬁnition and properties of F -spaces,
see [10], chapter 2.1. Remark that the set Mloc of continuous (Ft)-local mar-
tingales is a closed linear subspace of C, see for instance [24].
Denote by CBV the C subspace of processes whose paths are a.s. continuous
with bounded variation. The next observation is crucial.
Lemma 3. If H is an adapted process in CBV then
· Hd−M is an (F
0
t)-
local martingale whose quadratic variation is given by
·
·
<
Hd−M >t=<
H2sd < M >s .
0
0
Proof. We only sketch the proof. We restrict ourselves to prove that if M is
·
a local martingale then Y =
Hd−M is a local martingale.
0
By localization, we can suppose that H, its total variation H and M
are bounded processes.
Let 0 ≤ s < t. Since Ht = H0 + t dH
s
u, (11) implies
s
t
Yt = HsMt − H0M0 −
MudHu +
(Mt − Mu)dHu.
(12)
0
s
Let (πn) be a sequence of subdivisions of [s, t], such that the mesh of (πn)
goes to zero when n → +∞. Since M is continuous, M and H are bounded,
∆n :=
(Mt − Mu
)(H
− H ),
i+1
ui+1
ui
πn
goes to t(M
s
t − Mu)dHu a.s. and in L1. Consequently,

Stochastic calculus via regularisation
13
t
E
(Mt − Mu)dHu = lim E(∆n|Fs)
s
n→∞
and
E(∆n|Fs) =
E (Mt − Mu
)(H
− H )|F
i+1
ui+1
ui
s .
πn
But one has
E (Mt − Mu
)(H
− H )|F
i+1
ui+1
ui
s
= E E((Mt − Mu
)(H
− H )|F
)|F
i+1
ui+1
ui
ui+1
s
(13)
= E (Hu
− H )E(M
|F
)|F
i+1
ui
t − Mui+1
ui+1
s
= 0,
(14)
since H is adapted and M is a martingale.
Finally, taking the conditional expectation with respect to Fs in (12) yields
s
E Yt|Fs = HsMs − H0M0 −
MudHu = Ys.
0
·
Similar arguments show that Y 2 −
H2d < M > is a martingale.
0
The previous lemma allows to extend the map H →
t Hd−M. Let
0
L2(d < M >) denote the set of progressively measurable processes such that
T
H2d < M >< ∞ a.s.
(15)
0
L2(d < M >) is an F -space with respect to the metrizable topology d2 deﬁned
as follows: (Hn) converges to H when n → ∞ if T (Hn
0
s − Hs)2d < M >s→ 0
in probability, when n → ∞.
Remark 8. CBV is dense in L2(d < M >). Indeed, according to [30], lemma
2.7 section 3.2, simple processes are dense into L2(d < M >). On the other
hand, a simple process of the form Ht = ξ1]a,b], ξ being Fa measurable, can
be expressed as a limit of Hn
t = ξφn where φn are continuous functions with
bounded variation.
Let Λ : CBV → Mloc be the map deﬁned by ΛH = · Hd−M.
0
Lemma 4. If CBV (resp. Mloc) is equipped with d2 (resp. the ucp topology)
then Λ is continuous.
Proof. Let Hk be a sequence of processes in CBV , converging to 0 for d2 when
k → ∞. Set N k =
· Hkd−M. Lemma 3 implies that < Nk >
0
T converges
to 0 in probability. Finally Lemma 2 concludes the proof.

14
Francesco Russo and Pierre Vallois
We can now easily deﬁne the Itˆo integral. Since CBV is dense in L2(d < M >)
for d2, Lemma 4 and standard functional analysis arguments imply that Λ
uniquely and continuously extends to L2(d < M >).
Deﬁnition 8. If H belongs to L2(d < M >), we put · HdM := ΛH and we
0
call this the Itˆ
o integral of H with respect to M .
Proposition 4. If H belongs to L2(d < M >), then ( · HdM ) is an (F
0
t)-
local martingale with bracket
·
·
<
HdM >=
H2d < M > .
(16)
0
0
Proof. Let H ∈ L2(d < M >). From Deﬁnition 8, ( · HdM ) is an (F
0
t)-local
martingale. It remains to prove (16).
Since H belongs to L2(d < M >), then there exists a sequence (Hn) of
elements in CBV , such that Hn → H in L2(d < M >).
·
Introduce Nn =
HndM and Nn = N2n −<Nn>. According to lemma 4,
0
·
< Nn >=
H2nd < M >; now Nn → N, ucp, n → ∞ and < Nn > goes to
0
·
H2d < M > in the ucp sense, as n → ∞. Therefore Nn converges with
0
·
respect to the ucp topology, to the local martingale N 2 −
H2d < M >.
0
This actually proves (16).
Remark 9. 1. Recall that whenever H ∈ CBV
·
·
HdM =
Hd−M.
0
0
This property will be generalized in Propositions 6 and 2.
2. We emphasize that Itˆo stochastic integration based on adapted simple step
processes and the previous construction, ﬁnally lead to the same object.
If H is of the type Y 1]a,b] where Y is an Fa measurable random variable,
it is easy to show that
t HdM = Y (M
0
t∧b − Mt∧a). Since the class of
elementary processes obtained by linear combination of previous processes
is dense in L2(d < M >) and the map Λ is continuous, then
· HdM
0
equals the classical Itˆo integral.
In Proposition 5 below we state the chain rule property.
Proposition 5. Let (Mt, t ≥ 0) be an (Ft)-local martingale, (Ht, t ≥ 0) be
·
in L2(d < M >), N :=
HsdMs and (Kt, t ≥ 0) be a (Ft)-progressively
0
T
measurable process such that
(HsKs)2d < M >s< ∞ a.s. Then
0

Stochastic calculus via regularisation
15
t
t
KsdNs =
HsKsdMs,
0 ≤ t ≤ T.
(17)
0
0
Proof. Since the map Λ : H ∈ L2(d < M >) →
· HdM is continuous, it
0
suﬃces to prove (17) for H and K continuous and with bounded variation.
For simplicity we suppose M0 = H0 = K0 = 0.
One has
t
t
KdN =
(Nt − Nu)dKu,
0
0
and
t
u
Nt − Nu =
(Mt − Mv)dHv −
(Mu − Mv)dHv
0
0
t
= (Mt − Mu)Hu +
(Mt − Mv)dHv,
u
where 0 ≤ u ≤ t.
Using Fubini’s theorem one gets
t
t
KdN =
(Mt − Mu)(HudKu + KudHu)
0
0
t
t
=
(Mt − Mu)d(HK)u =
HKdM.
0
0
3.3 Connections with calculus via regularizations
The next Proposition will show that, under suitable conditions, the Itˆo integral
is a forward integral.
Proposition 6. Let X be an (Ft)-local martingale and suppose that (Ht) is
progressively measurable and locally bounded.
·
·
1. If H has a left limit at each point then
Hsd−Xs =
Hs−dXs.
0
0
2. If Ht = Ht−, d < X >t a.e. (in particular if H is c`adl`ag), then
·
·
Hsd−Xs =
HsdXs.
0
0
Proof.
s
Since s →
Hudu is continuous with bounded variation,
s−ε
t
1
s
t
1
s
Hudu dXs =
Hudu d−Xs
0
ε s−ε
0
ε s−ε
1
t
1
t
= Xt
H
(H
ε
udu
− H0X0 −
s − Hs−ε)Xsds.
t−ε
ε 0

16
Francesco Russo and Pierre Vallois
The second integral in the right-hand side can be modiﬁed as follows
t
t
t
−
(Hs − Hs−ε)Xsds =
Hs(Xs+ε − Xs)ds −
HsXs+εds
0
0
t−ε
ε
+ H0
Xsds.
0
Consequently
t
1
s
1
t
Hudu dXs =
Hs(Xs+ε − Xs)ds + Rε(t),
(18)
0
ε s−ε
ε 0
where
1
t
1
t
1
ε
Rε(t) = Xt
H
H
X
ε
sds
−
sXs+εds + H0
sds − X0
t−ε
ε t−ε
ε 0
(19)
1
t
1
ε
=
H
X
ε
s(Xt − Xs+ε)ds + H0
sds − X0
t−ε
ε 0
converges to zero ucp.
Under assumption 1, Lebesgue’s dominated convergence theorem implies
that 1 · H
ε
·−ε
sds converges to H− according to L2(d < M >), so the left-hand
·
side of equality (18) converges to the Itˆo integral
Hs−dXs. This forces the
0
·
right-hand side to converge to
Hsd−Xs.
0
The proof of 2 is similar, remarking that Hs = Hs−, for d < M >s a.e.
When the integrator is a Brownian motion W , we will see in Theorem 2
below that the forward integral coincides with the Itˆo integral for any inte-
grand in L2(d < W >). This is no longer true when the integrator is a general
semimartingale. The following example provides a martingale (Mt) and a de-
t
terministic integrand h such that the Itˆo integral
hdM and the forward
0
t
integral
hd−M exist, but are diﬀerent.
0
Example 1. Let ψ : [0, ∞[→ R verify ψ(0) = 0, ψ is continuous, increasing,
and ψ (t) = 0 a.e. (with respect to the Lebesgue measure). Let (Mt) be the
process: Mt = Wψ(t), t ≥ 0, and h be the indicator function of the support of
the positive measure dψ. Since W 2
t − t is a martingale, < W >t= t. Clearly
(Mt) is a martingale and < M >t= ψ(t), t ≥ 0. Observe that h = 0 a.e. with
·
M (s + ε) − M (s)
respect to Lebesgue measure. Then
h(s)
ds = 0 and so
0
ε
·
hd−M = 0.
0

Stochastic calculus via regularisation
17
t
On the other hand, h = 1, dψ a.e., implies
hdM = Mt, t ≥ 0.
0
Remark 10. A signiﬁcant result of classical stochastic calculus is the Bichteler-
Dellacherie theorem, see [43] Th. 22, Section III.7. In the regularization ap-
proach, an analogous property occurs: if the forward integral exists for a rich
class of adapted integrands, then the integrator is forced to be a semimartin-
gale. More precisely we recall the signiﬁcant statement of [47], Proposition
1.2.
Let (Xt, t ≥ 0) be an (Ft)-adapted and continuous process such that for
·
any c`adl`ag, bounded and adapted process (Ht), the forward integral
Hd−X
0
exists. Then (Xt) is an (Ft)-semimartingale.
From Proposition 6 we deduce the relation between skew and square
bracket.
Corollary 2. Let M be an (Ft)-local martingale. Then < M >= [M] and
t
M 2t = M20 + 2
M d−M + < M >t .
(20)
0
Proof. The proof of (20) is very simple and is based on the following identity
(Ms+ε − Ms)2 = M2s+ε − M2s − 2Ms(Ms+ε − Ms).
Integrating on [0, t] leads to
1
t
1
t
1
t
2
t
(M
M 2
M 2
M
ε
s+ε − Ms)2ds =
s+εds −
s ds −
s(Ms+ε − Ms)ds
0
ε 0
ε 0
ε 0
1
t+ε
1
ε
2
t
=
M 2
M 2
M
ε
s ds −
s ds −
s(Ms+ε − Ms)ds.
t
ε 0
ε 0
Therefore, taking the limit when ε → 0, one obtains
t
[M ]t = M2t − M20 − 2
Msd−Ms.
0
·
Since t → Mt is continuous, the forward integral
M d−M coincides with the
0
corresponding Itˆo integral. Consequently M 2t−M20−[M]t is a local martingale.
This proves both [M ] =< M > and (20).
Corollary 3. Let M, M be two (Ft)-local martingales. Then (M, M ) has all
its mutual covariations.
Proof. Since M, M and M +M are continuous local martingales, Corollary 2
directly implies that they have ﬁnite quadratic variation. The bilinearity prop-
erty of the covariation directly implies that [M, M ] exists and equals
1 ([M + M ] − [M] − [M ]).
2

18
Francesco Russo and Pierre Vallois
Proposition 7. Let M and M be two (Ft)-local martingales, H and H be
two progressively measurable processes such that
·
·
H2d < M >< ∞,
H2d < M >< ∞.
0
0
Then
·
·
t
HdM,
H dM
=
HH d[M, M ]t.
0
0
t
0
The next proposition provides a simple example of two processes (Mt) and
(Yt) such that [M, Y ] exists even though the vector (M, Y ) has no mutual
covariation.
Proposition 8. Let (Mt) be an continuous (Ft)-local martingale, (Yt) a
c`adl`ag and an (Ft)-adapted process. If M and Y are independent then
[M, Y ] = 0.
Proof. Let Y be the σ-ﬁeld generated by (Yt), and denote by ( ˜
Mt) the small-
est ﬁltration satisfying the usual conditions and containing (Ft) and Y, i.e.,
σ(Ms, s ≤ t) ∨ Y ⊂ ˜
Mt, ∀t ≥ 0. It is not diﬃcult to show that (Mt) is also an
( ˜
Mt)-martingale.
Thanks to Proposition 1 1., it is suﬃcient to prove that
t
t
Y d−M =
Y d+M.
(21)
0
0
Proposition 6 implies that the left-hand side coincides with the (Mt)-Itˆo in-
tegral t Y dM .
0
Without restricting generality we suppose M0 = 0. We proceed as in the
s+ε
proof of Proposition 6. Since a.s. s →
Yudu is continuous with bounded
s
variation,
t
1
s+ε
1
s+ε
1
t
Yudu d−Ms = Mt
Yudu −
(Ys+ε − Ys)Msds.
0
ε s
ε s
ε 0
As the processes Y and M are independent, the forward integral in the left-
hand side above is actually an Itˆo integral. Therefore, taking the limit when
ε → 0 and using Proposition 6, one gets
t
t
t
Y dM =
Y d−M = YtMt −
M d−Y.
0
0
0
According to point 4) of Proposition 1, the right-hand side is equal to
t
Y d+M ; this proves (21).
0

Stochastic calculus via regularisation
19
3.4 The semimartingale case
We begin this section with a technical lemma which implies that the decom-
position of a semimartingale is unique.
Lemma 5. Let (Mt, t ≥ 0) be a (Ft)-local martingale with bounded variation.
Then (Mt) is constant.
Proof. Since M has bounded variation, then Proposition 1, 7) implies that
[M ] = 0. Consequently Corollaries 1 and 2 imply that Mt = M0, t ≥ 0.
It is now easy to deﬁne stochastic integration with respect to continuous
semimartingales.
Deﬁnition 9. Let (Xt, t ≥ 0) be an (Ft)-semimartingale with canonical de-
composition X = M +V , where M (resp. V ) is a continuous (Ft)-local martin-
gale (resp. bounded variation, continuous and (Ft)-adapted process) vanishing
at 0. Let (Ht, t ≥ 0) be an (Ft)-progressively measurable process, satisfying
T
T
H2sd[M, M]s < ∞, and
|Hs|d V s < ∞,
(22)
0
0
where V t is the total variation of V over [0, t].
We set
t
t
t
HsdXs =
HsdMs +
HsdVs,
0 ≤ t ≤ T.
0
0
0
Remark 11. 1. In the previous deﬁnition, the integral with respect to M
(resp. V ) is an Itˆo-type (resp. Stieltjes-type) integral.
·
2. It is clear that
HsdXs is again a continuous (Ft)-semimartingale, with
0 ·
·
martingale part
HsdMs and bounded variation component
HsdVs.
0
0
Once we have introduced stochastic integrals with respect to continuous
semimartingales, it is easy to deﬁne Stratonovich integrals.
Deﬁnition 10. Let (Xt, t ≥ 0) be an (Ft)-semimartingale and (Yt, t ≥ 0) an
(Ft)-progressively measurable process. The Stratonovich integral of Y with
respect to X is deﬁned as follows
t
t
1
Ys ◦ dXs =
YsdXs + [Y, X]t;
t ≥ 0,
(23)
0
0
2
if [Y, X] and · Y
0
sdXs exist.
Remark 12. 1. Recall that conditions of type (22) ensure existence of the
stochastic integral with respect to X.

20
Francesco Russo and Pierre Vallois
2. If (Xt) and (Yt) are (Ft)-semimartingales, then · Y
0
s ◦ dXs exists and is
called the Fisk-Stratonovich integral.
3. Suppose that (Xt) is an (Ft)-semimartingale and (Yt) is a left continu-
ous and (Ft)-adapted process such that [Y, X] exists. We already have
·
·
observed (see Proposition 6) that
YsdXs coincides with
Ysd−Xs.
0
0 ·
Proposition 1 1) and 2) imply that the Stratonovich integral
Ys ◦ dXs
0
·
is equal to the symmetric integral
Ysd◦Xs.
0
At this point we can easily identify the covariation of two semimartingales.
Proposition 9. Let Si = M i + V i be two (Ft)-semimartingales, i = 1, 2,
where M i are local martingales and V i bounded variation processes. One has
[S1, S2] = [M 1, M 2].
Proof. The result follows directly from Corollary 3, Proposition 1 7), and the
bilinearity of the covariation.
Corollary 4. Let S1, S2 be two (Ft)- semimartingales such that their mar-
tingale parts are independent. Then [S1, S2] = 0.
Proof. It follows from Proposition 8.
The statement of Proposition 6 can be adapted to semimartingale integra-
tors as follows.
Proposition 10. Let X be an (Ft)-semimartingale and suppose that (Ht) is
·
·
adapted, with left limits at each point. Then
Hsd−Xs =
Hs−dXs. If H
0
0
·
·
is c`adl`ag then
Hd−X =
HdX.
0
0
Remark 13. 1. Forward integrals generalize not only classical Itˆo integrals
but also the integral obtained from the theory of enlargements of ﬁltra-
tions, see e.g. [29]. Let (Ft) and (Gt) be two ﬁltrations fulﬁlling the usual
conditions with Ft ⊂ Gt for all t. Let X be a (Gt)-semimartingale which
is (Ft)-adapted. By Stricker’s theorem, X is also an (Ft)-semimartingale.
Let H be a c`adl`ag bounded (Ft)-adapted process. According to Proposi-
tion 10, the (Ft)-Itˆo integral · HdX equals the (G
0
t)-Itˆ
o integral and it
coincides with the forward integral · Hd−X.
0
2. The result stated above is false when H has no left limits at each point.
Using a tricky example in [42], it is possible to exhibit a ﬁltration (Gt),
a (Gt)-semimartingale (Xt)
with natural ﬁltration FX
t≥0
t , a bounded and
(FX
t )-progressively measurable process H , such that
· Hd−X equals the
0
(FX
t )-Itˆ
o integral but diﬀers from the (Gt)-Itˆo integral. More precisely one
has:

Stochastic calculus via regularisation
21
a) X is a 3-dimensional Bessel process with decomposition
t 1
Xt = Wt +
ds,
(24)
0 Xs
where W is an (FX
t )-Brownian motion,
b) X is a (Gt)-semimartingale with decomposition M + V where M is
the local martingale part,
c) Ht(ω) = 1 for dt⊗dP -almost all (t, ω) ∈ [0, T ] × Ω,
d) βt = t HdX is a (G
0
t)-Brownian motion.
Property (d) implies that I−(ε, H, dX) = I−(ε, 1, dX) so that t Hd−X =
0
X
Hs
t. The (F X
t )-Itˆ
o integral
t HdX equals t HdW + t
ds; Theo-
0
0
0 Xs
rem 2 below and Proposition 1 8) imply that this integral coincides with
t Hd−X. Since a Bessel process cannot be equal to a Brownian motion,
0
the (Gt)-Itˆo integral t HdX diﬀers from the (FX
HdX.
0
t )-Itˆ
o integral t0
Indeed, the pathology comes from the integration with respect to the
bounded variation process. In fact, according to ii), [X]t = [W ]t = t;
therefore M is a (Gt)-Brownian motion. Theorem 2 below says that
· Hd−M = · HdM; the additivity of forward integrals and Itˆo inte-
0
0
grals imply that
· Hd−V = · HdV . Consequently it can be deduced
0
0
from Proposition 1 7) a) that the discontinuities of H are not a.s. count-
able. It can even be shown that the discontinuities of H are not negligible
with respect to dV .
3.5 The Brownian case
In this section we will investigate the link between forward and Itˆo integration
with respect to a Brownian motion. In this section (Wt) will denote a (Ft)-
Brownian motion.
The main result of this subsection is the following.
Theorem 2. Let (Ht, t ≥ 0) be an (Ft)-progressively measurable process sat-
T
·
isfying
H2sds < ∞ a.s. Then the Itˆo integral
HsdWs coincides with the
0
0
·
forward integral
Hsd−Ws.
0
Remark 14. 1. We would like to illustrate the advantage of using regulariza-
tion instead of discretization ([20]) through the following example.
Let g be the indicator function of Q ∩ R+.
Let Π = {t0 = 0, t1, · · · , tN = T } be a subdivision of [0, T ] and
I(Π, g, dW )t :=
g(ti) W (ti+1 ∧ t) − W (ti ∧ t) ;
0 ≤ t ≤ T.
i

22
Francesco Russo and Pierre Vallois
We remark that
0 if Π ⊂ R \ Q
I(Π, g, dW )t =
Wt if Π ⊂ Q.
t
Therefore there is no canonical deﬁnition of
gdW through discretiza-
0
tion. This is not surprising since g is not a.e. continuous and so is not
Riemann integrable. On the contrary, integration via regularization seems
t
drastically more adapted to deﬁne
gd−W , for any g ∈ L2([0, T ]), since
0
this integral coincides with the classical Itˆo-Wiener integral.
2. In order to overcome this problem, McShane pointed out an alternative
approximation scheme, see [38] chap. 2 and 3. McShane’s stochastic in-
tegration makes use of the so-called belated partition; the integral is then
even more general than Itˆo’s one, and it includes in particular the function
g above.
Proof. (of Theorem 2) 1) First, suppose in addition that H is a continuous
process. Replacing X by W in (18) one gets
t
1
s
1
t
Hudu dWs =
Hs(Ws+ε − Ws)ds + Rε(t),
(25)
0
ε s−ε
ε 0
where the remainder term Rε(t) is given by (19).
Recall the maximal inequality ([52], chap. I.1): there exists a constant C
such that for any φ ∈ L2([0, T ]),
T
1
v
2
T
sup
φvdv
du ≤ C
φ2vdv.
(26)
0
0<η<1
η (v−η)+
0
2) We claim that (25) may be extended to any progressively measurable
·
process (Ht) satisfying
H2sds < ∞.
0
t
Set Hn
t = n
Hudu for t ≥ 0. It is clear that as n → ∞
t−1/n
• for a.e. t, Hn
t converges to Ht,
·
• (Hn
t ) converges to (Ht) in L2(d < W >) (i.e.
(Hn
s − Hs)2ds goes to 0
0
in the ucp sense).
Since
·
1
s
·
1
s
2
<
Hudu dWs >t=
Hudu ds,
0
ε s−ε
0
ε s−ε
(26) and Lemma 2 imply that (25) and (19) are valid.
3) Letting ε → 0 in (25) and using once more (26), Lemma 2 allows to
conclude the proof of Theorem 2.

Stochastic calculus via regularisation
23
3.6 Substitution formulae
We conclude Section 3 by observing that discretization makes it possible to
integrate non adapted integrands in a context which is covered neither by Sko-
rohod integration theory nor by enlargement of ﬁltrations. A class of examples
is the following.
Let (X(t, x), t ≥ 0, x ∈ Rd) and (Y (t, x), t ≥ 0, x ∈ Rd) be two families
of continuous (Ft) semimartingales depending on a parameter x and (H(t, x),
t ≥ 0, x ∈ Rd) an (Ft) progressively measurable processes depending on x.
Let Z be a FT -measurable r.v., taking its values in Rd.
Under some minimal conditions of Garsia-Rodemich-Rumsey type, see for
instance [49, 50], one has
t
t
H(s, Z) d−X(s, Z) =
H(s, x) dX(s, x)
,
0
0
x=Z
[X(·, Z), Y (·, Z)] = [X(·, x), Y (·, x)]
.
x=Z
The ﬁrst result is useful to prove existence results for SDEs driven by semi-
martingales, with anticipating initial conditions.
It is signiﬁcant to remark that these substitution formulae give rise to
anticipating calculus in a setting which is not covered by Malliavin non-causal
calculus since our integrators may be general semimartingales, while Skorohod
integrals apply essentially to Gaussian integrators or eventually to Poisson
type processes. Note that the usual causal Itˆo calculus does not apply here
since (X(s, Z))s is not a semimartingale (take for instance a r.v. Z which
generates FT .)
4 Calculus for ﬁnite quadratic variation processes
4.1 Stability of the covariation
A basic tool of calculus via regularization is the stability of ﬁnite quadratic
variation processes under C1 transformations.
Proposition 11. Let (X1, X2) be a vector of processes having all its mutual
covariations and f, g ∈ C1(R). Then [f (X1), g(X2)] exists and is given by
t
[f (X1), g(X2)]t =
f (X1s)g (X2s)d[X1, X2]s
0
Proof. By polarization and bilinearity, it suﬃces to consider the case when
X = X1 = X2 and f = g. Using Taylor’s formula, one can write
f (Xs+ε) − f(Xs) = f (Xs)(Xs+ε − Xs) + R(s, ε)(Xs+ε − Xs),
s ≥ 0, ε > 0,

24
Francesco Russo and Pierre Vallois
where R(s, ε) denotes a process which converges in the ucp sense to 0 when
ε → 0. Since f is unifomly continuous on compacts,
(f (Xs+ε) − f(Xs))2 = f (Xs)2(Xs+ε − Xs)2 + R(s, ε)(Xs+ε − Xs)2.
Integrating from 0 to t yields
1
t
(f (X
ε
s+ε) − f (Xs))2ds = I1(t, ε) + I2(t, ε)
0
where
t
I1(t, ε) =
f (Xs)2 (Xs+ε − Xs)2 ds,
0
ε
1
t
I2(t, ε) =
R(s, ε)(X
ε
s+ε − Xs)2ds.
0
Clearly one has
1
T
sup |I2(t, ε)| ≤ sup |R(s, ε)|
(Xs+ε − Xs)2ds.
t≤T
s≤T
ε 0
Since [X] exists, I2(·, ε) ucp
−→ 0. The result will follow if we establish
1
·
·
Y
Y
ε
sdµε(s) ucp
−→
sd[X, X ]s
(27)
0
0
where µ
ds
ε(t) =
t
(X
0 ε
s+ε − Xs)2 and Y is a continuous process. It is not
diﬃcult to verify that a.s., µε(dt) converges to d[X, Y ], when ε → 0; this
ﬁnally implies (27).
4.2 Itˆ
o formulae for ﬁnite quadratic variation processes
Even though all Itˆo formulae that we will consider can be stated in the multi-
dimensional case, see for instance [49], we will only deal here with dimension 1.
Let X = (Xt)
be a continuous process.
t≥0
Proposition 12. Suppose that [X, X] exists and let f ∈ C2(R). Then
·
·
f (X)d−X
and
f (X)d+X
exist.
(28)
0
0
Moreover
a) f (X
t
t) = f (X0) +
t f (X)d X ± 1
f (X
0
2
0
s)d[X, X ]s,
t
1
b) f (Xt) = f(X0) +
f (X)d X ± [f (X), X]t,
0
2

Stochastic calculus via regularisation
25
t
c) f (Xt) = f(X0) +
f (X)d◦X.
0
Proof.
c) follows from b) summing up + and −.
b) follows from a), since Proposition 11 implies that
t
[f (X), X]t =
f (X)d[X, X].
0
The proof of a) and (28) is similar to that of Proposition 11, but with a
second-order Taylor expansion.
The next lemma emphasizes that the existence of a quadratic variation is
closely connected with the existence of some related forward and backward
integrals.
·
Lemma 6. Let X be a continuous process. Then [X, X] exists ⇐⇒
Xd−X
0
·
exists ⇐⇒
Xd+X exists.
0
Proof. Start with the identity
(Xs+ε − Xs)2 = X2s+ε − X2s − 2Xs(Xs+ε − Xs)
(29)
and observe that, when ε → 0,
1
t
(X2
ε
s+ε − X 2
s )ds → X 2
t − X 2
0 .
0
Integrating (29) from 0 to t and dividing by ε easily gives the equivalence
between the ﬁrst two assertions.
The equivalence between the ﬁrst and third ones is similar, replacing ε
with −ε in (29).
Lemma 6 admits the following generalization.
Corollary 5. Let X be a continuous process. The following properties are
equivalent
a) [X, X] exists; ·
b) ∀g ∈ C1,
g(X)d−X exists;
0·
c) ∀g ∈ C1,
g(X)d+X exists.
0

26
Francesco Russo and Pierre Vallois
Proof.
The Itˆo formula stated in Proposition 12 1) implies a) ⇒ b). b) ⇒
a) follows setting g(x) = x and using Lemma 6.
b) ⇔ c) because of Proposition 1 1) which states that
·
·
g(X)d+X =
g(X)d−X + [g(X), X],
0
0
and Proposition 11 saying that [g(X), X] exists.
When X is a semimartingale, the Itˆo formula seen above becomes the
following.
Proposition 13. Let (St)t≥0 be a continuous (Ft)-semimartingale and f a
function in C2(R). One has the following.
1.
t
1
t
f (St) = f(S0) +
f (Su)dSu +
f (Su)d[S, S]u.
0
2 0
2. Let (S0t) be another continuous (Ft)-semimartingale. The following inte-
gration by parts holds:
t
t
StS0t = S0S00 +
SudS0u +
S0udSu + [S, S0]t.
0
0
Proof. We recall that Itˆo and forward integrals coincide, see Proposition 6;
therefore point 1 is a consequence of Proposition 12.
2 stems from the integration by parts formula in Proposition 1 4).
4.3 L´
evy area
In Corollary 5, we have seen that
t g(X)d−X exists when X is a one-
0
dimensional ﬁnite quadratic variation process and g ∈ C1(R).
If X = (X1, X2) is two-dimensional and has all its mutual covariations,
consider g ∈ C1(R2; R2). We naturally deﬁne, if it exists,
t
g(X) · d−X = lim I−(ε, g(X) · dX)(t),
0
ε→0+
where
t
X(s + ε) − X(s)
I−(ε, g(X) · dX)(t) =
g(X)(s) ·
ds;
0 ≤ t ≤ T, (30)
0
ε
and · denotes the scalar product in R2.
With a 2-dimensional Itˆo formula of the same type as in Proposition 12, it
is possible to show that t g(X) · d−X exists if g =
u, where u is a potential
0
of class C2. If g is a general C1(R2) function, one cannot expect in general
that t g(X) · d−X exists.
0

Stochastic calculus via regularisation
27
T. Lyons’ rough paths approach, see for instance [36, 35, 31, 28, 8] has
considered in detail the problem of the existence of integrals of the type
t g(X) · dX. In this theory, the concept of L´evy area plays a signiﬁcant
0
role. Translating this in the present context one would say that the essential
assumption is that X = (X1, X2) has a L´evy area type process. This section
will only make some basic observations on that topic from the perspective of
stochastic calculus via regularization.
Given two classical semimartingales S1, S2, the classical notion of L´evy
area is deﬁned by
t
t
L(S1, S2)t =
S1dS2 −
S2dS1,
0
0
where both integrals are of Itˆo type.
Deﬁnition 11. Given two continuous processes X and Y , we put
t X
L(X, Y )
sYs+ε − Xs+εYs
t = lim
ds.
ε→0+
0
ε
where the limit is understood in the ucp sense. L(X, Y ) is called the L´
evy
area of the processes X and Y .
Remark 15. The following properties are easy to establish.
1. L(X, X) ≡ 0.
2. The L´evy area is an antisymmetric operation, i.e.
L(X, Y ) = −L(Y, X).
Using the approximation of symmetric integral we can easily prove the fol-
lowing.
Proposition 14. · Xd◦Y exists if and only if L(X, Y ) exists. Moreover
0
t
2
Xd◦Y = XtYt − X0Y0 + L(X, Y )t
0
Recalling the convention that an equality among three objects implies that
at least two among the three are deﬁned, we have the following.
t
t
Proposition 15. 1.
L(X, Y )t =
Xd◦Y −
Y d◦X.
0
0
t
t
2.
L(X, Y )t =
Xd−Y −
Y d−X.
0
0

28
Francesco Russo and Pierre Vallois
Proof. 1. From Proposition 14 applied to X, Y and Y, X, and by antisym-
metry of L´evy areas we have
t
2
Xd◦Y = XtYt − X0Y0 + L(X, Y )t,
0
t
2
Y d◦X = XtYt − X0Y0 − L(X, Y )t.
0
Taking the diﬀerence gives 1.
2. follows from the deﬁnition of forward integrals.
Remark 16. If [X, Y ] exists, point 2 of Proposition 15 is a consequence of
point 1 and of Proposition 1 1, 2.
For a real-valued process (Xt)
, Lemma 6 says that
t≥0
·
[X, X] exists ⇔
Xd−X
exists.
0
Given a vector of processes X = (X1, X2) we may ask wether the following
statement is true:
(X1, X2) has all its mutual brackets if and only if
·
Xid−Xj
exists,
0
for i, j = 1, 2. In fact the answer is negative if the two-dimensional process X
does not have a L´evy area.
Remark 17. Suppose that (X1, X2) has all its mutual covariations. Let ∗ stand
for ◦, or −, or +. The following are equivalent.
1. The L´evy area L(X1, X2) exists.
2. · Xid∗Xj exists for any i, j = 1, 2.
0
By Lemma 6, we ﬁrst observe that
Xid◦Xi exists since Xi is a ﬁnite
quadratic variation process. In point 2, equivalence between the three cases
◦,− and + is obvious using Proposition 1 1 2. Equivalence between the exis-
tence of · X1d◦X2 and L(X1, X2) was already established in Proposition 14.
0
5 Weak Dirichlet processes
5.1 Generalities
Weak Dirichlet processes constitute a natural generalization of Dirichlet
processes, which in turn naturally extend semimartingales. Dirichlet processes
have been considered by many authors, see for instance [21, 2].
Let (Ft)t≥0 be a ﬁxed ﬁltration fulﬁlling the usual conditions. In the
present section 5, (Wt) will denote a classical (Ft)-Brownian motion. For
simplicity, we shall stick to the framework of continuous processes.

Stochastic calculus via regularisation
29
Deﬁnition 12. 1. An (Ft)-Dirichlet process is the sum of an (Ft)-local
martingale M and a zero quadratic variation process A.
2. An (Ft)-weak Dirichlet process is the sum of an (Ft)-local martingale
M and a process A such that [A, N ] = 0 for every continuous (Ft)- local
martingale N .
In both cases, we will suppose A0 = 0 a.s.
Remark 18. 1. The process (At) in the latter decomposition is (Ft)-adapted.
2. Any (Ft)-semimartingale is an (Ft)-Dirichlet process.
The statement of the following proposition is essentially contained in [13].
Proposition 16. 1. Any (Ft)-Dirichlet process is an (Ft)-weak Dirichlet
process.
2. The decomposition M + A is unique.
Proof. Point 1 follows from Proposition 1 6).
Concerning point 2, let X be a weak Dirichlet process with decompositions
X = M 1+A1 = M 2+A2. Then 0 = M +A where M = M 1−M 2, A = A1−A2.
We evaluate the covariation of both members against M to obtain
0 = [M ] + [M, A1] − [M, A2] = [M ].
Since M0 = A0 = 0 and M is a local martingale, Corollary 1 gives M = 0.
The class of semimartingales with respect to a given ﬁltration is known
to be stable with respect to C2 transformations, as Proposition 13 implies.
Proposition 11 says that ﬁnite quadratic variation processes are stable under
C1 transformations.
It is possible to show that the class of weak Dirichlet processes with ﬁnite
quadratic variation (as well as Dirichlet processes) is stable with respect to
the same type of transformations. We start with a result which is a slight
improvement (in the continuous case) of a result obtained by [7].
Proposition 17. Let X be a ﬁnite quadratic variation process which is (Ft)-
weak Dirichlet, and f ∈ C1(R). Then f (X) is also weak Dirichlet.
Proof. Let X = M + A be the corresponding decomposition. We express
f (Xt) = Mf + Af where
t
M ft = f(X0) +
f (X)dM,
Aft = f(Xt) − Mft .
0
Let N be a local martingale. We have to show that [f (X) − M f , N ] = 0.
By additivity of the covariation, and the deﬁnition of weak Dirichlet
process, [X, N ] = [M, N ] so that Proposition 11 implies [f (X), N ]t =
t f (X
0
s)d[M, N ]s.
On the other hand, Proposition 7 gives

30
Francesco Russo and Pierre Vallois
t
[M f , N ]t =
f (Xs)d[M, N]s,
0
and the result follows.
Remark 19. 1. If X is an (Ft)- Dirichlet process, it can be proved similarly
that f (X) is an (Ft)- Dirichlet process; see [2] and [51] for details.
2. The class of Lyons-Zheng processes introduced in [51] consitutes a nat-
ural generalization of reversible semimartingales, see Deﬁnition 13. The
authors proved that this class is also stable through C1 transformation.
3. Suppose that (Ft) is the canonical ﬁltration associated with a Brownian
motion W . Then a continuous (Ft)-adapted process D is weak Dirichlet
if and only if D is it is the sum of an (Ft)-local martingale and a process
A such that [A, W ] = 0. See [9], Corollary 3.10.
We also report a Girsanov type theorem established by [7] at least in a
discretization framework.
Proposition 18. Let X = (Xt)
be an (F
t∈[0,T ]
t)-weak Dirichlet process, and
Q a probability equivalent to P on FT . Then X = (Xt)
is an (F
t∈[0,T ]
t)-weak
Dirichlet process with respect to Q.
Proof. We set Dt = dQ | ; D is a positive local martingale.
dP Ft
Let L be the local martingale such that Dt = exp(Lt− 1 [L]
2
t). Let X = M +
A be the corresponding decomposition. It is well-known that ˜
M = M − [M, L]
is a local martingale under Q. So, X is a Q-weak Dirichlet process.
As mentioned earlier, Dirichlet processes are stable with respect to C1
transformations. In applications, in particular to control theory, one often
needs to know the nature of process (u(t, Dt)) where u ∈ C0,1(R+ × R) and
D is a Dirichlet process. The following result was established in [24].
Proposition 19. Let (St) be a continuous (Ft)-weak Dirichlet process with
ﬁnite quadratic variation; let u ∈ C0,1(R+ × R). Then (u(t, St)) is a (Ft)-
weak Dirichlet process.
Remark 20. There is no reason for (u(t, St)) to have a ﬁnite quadratic variation
since the dependence of u on the ﬁrst argument t may be very rough. A fortiori
(u(t, St)) will not be Dirichlet. Consider for instance u only depending on time,
deterministic, with inﬁnite quadratic variation.
Examples of Dirichlet processes (respectively weak Dirichlet processes)
arise directly from classical Brownian motion W .
Example 2. Let f be of class C0(R), u ∈ C0,1(R+ × R).
1. If f is C1, then X = f (W ) is a (Ft)-Dirichlet process.
2. u(t, Wt) is an (Ft)-weak Dirichlet process, but not Dirichlet in general.

Stochastic calculus via regularisation
31
3. f (W ) is not always a Dirichlet process, not even of ﬁnite quadratic vari-
ation as shown by Proposition 20.
The Example and Remark above easily show that the class of (Ft)-
Dirichlet processes strictly includes the class of (Ft)-semimartingales.
More sophisticated examples of weak Dirichlet processes may be found in
the class of the so called Volterra type processes, se e.g. [12, 13]
Example 3. Let (Nt)t≥0 be an (Ft)-local martingale, G : R+ × R+ × Ω −→ R
a continuous random ﬁeld such that G(t, ·) is (Fs)-adapted for each t. Set
t
Xt =
G(t, s)dNs.
0
Then (Xt) is an (Ft)-weak Dirichlet process with decomposition M +A, where
t
Mt =
G(s, s)dNs.
0
Suppose that [G(·, s1); G(·, s2)] exists for any s1, s2. With some additional
technical assumption, one can show that A is a ﬁnite quadratic variation
process with
t
s2
[A]t = 2
[G(·, s1); G(·, s2)] ◦ dMs
◦ dM ;
1
s2
0
0
this iterated Stratonovich integral can be expressed as the sum C1(t) + C2(t)
where
t
C1(t) =
[G(·, s); G(·, s)]d[M ]s,
0
t
s2
C2(t) = 2
[G(·, s1); G(·, s2)]dMs
dM .
1
s2
0
0
Example 4. Take for N a Brownian motion W and G(t, s) = B(t−s)∨0 where
t
t2
B is a Brownian motion independent of W . Then [A] =
(t − s)ds =
.
0
2
One signiﬁcant motivation for considering Dirichlet (respectively weak
Dirichlet) processes comes from the study of generalized diﬀusion processes,
typically solutions of stochastic diﬀerential equations with distributional drift.
Such processes were investigated using stochastic calculus via regulariza-
tion by [18, 19]. We try to express here just a guiding idea. The following
particular case of such equations is motivated by random media modeliza-
tion:
dXt = dWt + b (Xt)dt,
X0 = x0
(31)
where b is a continuous function. Typically, b could be the realization of a
continuous process, independent of W , stopped outside a ﬁnite interval.

32
Francesco Russo and Pierre Vallois
We shall not recall the precise meaning of the solution of (31). In [18, 19]
a rigorous sense is given to a solution (in the distribution laws) and existence
and uniqueness are established for any initial conditions.
Here we shall just attempt to convince the reader that the solution is a
Dirichlet process. For this we deﬁne the real function h of class C1 by
x
h(x) =
e−b(y)dy.
0
We set σ0 = h ◦h−1. We consider the unique solution in law of the equation
dYt = σ0(Yt)dWt, Y0 = h(x0)
which exists because of classical Stroock-Varadhan arguments ([53]); so Y is
clearly a semimartingale, thus a Dirichlet process. The process X = h−1(Y )
is a Dirichlet process since h−1 is of class C1. If b were of class C1, (31) would
be an ordinary stochastic diﬀerential equation, and it could be shown that X
is the unique solution of that equation. In the present case X will still be the
solution of (31), considered as a generalized stochastic diﬀerential equation.
We now consider the case when the drift is time inhomogeneous as follows:
dXt = dWt + ∂xb(t, Xt)dt, X0 = x0
(32)
where b : R+ × R → R is a continuous function of class C1 in time. Then it is
possible to ﬁnd a k : R+ × R → R of class C0,1 such that the solution (Xt) of
(32) can be expressed as (k(t, Yt)) for some semimartingale Y ; so X will be
an (Ft)-weak Dirichlet process. For this and more general situations, see [44].
5.2 Itˆ
o formula under weak smoothness assumptions
In this section, we formulate and prove an Itˆo formula of C1 type. As for
the C2 type Itˆo formula, the next Theorem is stated in the one-dimensional
framework only in spite of its validity in the multidimensional case.
Let (St)
be a semimartingale and f ∈ C2. We recall the classical Itˆo
t≥0
formula, as a particular case of Proposition 13: :
t
1
t
f (St) = f(S0) +
f (Ss)dSs +
f (Ss)d[S, S]s.
0
2 0
Using Proposition 6 and Deﬁnition 10 (Stratonovich integrals), we obtain
t
1
f (St) = f(S0) +
f (Ss)dSs + [f (S), S]t
0
2
(33)
t
= f (S0) +
f (S) ◦ dS.
0

Stochastic calculus via regularisation
33
We observe that in formulae (33), only the ﬁrst derivative of f appears. Be-
sides, we know that f (S) is a Dirichlet process if f ∈ C1(R).
At this point we may ask if formulae (33) remains valid when f is in C1(R)
only; a partial answer will be given in Theorem 3 below.
Deﬁnition 13. Let (St) be a continuous semimartingale; set ˆ
St = ST−t for
t ∈ [0, T ]. S is called a reversible semimartingale if ( ˆ
St)t∈[0,T] is again a
semimartingale.
Theorem 3. ([45]) Let S be a reversible semimartingale indexed by [0, T ] and
f ∈ C1(R). Then one has
t
t
f (St) = f(S0) +
f (S)dS + Rt = f(S0) +
f (S) ◦ dS
0
0
where R = 1 [f (S), S].
2
Remark 21. After the pioneering work of [5], which expressed the remainder
term (Rt) with the help of generalized integral with respect to local time, two
papers appeared: [22] in the case of Brownian motion and [22] and [45] for
multidimensional reversible semimartingales. Later, an incredible amount of
contributions on that topic have been published. We cannot give the precise
content of each paper; a non-exhaustive list is [1, 14, 15, 23, 24, 39, 40].
Among the C1-type Itˆo formulae in the framework of generalized Stratonovich
integral with respect to Lyons-Zheng processes, it is also important to quote
[33, 34, 51].
Example 5. i) Classical (Ft)-Brownian motion W is a reversible semimar-
tigale, see for instance [22, 41, 19]. More precisely ˆ
Wt = WT + βt +
t
ˆ
Ws ds, where β is a (Gt)-Brownian motion and (Gt) is the natural
0 T − s
ﬁltration associated with ˆ
Wt.
ii) Let (Xt) be the solution of the stochastic diﬀerential equation
dXt = σ(t, Xt)dWt + b(t, Xt)dt,
with σ, b : R × R → R Lipschitz with at most linear growth, σ ≥ c > 0.
Then (Xt) is a reversible semimartingale; see for instance [19]. Moreover
if f ∈ W 1,2, it is proved in [19] that (f (X
loc
t)) is an (Ft)-Dirichlet process.
Proof. (of Theorem 3). We use in an essential way the Banach-Steinhaus the-
orem for F -spaces; see for instance [10] chap. 2.1.
Deﬁne two maps T ±
ε
from the F -space C0(R) to the F -space C([0, T ]),
which consists of all continuous processes indexed by [0, T ], by
·
S
T −
s+ε − Ss
ε g =
g(Ss)
ds,
0
ε

34
Francesco Russo and Pierre Vallois
·
S
T +
s − Ss−ε
ε g =
g(Ss)
ds.
0
ε
These operators are linear and continuous. Moreover, for each g ∈ C0 we have
·
lim T −
ε g =
g(S)dS,
ε→0
0
because of Proposition 6 which says that t g(S)dS is also an Itˆo integral.
0
Since ˆ
S is a semimartingale, for the same reasons as above,
T
g( ˆ
S)d− ˆ
S
(34)
T −t
also exists and equals an Itˆo integral.
·
Using Proposition 1 3), it follows that
g(S)d+S also exists.
0
Therefore the Banach-Steinhaus theorem implies that
·
·
g →
g(S)d−S,
g →
g(S)d+S,
0
0
are continuous maps from C0(R) to C([0, T ]); by additivity, so are also
·
g → [g(S), S],
g →
g(S)d◦S.
0
Let f ∈ C1(R), (ρε)ε>0 be a family of molliﬁers converging to the Dirac
measure at zero. We set fε = f
ρε where
denotes convolution. Since fε
is of class C2, by the “smooth” Itˆo formula stated at Proposition 13 and by
Proposition 1 1) and 2), we have
t
1
fε(St) = fε(S0) +
fε(S)dS + [fε(S), S],
0
2
t
fε(St) = fε(S0) +
fε(S)d◦S.
0
Since fε goes to f in C0(R), we can take the limit term by term and
t
1
f (St) = f(S0) +
f (S)dS + [f (S), S],
0
2
(35)
t
f (St) = f(S0) +
f (S)d◦S.
0
Remark 12 says that the latter symmetric integral is in fact a Stratonovich
integral.

Stochastic calculus via regularisation
35
Corollary 6. If (St)
is a reversible semimartingale and g ∈ C0(R), then
t∈[0,T ]
[g(S), S] exists and has zero quadratic variation.
Proof. Let g ∈ C0(R) and let S = M + V be the decomposition of S as a sum
of a local martingale M and a ﬁnite variation process V , such that V0 = 0.
Let f ∈ C1(R) such that f = g. We know that f (S) is a Dirichlet process
with local martingale part
t
M ft = f(S0) +
g(S)dM.
0
Let Af be its zero quadratic variation component. Using Thereom 3, we have
t
1
Aft =
g(S)dV + [g(S), S].
0
2
· g(S)dV has ﬁnite variation, therefore it has zero quadratic variation; since
0
so does also Af , the result follows immediately.
Proposition 20. Let g ∈ C0(R) such that g(W ) is a ﬁnite quadratic variation
process. Then g has bounded variation on compacts.
Proof. Suppose that g(W ) is of ﬁnite quadratic variation. We already know
that W is a reversible semimartingale. By Corollary 6, [W, g(W )] exists
and it is a zero quadratic variation process. Since [W ] exists, we deduce
that (g(W ), W ) has all its mutual covariations. In particular [g(W ), W ] has
bounded variation because of Remark 2. Let f be such that f = g; Theorem 3
implies that f (W ) is a semimartingale. A celebrated result of C
¸ inlar, Jacod,
Protter and Sharpe [6] asserts that f (W ) is a (Ft)-semimartingale if and only
if f is a diﬀerence of two convex functions; this ﬁnally allows to conclude that
g has bounded variation on compacts.
Remark 22. Given two processes X and Y , the covariations [X] and [X, Y ]
may exist even if Y is not of ﬁnite quadratic variation. In particular (X, Y )
may not have all its mutual covariations. For instance, if X has bounded
variation, and Y is any continuous process, then [X, Y ] = 0, see Proposition
1 7 b). A less trivial exemple is provided by X = W , Y = g(W ) where g is
continuous but not of bounded variation, see Proposition 20.
Remark 23. ([22]). When S is a Brownian motion, Theorem 3 and Corollary
6 are in fact respectively valid for f ∈ W 1,2(R) and g ∈ L2 (R).
loc
loc
6 Final remarks
We conclude this paper with some considerations about calculus related to
processes having no quadratic variation. On this, the reader can consult
[13, 27, 26]. In [13] one deﬁnes a notion of n− covariation [X1, . . . , Xn] of
n processes X1, . . . , Xn and the n-variation of a process X.
We recall some basic signiﬁcant results related to those papers.

36
Francesco Russo and Pierre Vallois
1. For a process X having a 3-variation, it is possible to write an Itˆo formula
of the type
t
1
t
f (Xt) = f(X0) +
f (Xs)d◦Xs −
f (3)(Xs)d[X, X, X]s.
0
12 0
Moreover one-dimensional stochastic diﬀerential equations driven by a
strong 3-variation were considered in [13].
2. Let B = BH be a fractional Brownian motion with Hurst index H > 16
and f a function of class C6. It is shown in [27, 26] that
t
f (Bt) = f(B0) +
f (B)d◦B.
0
3. Using more sophisticated integrals via regularization, other types of Itˆo
formulae can be written for any H in ]0, 1[; see [26].
4. In [25], it is shown that stochastic calculus via regularization is almost
pathwise. Suppose for instance that X is a semimartingale or a fractional
Brownian motion, with Hurst index H > 1 ; then its quadratic variation
2
[X] is a limit of C(ε, X, X) not only ucp as in (5), but also uniformly a.s.
Similarly, if X is semimartingale and Y is a suitable integrand, the Itˆo
integral · Y dX is approximated by I−(ε, Y, dX) not only ucp as in (2),
0
but also uniformly a.s.
Acknowledgement. We wish to thank an anonymous referee and the R´edac-
tion of the S´eminaire for their careful reading of a preliminary version, which
motivated us to improve it considerably.
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