Existence of L\'evy's area and pathwise integration

Abstract

Rough path analysis can be developed using the concept of controlled paths
and with respect to a topology in which besides the uniform distance L\'evy's
area plays a role. For vectors of rough paths we investigate the relationship
between the property of being controlled and the existence of associated L\'evy
areas. Given a path which is controlled by another one, the L\'evy area in the
It\^o or Stratonovitch sense can be defined pathwise, eventually provided the
existence of quadratic variation along a sequence of partitions is guaranteed.
This leads us to a study of the pathwise change of variable (It\^o) formula in
the spirit of F\"ollmer, from the perspective of controlled paths.

Full text

arXiv:1404.3645v1 [math.PR] 14 Apr 2014
Existence of L´evy’s area and pathwise integration
Peter Imkeller and David J. Pr¨omel1
Humboldt-Universit¨at zu Berlin
Institut f¨ur Mathematik
May 7, 2014
Abstract
Rough path analysis can be developed using the concept of controlled paths and with
respect to a topology in which besides the uniform distance L´evy’s area plays a role.
For vectors of rough paths we investigate the relationship between the property of being
controlled and the existence of associated L´evy areas. Given a path which is controlled
by another one, the L´evy area in the Itˆo or Stratonovitch sense can be defined pathwise,
eventually provided the existence of quadratic variation along a sequence of partitions is
guaranteed. This leads us to a study of the pathwise change of variable (Itˆo) formula in
the spirit of F¨ollmer, from the perspective of controlled paths.
Key words: Controlled path, F¨ollmer integration, Itˆo’s formula, Levy’s area, rough path,
Stratonovich integral.
MSC 2010 Classification: 26A42, 60H05.
1 Introduction
The theory of rough paths (see [LCL07], [Lej09] or [FH13]) has established an analytical
frame in which stochastic differential- and integral calculus beyond Young’s classical notions
is traced back to properties of the trajectories of processes involved without reference to a
particular probability measure. For instance, in the simplest non-trivial setting it provides a
topology on the set of continuous functions enhanced with an ”area”, with respect to which the
(Itˆo) map associating the trajectories of a solution process of a stochastic differential equation
driven by trajectories of continuous martingale is continuous. In this topology, convergence of
a sequence of functions Xn= (X1,n,··· , Xd,n)n∈Ndefined on the time interval [0, T ] involves
besides uniform convergence also the convergence of the L´evy areas associated to the vector
of trajectories, formally given by
Li,j,n
t=Zt
0
(Xi,n
sdXj,n
s−Xj,n
sdXi,n
s),1≤i, j ≤d, t ∈[0, T ].
1We are grateful to Randolf Altmeyer and Nicolas Perkowski for helpful comments and discussions on the
subject matter. D.J.P. was financially supported by a Ph.D. scholarship of the DFG Research Training Group
1845 ”Stochastic Analysis with Applications in Biology, Finance and Physics”.
Email: imkeller@math.hu-berlin.de
Email: proemel@math.hu-berlin.de
1

In probability theory the concept of L´evy’s area was already studied in the 1940s. It
was first introduced by P. L´evy in [L´ev40] for a two dimensional Brownian motion (B1, B2).
For time Tfixed, and any trajectory of the process it is defined as the area enclosed by the
trajectory (B1, B 2) and the chord given by the straight line from (0,0) to (B1
T, B2
T), and may
be expressed formally by
1
2ZT
0
B1
tdB2
t−ZT
0
B2
tdB1
t,
provided the integrals make sense.
More recently, an alternative calculus with a more (Fourier) analytic touch has been de-
signed (see [GIP14], [Per14]) in which an older idea by Gubinelli [Gub04] is further developed.
It is based on the concept of controlled paths. In this calculus, rough integrals are described
in terms of Fourier series for instance in the Haar-Schauder wavelet, and are seen to decom-
pose into different parts, one of them representing L´evy’s area. The existence of a stochastic
integral in this approach is seen to be linked to the existence of the corresponding L´evy area,
and both can be approximated along a Schauder development in which H¨older functions are
limits of their finite degree Schauder expansions. In its simplest (one-dimensional) form a
path of bounded variation Yon [0, T ] is controlled by another path Xof bounded variation
on [0, T ], if the associated signed measures µX, µYon the Borel sets of [0, T ] satisfy that µY
is absolutely continuous with respect to µX. In its version relevant here two rough (vector
valued) functions Xand Yon [0, T ] are considered, both with finite p-variation for some
p≥1. In the simplest setting, Yis controlled by Xif there exists a rough function Y′of
finite p-variation such that the first order Taylor expansion errors
RY
s,t =Yt−Ys−Y′
s(Xt−Xs)
are bounded in r-norm, i.e. P[s,t]∈π|RY
s,t|ris bounded over all possible partitions πof [0, T ].
Here 1
r=2
p.Since for a path XH¨older continuity of order 1
pis closely related to finite p-
variation, the control relation can be seen as expressing a type of fractional Taylor expansion
of first order: the first order Taylor expansion error of Ywith respect to X- both of H¨older
order 1
pand ”derivative” Y′- is of double H¨older order 2
p.In its para-controlled refinement
as developed by Gubinelli et. al. in [GIP13] this notion has been seen to give an alternative
approach to classical rough path analysis and is essentially suitable for the application to
singular PDEs. In the comparison of the two approaches, to make the Itˆo map continuous,
information stored in the Levy areas of vector paths has to be replaced by information con-
veyed by path control or vice versa. This raises the problem about the relationship between
the existence of L´evy’s area and the control relationship between vector trajectories or the
components of such.
In the first part of this paper (section 2) we shall deal with this question. In fact we shall
show that for a vector Xof functions a particular version of control which we will call self-
control (roughly, for pairs of components (Xi, Xj) one has to control the other one) is sufficient
for the existence of the L´evy areas, and thus for the existence of related pathwise integrals.
We shall also show by an example that missing control can lead to the failure of possessing a
L´evy area. In the second part (section 3) we shall study the question how control concepts
and the existence of different kinds of integrals (Itˆo type, Stratonovich type) are related, and
in particular in which way control leads to versions of F¨ollmer’s pathwise Itˆo formula. We will
decompose Riemann approximations of different versions of integrals into a symmetric and
2

an antisymmetric component. While the convergence of the antisymmetric component, very
closely related to L´evy areas, does not require to go along prescribed sequences of partitions,
this will be the case for the symmetric part. In F¨ollmer’s approach the entire integral is
constructed along a given sequence of partitions, and the existence of quadratic variation
along this sequence is assumed. We shall show that for the classical Stratonovich integral just
the antisymmetric Riemann sums have to converge, while for more general Stratonovich or
Itˆo type integrals the existence of limits for the symmetric part has to be guaranteed along
fixed sequences of partitions, as in F¨ollmer’s approach. The versions of pathwise Itˆo’s formula
derived therefrom will exhibit this variety of integrals.
2 L´evy’s area and controlled paths
We know that both the control of a (one-dimensional) path X1with respect to another
path X2, as well as the existence of L´evy’s area for (X1, X2) ([Lyo98]) entails the existence
of the rough path integral of X1with respect to X2. This raises the question about the
strengths of the hypotheses leading to the existence of the integral. This question will be
answered here. We will show that control entails the existence of L´evy’s area. The analysis
we present, as usual, is based on d−dimensional rough paths, and corresponding notions of
areas. For a continuous path X: [0, T ]→Rd, say X= (X1, ..., Xd)∗, we recall that L´evy’s
area L(X) = (Li,j (X))i,j is given by
L(X)i,j := ZT
0
Xi
tdXj
t−ZT
0
Xj
tdXi
t,1≤i, j ≤d,
where X∗denotes the transpose of the vector X, if the respective integrals exist. There are
pairs of H¨older continuous paths X1and X2for which L´evy’s area does not exist (see example
7 below). To answer this question, we need the basic setup of rough path analysis, starting
with the notion of power variation.
Apartition π:= {[ti−1, ti]|i= 1, ..., N}of an interval [0, T ] is a family of essentially
disjoint intervals such that SN
i=1[ti−1, ti] = [0, T ]. For any 1 ≤p < ∞, a continuous function
X: [0, T ]→Rdis of finite p-variation if
||X||p:= sup
π∈P X
[s,t]∈π
|Xs,t|p
1
p
<∞,
where the supremum is taken over the set Pof partitions of [0, T ] and Xs,t := Xt−Xsfor
s, t ∈[0, T ], s≤t. We write Vp([0, T ],Rd) for the set (linear space) of continuous functions of
finite p-variation. Let, more generally, R: [0, T ]2→Rd×dbe a continuous function, 1 ≤r <
∞. In this case we consider the functional
||R||r:= sup
π∈P X
[s,t]∈π
|Rs,t|r1
r
.
An equivalent way to characterize the property of finite p-variation is by the existence of a
control function. Denoting by ∆T:= {(s, t)∈[0, T ]2|0≤s≤t≤T}, we call a continuous
3

function ω: ∆T→R+vanishing on the diagonal control function if it is superadditive, i.e. if
for (s, u, t)∈[0, T ]3such that 0 ≤s≤u≤t≤T
ω(s, u) + ω(u, t)≤ω(s, t).
Note that a function is of finite p-variation if and only if there exists a control function ω
such that |Xs,t|p≤ω(s, t) for (s, t)∈∆T. For a more detailed discussion of p-variation and
control functions see Chapter 1.2 in [LCL07].
A fundamental insight due to Gubinelli [Gub04] was that an integral RY dX exists if “Y
looks like Xin the small scale”. This leads to the concept of controlled paths which we recall
in its general form.
Definition 1. Let p, q , r ∈R+be such that 2/p + 1/q > 1 and 1/r = 1/p +1/q. Suppose X∈
Vp([0, T ],Rd). We call Y∈ Vp([0, T ],Rd)controlled by Xif there exists Y′∈ Vq([0, T ],Rd×d)
such that the remainder term RYgiven by the relation Ys,t =Y′
sXs,t +RY
s,t satisfies ||RY||r<
∞. In this case we write Y∈Cq
X, and call Y′Gubinelli derivative.
See Theorem 1 in [Gub04] for the case of H¨older continuous paths, or Theorem 24 in
[PP13] for precise existence results of RY dX. Let us now modify this concept to a notion of
control of a path by itself.
Definition 2. Let p, q, r ∈R+be such that 2/p + 1/q > 1 and 1/r = 1/p + 1/q. We call
X∈ Vp([0, T ],Rd)self-controlled if we have Xi∈Cq
Xjor Xj∈Cq
Xifor all 1 ≤i, j ≤dwith
i6=j.
With this notion we are now able to deal with the main task of this section, the construc-
tion of the L´evy area of a self-controlled path X. In fact, the integrals arising in L´evy’s area
will be obtained via left-point Riemann sums as
L(X)i,j =ZT
0
Xi
tdXj
t−ZT
0
Xj
tdXi
t:= lim
|π|→0X
[s,t]∈π
(Xi
sXj
s,t −Xj
sXi
s,t),(1)
for 1 ≤i, j ≤d, where |π|denotes the mesh of a partition π. Our approach uses the abstract
version of classical ideas due to Young [You36] comprised in the so-called sewing lemma.
Lemma 3. [Corollary 2.3, Corollary 2.4 in [FD06]] Let Ξ : ∆T→Rdbe a continuous function
and K > 0some constant. Assume that there exist a control function ωand a constant θ > 1
such that for all (s, u, t)∈[0, T ]3with 0≤s≤u≤t≤Twe have
|Ξs,t −Ξs,u −Ξu,t| ≤ K ω(s, t)θ.(2)
Then there exists a unique function Φ: [0, T ]→Rdsuch that Φ(0) = 0 and
|Φ(t)−Φ(s)−Ξs,t| ≤ C(θ)ω(s, t)θ,0≤s≤t≤T ,
where C(θ) := K(1 −21−θ)−1, and we have
lim
|π(s,t)|→0X
[u,v]∈π(s,t)
Ξu,v = Φ(t)−Φ(s),
where π(s, t)denotes a partition of [s, t].
4

With this tool we now derive the existence of L´evy’s area for self-controlled paths of finite
p-variation with p≥1.
Theorem 4. Let 1≤p < ∞and suppose that X∈ Vp([0, T ],Rd)is self-controlled, then
L´evy’s area as defined in (1) exists.
Proof. Let X∈ Vp([0, T ],Rd) for 1 ≤p < ∞be self-controlled and fix 1 ≤i, j ≤d, i 6=j.
We may assume without loss of generality that Xi∈Cq
Xj, i.e. Xi
s,t =X′
s(i, j)Xj
s,t +Ri,j
s,t and
||X′(i, j)||q,||Ri,j ||r<∞. In order to apply Lemma 3, we set Ξi,j
s,t := Xi
sXj
s,t −Xj
sXi
s,t for
(s, t)∈∆Tand observe that for (s, u, t)∈[0, T ]3with 0 ≤s≤u≤t≤Twe have
Ξi,j
s,t −Ξi,j
s,u −Ξi,j
u,t =Xi
sXj
s,t −Xj
sXi
s,t −Xi
sXj
s,u +Xj
sXi
s,u −Xi
uXj
u,t +Xj
uXi
u,t
=Xi
s,uXj
u,t −Xj
s,uXi
u,t
= (X′
s(i, j)Xj
s,u +Ri,j
s,u)Xj
u,t −Xj
s,u(X′
u(i, j)Xj
u,t +Ri,j
u,t)
=Ri,j
s,uXj
u,t −Xj
s,uRi,j
u,t + (X′
s(i, j)−X′
u(i, j))Xj
s,uXj
u,t.
Since the finite sum of control functions is again a control function, we can choose the same
control function ωfor X, X′(i, j ) and Ri,j, and therefore we get
|Ξi,j
s,t −Ξi,j
s,u −Ξi,j
u,t| ≤ ω(s, t)
1
p+1
r+ω(s, t)
1
p+1
r+ω(s, t)
2
p+1
q≤3ω(s, t)θ
with θ:= 2
p+1
q>1.
We will next show that Riemann sums with arbitrary choices of base points for the inte-
grand functions lead to the same L`evy area as just constructed.
Lemma 5. Let X∈ Vp([0, T ],Rd)for some 1≤p < ∞. Suppose Xis self-controlled. Denote
by s′∈[s, t]an arbitrary point chosen in a partition interval [s, t]∈π. Then L´evy’s area from
the preceding theorem is also given by
L(X)i,j = lim
|π|→0X
[s,t]∈π
(Xi
s′Xj
s,t −Xj
s′Xi
s,t),1≤i, j ≤d.
Proof. From Theorem 4 we already know that the left-point Riemann sums converge. Hence,
we only need to show that
X
[s,t]∈πn
(Xi
sXj
s,t −Xj
sXi
s,t)−X
[s,t]∈πn
(Xi
s′Xj
s,t −Xj
s′Xi
s,t) (3)
tends to zero along every sequence of partitions (πn) such that the mesh |πn|converges to
zero. Indeed, we may write for a partition interval [s, t]
Xi
sXj
s,t −Xj
sXi
s,t −(Xi
s′Xj
s,t −Xj
s′Xi
s,t) = −Xi
s,s′Xj
s,t +Xj
s,s′Xi
s,t
=−(X′
s(i, j)Xj
s,s′+Ri,j
s,s′)Xj
s,t +Xj
s,s′(X′
s(i, j)Xj
s,t +Ri,j
s,t)
=−Ri,j
s,s′Xj
s,t +Xj
s,s′Ri,j
s,t.
Taking again the same control function ωfor Xand Ri,j , we estimate
|Xi
sXj
s,t −Xj
sXi
s,t −(Xi
s′Xj
s,t −Xj
s′Xi
s,t)|=| − Ri,j
s,s′Xj
s,t +Xj
s,s′Ri,j
s,t| ≤ 2ω(s, t)θ
5

with θ:= 2
p+1
p>1. Recalling the superadditivity of ω, we get for n∈N



X
[s,t]∈πn
(Xi
s,s′Xj
s,t −Xj
s,s′Xi
s,t)



≤X
[s,t]∈πn
ω(s, t)θ≤max
[s,t]∈πn
ω(s, t)θ−1ω(0, T ),
which means that (3) tends to zero as n→ ∞.
Example 6. Let (Bt;t∈[0, T ]) be a standard Brownian motion on a probability space
(Ω,F,P) and f∈C2(R,R) a twice continuously differentiable function. The trajectories of
Bare of finite p-variation for all p > 2 outside a null set Nand thus we can deduce from
Theorem 4 that L´evy’s area of (B , f(B)) exists outside the same null set N.
The following example illustrates that for p≥2 things are essentially different. It will in
particular show that in this case self-control of a path is necessary for the existence of L´evy’s
area.
Example 7. Let us consider the function X: [−1,1] →R2with components given by
X1
t:=
∞
X
k=1
aksin(2kπt) and X2
t:=
∞
X
k=1
akcos(2kπt),
where ak:= 2−αk and α∈[0,1]. For m∈Nlet Xm: [−1,1] →R2be the mth finite partial
sum of the infinite series with components
X1,m
t:=
m
X
k=1
aksin(2kπt) and X2,m
t:=
m
X
k=1
akcos(2kπt),
for t∈[−1,1]. These functions are α-H¨older continuous uniformly in m. Indeed, let s, t ∈
[−1,1] and choose k∈Nsuch that 2−k−1≤ |s−t| ≤ 2−k. Then we can estimate as follows
|X1,m
t−X1,m
s|=



m
X
l=1
al(sin(2lπt)−sin(2lπs))



=



m
X
k=1
al2 cos(2l−1π(s+t)) sin(2l−1π(s−t))



= 2
k
X
l=1
|al|| sin(2l−1π(s−t))|+ 2
∞
X
l=1
|al|
≤2
k
X
l=1
|al|2l−1π|s−t|+ 2
∞
X
l=k+1
|al|
≤
k
X
l=1
2l−αlπ|s−t|+ 2−α(k+1)+1 1
1−2−α
≤2(k+1)(1−α)−1
21−α−1π|s−t|+21−α
1−2−α|s−t|α
≤2(k+1)(1−α)−1
21−α−1π2−k(1−α)|s−t|α+21−α
1−2−α|s−t|α≤C|s−t|α.
6

for some constant C > 0 independent of m∈N. Analogously, we can get the α-H¨older
continuity of X2,m . Furthermore, it can be seen with the same estimate that (Xm) converges
uniformly to Xand thus also in α-H¨older topology. The limit function Xis not β-H¨older
continuous for every β > α. In order to see this, choose s= 0 and t=tn= 2−nfor n∈N
and observe that
|X1
tn−X1
0|
|tn−0|β=
n−1
X
k=1
2−αk+βn sin(2k−nπ)≥2(β−α)n+α,
which obviously tends to infinity with n. Since α-H¨older continuity is obviously related to
finite 1
α-variation, we can conclude that X∈ V 1
α([−1,1],R2),and X6∈ Vγ([−1,1],R2) for
γ < 1
α.Let us now show that Xpossesses no L´evy area. For this purpose, fix α∈[0,1] and
m∈N. Then L´evy’s area for Xmis given by
Z1
−1
X1,m
sdX2,m
s−Z1
−1
X2,m
sdX1,m
s
=−
m
X
k,l=1
akalZ1
−1sin(2kπs) sin(2lπs)2lπ+ cos(2lπs) cos(2kπs)2kπds
=−
m
X
k,l=1
akal2lπZ1
−1
1
2(cos((2k−2l)πs)−cos((2k+ 2l)πs)) ds
+ 2kπZ1
−1
(cos((2k−2l)πs) + cos((2k+ 2l)πs)) ds
=−2
m
X
k=1
a2
k2kπ=−2
m
X
k=1
2(1−2α)kπ.
This quantity diverges as mtends to infinity for 1
α≥2. Since (Xm) converges to Xin
the α-H¨older topology, we can use this result to choose partition sequences of [−1,1] along
which Riemann sums approximating the L´evy area of Xdiverge as well. This shows that
Xpossesses no L´eva area. In return Theorem 4 implies that Xcannot be self-controlled.
However, it is not to hard to see directly that no regularity is gained by controlling X1with
X2. For this purpose, note that for −1≤s≤t≤1, and 0 6=X′
s∈R
|X1
s,t −X′
sX2
s,t|=



∞
X
k=1
ak[(sin(2kπt)−sin(2kπs)) −X′
s(cos(2kπt)−cos(2kπs))]



=



2
∞
X
k=1
aksin(2k−1π(s−t)) cos(2k−1π(s+t)) + X′
ssin(2k−1π(s+t)) sin(2k−1π(s−t))



=



2
∞
X
k=1
aksin(2k−1π(s−t))p1 + (X′
s)2sin(2k−1π(s+t) + arctan((X′
s)−1))



.
Let us now consider the H¨older regularity for s= 0. First, assume X′
0>0, and take t= 2−n
to obtain
|X1
0,2n−X′
0X2
0,2n|
2−βn = 2βn 



2
n
X
k=1
aksin(2k−1−nπ)q1 + (X′
0)2sin(2k−1−nπ+ arctan((X′
0)−1))



≥2(β−α)nsin π
2+ arctan((X′
0)−1).
7

For X′
0<0 the same estimates work for tn=−2ninstead. Therefore, the H¨older regularity
at 0 cannot be better than αand in particular Xcannot be self-controlled for 1
α>2.
The sewing lemma (Lemma 3) not only guarantees the existence of L`evy’s area. As we
will now show, it also allows to conclude that the area is regular as a function of time.
Corollary 8. Let Ξ : ∆T→Rdbe a continuous function and K > 0some constant. Assume
that there exist a control function ω, a constant θ > 1, and r > 1such that
|Ξs,t|r≤ω(s, t)and |Ξs,t −Ξs,u −Ξu,t | ≤ Kω(s, t)θ,0≤s≤u≤t≤T .
Then there exists a unique continuous function Φ: [0, T ]→Rdsuch that Φ(0) = 0 and
|Φ(t)−Φ(s)−Ξs,t| ≤ C(θ)ω(s, t)θ,0≤s≤t≤T ,
with C(θ)from Lemma 3. Φalso satisfies for (s, t)∈∆T|Φ(t)−Φ(s)|r≤C2(r, θ , ω)ω(s, t)
with some constant C2(r, θ , ω)>0.
Proof. Under the given assumptions, Lemma 3 provides a function Φ: [0, T ]→Rdsuch that
Φ(0) = 0 and |Φ(t)−Φ(s)−Ξs,t | ≤ C(θ)ω(s, t)θfor (s, t)∈∆T. In order to prove the
r-variation inequality for Φ, we estimate for (s, t)∈∆T
|Φ(t)−Φ(s)|r≤|Ξs,t|+C(θ)ω(s, t)θr
≤2r−1|Ξs,t|r+ 2r−1C(θ)rω(s, t)rθ ≤2r−1+ 2r−1C(θ)rω(0, T )rθ −1ω(s, t).
Remark 9. For consistency we state Corollary 8 only for a continuous function Ξ: ∆T→Rd.
Yet, it still holds true without the continuity assumption and for a general Banach space
replacing Rd, since the sewing lemma also holds under these weaker conditions. See Theorem
1 and Remark 3 in [FDM08].
For instance, Corollary 8 implies that for p < 2 the mapping
L:Vp([0, T ],R2)→ Vp([0, T ],R2),
X7→ Lt(X) := lim
|π(0,t)|→0X
[u,v]∈π(0,t)
(X1
uX2
u,v −X2
uX1
u,v)
is locally Lipschitz continuous since it satisfies the estimate
||L·(X)||r≤C2(r, θ, ||X||p)||X||p.(4)
3 F¨ollmer integration
In his well-known paper F¨ollmer [F¨ol79] considered one dimensional pathwise integrals. He
was able to give a pathwise meaning to the limit
ZT
0
DF(Xt) dπnXt:= lim
n→∞ X
[s,t]∈πn
hDF(Xs), Xs,ti,
8

provided F∈C2(Rd,R). His starting point was the hypothesis that quadratic variation of
X∈C([0, T ],Rd) exists along a sequence of partitions (πn)n∈Nwhose mesh tends to zero.
Here h·,·i denotes the usual inner product on Rd. This is today known as F¨ollmer integration.
As indicated and discussed below, this construction of an integral depends strongly on the
chosen sequence of partitions (πn)n∈N.
Before coming back to an approach of F¨ollmer’s integral, we shall construct a Stratonovich
type integral, thereby discussing the problem of dependence on a chosen sequence of partitions.
As in the previous section, our approach is based on the notion of controlled paths. This will
also lead us on a route which does not require the existence of iterated integrals as in the
classical rough paths approach. We fix a γ∈[0,1], to discuss Stratonovich limits for Riemann
sums where integrands are taken at convex combinations γXs+ (1 −γ)Xtof the values of
Xat the extremes of a partition interval [s, t]. We start by decomposing these sums into
symmetric and antisymmetric parts. For p, q ∈[1,∞), X, Y ∈ Vp([0, T ],Rd) , Y∈Cq
Xwe
have
γ-ZT
0
YtdXt:= lim
|π|→0X
[s,t]∈π
hYs+γYs,t , Xs,ti
=1
2lim
|π|→0X
[s,t]∈π
(hYs+γYs,t , Xs,t +hXs+γXs,t, Ys,t i)
+1
2lim
|π|→0X
[s,t]∈π
(hYs+γYs,t , Xs,ti − hXs+γXs,t , Ys,ti)
=1
2γ-ZT
0
YtdXt+γ-ZT
0
XtdYt+1
2γ-ZT
0
YtdXt−γ-ZT
0
XtdYt
=1
2SγhX, Y i+1
2AγhX, Y i.(5)
Note that γ= 0 corresponds to the classical Itˆo integral and γ=1
2to the classical Stratonovich
integral.
If the variation orders of Xand Yfulfil 1/p + 1/q > 1, we are in the framework of Young’s
integration theory. Below 1, either the existence of the rough path or control is needed. To
illustrate this, we go back to Example 7.
Example 10. Let X= (X1, X 2) be given according to Example 7. In this case, we have
seen that X1and X2are of finite 1
α-variation. With decomposition (5), we see that
1
2-Z1
0
X2
tdX1
t:= lim
|π|→0X
[s,t]∈π
hX2
s+1
2X2
s,t, X 1
s,ti=1
2S1
2hX1, X2i+1
2A1
2hX1, X2i
=1
2lim
|π|→0X
[s,t]∈π
(hX2
s+X2
t, X1
t−X1
si+hX1
s+X1
t, X2
t−X2
si) + 1
2L1,2(X)
=1
2lim
|π|→0X
[s,t]∈π
hX1, X2is,t +1
2L1,2(X)
=1
2(X1
1X2
1−X1
0X2
0) + 1
2L1,2(X).
9

Therefore, the integral exists if and only if L´evy’s area exists, which is not the case for instance
if α=1
2.So beyond Young’s theory, the existence of the Stratonovich-1
2integral is closely
linked to the existence of L´evy’s area.
Using a suitable control concept, we will next construct the Stratonovich integral described
above, but not just with restriction to a particular sequence of partitions. This time, due to
the fact that for twice continuously differentiable Fthe Hessian matrix D2Fis symmetric,
the usual concept of controlled paths is sufficient.
Theorem 11. Let γ∈[0,1],X∈ Vp([0, T ],Rd)and Y∈Cq
X. If Y′
tis a symmetric matrix
for all t∈[0, T ], then the antisymmetric part
AγhX, Y i:= lim
|π|→0X
[s,t]∈πhYs+γYs,t , Xs,ti − hXs+γXs,t , Ys,ti,(6)
exists and satisfies
AγhX, Y i=AhX, Y i:= lim
|π|→0X
[s,t]∈πhYs′, Xs,ti − hXs′, Ys,t i
for every choice of points s′∈[s, t]∈π.
Proof. It is easy to verify that by definition the antisymmetric part, if it exists as a limit of
the Riemann sums considered, has to satisfy the second formula of the claim at least with the
choice s′=s, for all intervals [s, t] belonging to a partition. To prove that this limit exists,
we use Lemma 3. For this purpose, we set Ξs,t := hYs, Xs,t i − hXs, Ys,tifor (s, t)∈∆T. Since
Yis controlled by X, we obtain
Ξs,t −Ξs,u −Ξu,t =hYs, Xs,ti − hXs, Ys,ti − hYs, Xs,u i+hXs, Ys,ui − hYu, Xu,ti+hXu, Yu,t i
=hYu,t, Xs,u i − hXu,t, Ys,ui
=hY′
uXu,t +RY
u,t, Xs,u i − hXu,t, Y ′
sXs,u +RY
s,ui
=hRY
u,t, Xs,u i − hXu,t, RY
s,ui+hY′
uXu,t, Xs,u i − hXu,t, Y ′
sXs,ui
=hRY
u,t, Xs,u i − hXu,t, RY
s,ui+hXu,t , Y ′
uXs,u −Y′
sXs,ui
for 0 ≤s < u < t ≤T, where we used hY′
uXu,t, Xs,u i=hXu,t, Y ′
uXs,uiin the last line thanks
to symmetry. With the same control ωfor all functions involved as above, this gives
|Ξs,t −Ξs,u −Ξu,t| ≤ ω(s, t)
1
p+1
r+ω(s, t)
1
p+1
r+ω(s, t)
2
p+1
q≤3ω(s, t)θ
with θ:= 2
p+1
q>1. So from Lemma 3 we conclude that the left-point Riemann sums
converge. It remains to show that
X
[s,t]∈πnhYs, Xs,ti − hXs, Ys,t i−X
[s,t]∈πnhYs′, Xs,ti − hXs′, Ys,ti(7)
tends to zero along every sequence of partitions (πn) such that the mesh |πn|converges to
zero. Applying the symmetry of Y′, we get
hYs′, Xs,ti − hXs′, Ys,t i−hYs, Xs,ti − hXs, Ys,ti=hYs,s′, Xs,ti − hXs,s′, Ys,t i
=hY′
sXs,s′+RY
s,s′, Xs,ti − hXs,s′, Y ′
sXs,t +RY
s,ti
=hRY
s,s′, Xs,ti − hXs,s′, RY
s,ti,
10

and thus

hYs, Xs,ti − hXs, Ys,t i − hYs′, Xs,ti − hXs′, Ys,t i
≤ω(s, t)θ
with θ:= 1
p+1
r>1, where we choose the same control function ωfor Xand RY. Therefore,
the properties of ωimply



X
[s,t]∈πnhYs,s′, Xs,ti − hXs,s′, Ys,ti



≤X
[s,t]∈πn
ω(s, t)θ≤max
[s,t]∈πn
ω(s, t)θ−1ω(0, T ),
which means that (7) tends to zero as |πn|tends to zero.
Remark 12. 1. The proof of Theorem 11 analogously works under the assumption that X
is controlled by Yand X′
tis a symmetric matrix for all t∈[0, T ].
2. If Yis controlled by Xand Y′
tis an antisymmetric matrix for all t∈[0, T ], then an
analogous result to Theorem 11 holds true for the symmetric part SγhX, Y i.
In case γ=1
2as in the example above, the symmetric part simplifies considerably, and
therefore the preceding Theorem will already imply the existence of the 1
2-Stratonovich inte-
gral.
Corollary 13. Let X∈ V p([0, T ],Rd),Y∈Cq
Xand suppose Y′
tis a symmetric matrix for all
t∈[0, T ]. Then, the Stratonovich integral
ZT
0
Yt◦dXt:= lim
|π|→0X
[s,t]∈π
hYs+1
2Ys,t, Xs,t i(8)
exists and satisfies
1
2-ZT
0
YtdXt=ZT
0
Yt◦dXt=1
2hYT, XTi − hY0, X0i+1
2AhX, Y i.
Proof. By equation (5) we may separately deal with the symmetric part S1
2hX, Y iand the
antisymmetric part A1
2hX, Y iof the integral 1
2-RT
0YtdXt. The existence of the antisymmetric
part A1
2hX, Y ifollows from Theorem 11. For the symmetric part, note that as in Example 7
for (s, t)∈∆T
2hYs+1
2Ys,t, Xs,t i+hXs+1
2Xs,t, Ys,t i=hYs+Yt, Xs,ti+hXs+Xt, Ys,t i
=hYs, Xti − hYs, Xsi+hYt, Xti − hYt, Xsi+hXs, Yti − hXs, Ysi+hXt, Yti − hXt, Ysi
= 2hY, X is,t.
Therefore, S1
2hX, Y iis given by
S1
2hX, Y i= lim
|π|→0X
[s,t]∈πhYs+1
2Ys,t, Xs,t i+hXs+1
2Xs,t, Ys,t i=hYT, XTi − hX0, Y0i.(9)
11

The treatment of γ-Stratonovich integrals above has shown that the corresponding asym-
metric component can be treated by means of the concept of path control. In the case γ6=1
2,
a symmetric term is left to consider. This does not seem to be possible by means of the
ideas used for the asymmetric component. And this brings us back to F¨ollmer’s approach.
Our treatment of the symmetric part is much related to F¨ollmer’s treatment of quadratic
variation, and will therefore be strongly dependent on partition sequences. For this purpose
we define the quadratic variation in the sense of F¨ollmer (cf. [F¨ol79]) and call a sequence of
partitions (πn)increasing if for all [s, t]∈πnthere exist [ti, ti+1 ]∈πn+1,i= 1, ..., N , such
that [s, t] = SN
i=1[ti, ti+1 ].
Definition 14. Let (πn) be an increasing sequence of partitions such that limn→∞ |πn|= 0.
A continuous function f: [0, T ]→Rhas quadratic variation along (πn) if the sequence of
discrete measures on ([0, T ],B([0, T ])) given by
µn:= X
[s,t]∈πn
|fs,t|2δs(10)
converges weakly to a measure µ, where δsdenotes the Dirac measure at s∈[0, T ]. We write
[f]tfor the “distribution function” of the interval measure associated with µ. A continuous
path X= (X1, ..., Xd) has quadratic variation along (πn) if (10) holds for all Xiand Xi+Xj,
1≤i, j ≤d. In this case, we set
[Xi, Xj]t:= 1
2([Xi+Xj]t−[Xi]t−[Xj]t), t ∈[0, T ].
Remark 15. Since in our situation the limiting distribution function is continuous, the weak
convergence is equivalent to the uniform convergence of the distribution function. Hence,
X= (X1, ..., Xd)∈C([0, T ],Rd)has quadratic variation in the sense of F¨ollmer if and only
if
[Xi, Xj]n
t:= X
[u,v]∈πn
Xi
u∧t,v∧tXj
u∧t,v∧t
convergences uniformly to [Xi, Xj]·in C([0, T ],R)for all 1≤i, j ≤d, where u∧t:= min{u, t}.
See Lemma 32 in [PP13].
Remark 16. Let us emphasize here that quadratic variation should not be confused with the
notion of 2-variation: quadratic variation depends on the choice of a partition sequence (πn),
2-variation does not. In fact, for every continuous function f∈C([0, T ],R)there exits a
sequence of partitions (πn)with limn→∞ |πn|= 0 such that [f , f]t= 0 for all t∈[0, T ]. See
for instance Proposition 70 in [Fre83].
The existence of quadratic variation guaranteed, F¨ollmer was able to prove a pathwise
version of Itˆo’s formula. In his case, the construction of the integral is closely linked to the
partition sequence chosen for the quadratic variation. We will now aim at combining the
techniques of controlled paths with the quadratic variation hypothesis, and derive a pathwise
version of Itˆo’s formula for paths with finite variation, in which the quadratic variation term
may depend on a partition sequence, but the integral does not. As a first step, we derive the
existence of γ-Stratonovich integrals for any γ∈[0,1].To do so, we will need the following
technical lemma the easy proof of which is left to the reader.
12

Lemma 17. Let p≥1,(πn)be an increasing sequence of partitions such that limn→∞ |πn|=
0,X∈ Vp([0, T ],Rd)with quadratic variation along (πn)and Y∈Cq
X. In this case the
quadratic covariation of Xand Yexists and is given by
[Y, X ]t:= lim
n→∞ X
[s,t]∈πn
hXs,t, Ys,t i=X
1≤i,j≤dZT
0
Y′
t(i, j) dπn[Xi, X j]t,
where Y′
t= (Y′
t(i, j))1≤i,j ≤d,0≤t≤T.
Theorem 18. Let X∈ Vp([0, T ],Rd),Y∈Cq
Xand suppose Y′
tis a symmetric matrix for
all t∈[0, T ]. Let (πn)be a increasing sequence of partitions such that limn→∞ |πn|= 0 and
Xhas quadratic variation along (πn). Then for all γ∈[0,1] the γ-RYtdπnXtintegral exists
and is given by
γ-ZT
0
YtdπNXt=ZT
0
Yt◦dXt+1
2(2γ−1) X
1≤i,j≤dZT
0
Y′
t(i, j) dπn[Xi, X j]t,
where Y′
t= (Y′
t(i, j))1≤i,j ≤d.
Proof. Fix γ∈[0,1]. As before we split the sum in (5) into its symmetric and antisymmetric
parts:
X
[s,t]∈πn
hYs+γYs,t , Xs,ti=1
2X
[s,t]∈πnhYs+γYs,t , Xs,ti+hXs+γXs,t , Ys,ti
+1
2X
[s,t]∈πnhYs+γYs,t , Xs,ti − hXs+γXs,t , Ys,ti.
The second sum converges for every sequence of partitions (πn) such that limn→∞ |πn|= 0
and is independent of γthanks to Theorem 11. Taking γ= 1/2 we can apply Corollary 13
to see that 1
2AhX, Y i=ZT
0
Yt◦dXt−1
2hXT, YTi − hX0, Y0i.(11)
For the symmetric part, we note for (s, t)∈∆T
hYs+γYs,t , Xs,ti+hXs+γXs,t , Ys,ti
=(1 −γ)hYs, Xs,ti+hXs, Ys,t i+γhYt, Xs,ti+hXt, Ys,t i
=(1 −γ)hYt, Xti − hYs, Xsi+hYs, Xti − hYt, Xti+hXs, Ys,ti
+γhYt, Xti − hYs, Xsi+hYs, Xsi − hYt, Xsi+hXt, Ys,ti
=(1 −γ)hYt, Xti − hYs, Xsi − hXs,t, Ys,ti+γhYt, Xti − hYs, Xsi+hXs,t, Ys,ti
=hYt, Xti − hYs, Xsi+ (2γ−1)hXs,t, Ys,t i.
Thus the first sum reduces to
1
2X
[s,t]∈πnhYs+γYs,t , Xs,ti+hXs+γXs,t, Ys,ti
=1
2hYT, XTi − hY0, X0i+2γ−1
2X
[s,t]∈πn
hXs,t, Ys,t i.
Therefore, the symmetric part converges along (πn), and the assertion follows by (11) and
Lemma 17.
13

An application of Theorem 18 to the particular case Y= DF(X) for Fsmooth enough
provides the classical Stratonovich formula.
Lemma 19. Let 1≤p < 3,X∈ Vp([0, T ],Rd)and F∈C2(Rd,R). Suppose that the second
derivative D2Fis α-H¨older continuous of order α > max{p−2,0}. Then the Stratonovich
integral RhDF(Xt)◦dXtiexists and is given by
ZT
0
DF(Xt)◦dXt=F(XT)−F(X0).
Proof. Let X= (X1, ..., Xd)∗∈ Vp([0, T ],Rd) for 1 ≤p < 3. Then, with r=p
2in the
definition of controlled paths we easily see that DF(X)∈Cp
X. Thus by Corollary 13 the
(1
2-)Stratonovich integral is well-defined and independent of the chosen sequence of partitions
(πn) along which the limit is taken. Now choose an increasing sequence of partitions (πn) such
that limn→∞ |πn|= 0 and [X]t= 0 along (πn) for t∈[0, T ] (cf. Proposition 70 in [Fre83]).
Applying Taylor’s theorem to F, we observe that
F(XT)−F(X0) = 1
2X
[s,t]∈πn(F(Xt)−F(Xs)) −(F(Xs)−F(Xt))
=1
2X
[s,t]∈πn
hDF(Xs), Xs,ti+1
2X
1≤i,j≤d
Di,j(Xs)Xi
s,tXj
s,t +R(Xs, Xt)
+1
2X
[s,t]∈πn
hDF(Xt), Xs,ti − 1
2X
1≤i,j≤d
Di,j(Xt)Xi
s,tXj
s,t +˜
R(Xs, Xt)
=X
[s,t]∈πn
h1
2DF(Xs) + 1
2DF(Xt), Xs,ti+1
4X
[s,t]∈πnX
1≤i,j≤d
(Di,j (Xs)−Di,j (Xt))Xi
s,tXj
s,t
+X
[s,t]∈πn
(R(Xs, Xt) + ˜
R(Xs, Xt)),
where |R(x, y)|+|˜
R(x, y)| ≤ φ(|x−y|)|x−y|2,for some increasing function φ: [0,∞)→R
such that φ(c)→0 as c→0. Since Xis continuous and has zero quadratic variation along
(πn), the last two terms converge to 0 as n→ ∞, and we obtain
ZT
0
DF(Xt)◦dXt= lim
n→∞ X
[s,t]∈πn
hDF(Xs) + 1
2(DF(Xt)−DF(Xs)), Xs,ti=F(XT)−F(X0).
We can now state and prove the announced version of the pathwise formula by F¨ollmer
(cf. [F¨ol79]).
Corollary 20. Let 1≤p < 3,γ∈[0,1] and (πn)be an increasing sequence of partitions such
that limn→∞ |πn|= 0. Assume F∈C2(Rd,R)with α-H¨older continuous second derivative
D2Ffor some α > max{p−2,0}. If X∈ Vp([0, T ],Rd)has quadratic variation along (πn),
then the formula
F(XT) = F(X0) + γ-ZT
0
DF(Xt) dπnXt−1
2(2γ−1) X
1≤i,j≤dZT
0
DiDjF(Xs) dπn[Xi, Xj]s
14

holds.
Proof. Combine the results of Theorem 18 and Lemma 19.
The assumptions, that Xis of finite p-variation for some 1 ≤p < 3 and that the second
derivative D2Fis α-H¨older continuous for some α > max{p−2,0}can be considered the price
we have to pay for obtaining an integral the antisymmetric part of which does not depend on
the chosen partition sequence. F¨ollmer [F¨ol79] does not need these hypotheses. However, his
integral is only defined along the sequence of partitions for which the existence of quadratic
variation is claimed.
So far we have always assumed that the Gubinelli derivative of a controlled path is sym-
metric. This assumption can be avoided if the paths involved control each other.
Definition 21. Let X, Y ∈ Vp([0, T ],Rd). We say that Xand Yare similar if there exist
X′, Y ′∈ V q([0, T ],Rd×d) such that X∈Cq
Ywith Gubinelli derivative X′,Y∈Cq
Xwith
Gubinelli derivative Y′, and ((X′
t)∗)−1=Y′
tfor all t∈[0, T ]. In this case we write Y∈Sq
X.
Let us give an very simple example of two paths X, Y ∈ Vp([0, T ],Rd) such that Y∈Sq
X
but neither Y∈Cq
Xwith Y′symmetric nor X∈Cq
Ywith X′symmetric.
Example 22. For p∈[2,3) take X1∈ Vp([0, T ],R) and X2, X3∈ V p
2([0, T ],R). If we set
X= (X1, X2, X3) and Y= (X1,0,0), we obviously have X, Y ∈ Vp([0, T ],R3). In this case
we could choose X′and Y′identical to


1 0 0
0 0 −1
0 1 0

.
We see that Y∈Sp
X, but X′and Y′are not symmetric matrices.
For similar paths the antisymmetric part of the integral also exists and is independent of
the choice of a particular sequence of partitions along which Riemann sums are defined.
Theorem 23. Let 1≤p < ∞and X∈ Vp([0, T ],Rd). If Y∈Sq
X, then for every γ∈[0,1]
the antisymmetric part of the integral as defined in (6) exists and satisfies
AγhX, Y i=AhX, Y i= lim
|π|→0X
[s,t]∈πhYs′, Xs,ti − hXs′, Ys,t i,
where s′is an arbitrary point in [s, t]for each [s, t]∈π.
Proof. By Lemma 3 it is again sufficient to check condition (2) for Ξs,t := hYs, Xs,ti−hXs, Ys,ti.
Because Xand Yare similar, we obtain
Ξs,t−Ξs,u −Ξu,t =hYs, Xs,t i − hXs, Ys,ti − hYs, Xs,ui+hXs, Ys,u i − hYu, Xu,ti+hXu, Yu,ti
=hXs,u, Yu,t i − hYs,u, Xu,ti
=hX′
sYs,u +RX
s,u, Y ′
uXu,t +RY
u,ti − hYs,u , Xu,ti
=hX′
sYs,u, RY
u,ti+hRX
s,u, Y ′
uXu,ti+hRX
s,u, RY
u,ti+hX′
sYs,u, Y ′
uXu,ti − hYs,u , Xu,ti
15

for 0 ≤s≤u≤t≤T. The second last term in the preceding formula can be rewritten as
hX′
sYs,u, Y ′
uXu,ti − hYs,u , Xu,ti=hYs,u ,(X′
t)∗Y′
uYu,t −Xu,ti=hYs,u ,(X′
t)∗(Y′
u−Y′
t)Xu,ti,
where we applied ((X′
t)∗)−1=Y′
t. Since the finite sum of control functions is again a control
function, we can choose the same control function ωfor X, X′, RXand Y, Y ′, RY, and therefore
|Ξs,t −Ξs,u −Ξu,t|
≤||X′||∞ω(s, u)
1
pω(u, t)1
r+||Y′||∞ω(s, u)1
rω(u, t)
1
p+ω(s, u)1
rω(u, t)1
r
+||X′||∞ω(s, u)
1
qω(u, t)
1
pω(s, u)
1
p
≤||X′||∞ω(s, t)
1
p+1
r+||Y′||∞ω(s, t)
1
r+1
p+ω(s, t)1
r+1
r+||X′||∞ω(s, t)
1
q+2
p,
where || · ||∞denotes the supremum norm. Setting θ:= 2/p + 1/q > 1, we conclude that
|Ξs,t −Ξs,u −Ξu,t| ≤ 2||X′||∞+||Y′||∞+ω(0, T )
1
qω(s, t)θ.
We therefore have shown that the left-point Riemann sums converge. It remains to prove
that (7) goes to zero along every sequence of partitions (πn) such that the mesh |πn|tends to
zero. Since Xand Yare similar, we observe that for (s, t)∈∆T, and s′∈[s, t]
hYs, Xs,ti − hXs, Ys,t i − hYs′, Xs,ti − hXs′, Ys,ti=hYs,s′, Xs,t i − hXs,s′, Ys,t i
=hY′
sXs,s′+RY
s,s′, X′
sYs,t +RX
s,ti − hXs,s′, Ys,ti
=hY′
sXs,s′, X′
sYs,ti+hY′
sXs,s′, RX
s,ti+hRY
s,s′, X′
sYs,ti+hRY
s,s′, RX
s,ti − hYs,t , Xs,s′i
=hY′
sXs,s′, RX
s,ti+hRY
s,s′, X′
sYs,ti+hRY
s,s′, RX
s,ti.
To obtain the last line, we once again use ((X′
s)∗)−1=Y′
s. Taking again the same control
function ωfor X, X′, RXand Y, Y ′, RY, we estimate

hYs, Xs,ti − hXs, Ys,ti − hYs′, Xs,t i − hXs′, Ys,ti

≤ ||Y′||∞ω(s, t)
1
p+1
r+||X′||∞ω(s, t)
1
p+1
r+ω(s, t)1
r+1
r+||Y′||∞ω(s, t)
1
p+1
p+1
q
≤||X′||∞+ 2||Y′||∞+ω(0, T )
1
qω(s, t)θ,
with θ:= 2
p+1
p>1. Superadditivity of ωfinally gives



X
[s,t]∈πnhYs,s′, Xs,ti − hXs,s′, Ys,ti



≤||X′||∞+ 2||Y′||∞+ω(0, T )
1
qmax
[s,t]∈πn
ω(s, t)θ−1ω(0, T ),
which means that (7) tends to zero as |πn|tends to zero.
We conclude this section by presenting an analogue of Theorem 18 for similar paths.
Theorem 24. Let (πn)be an increasing sequence of partitions such that limn→∞ |πn|= 0,
X∈ Vp([0, T ],Rd)and Y∈Sq
X.
16

1. Then the Stratonovich integral as defined in (8) exists and fulfills
1
2-ZT
0
YtdXt=ZT
0
Yt◦dXt=1
2hXT, YTi − hX0, Y0i+1
2AhX, Y i.
2. If Xhas quadratic variation along (πn), then the limit γ-RT
0YtdXtas defined in (5)
exists for all γ∈[0,1] and satisfies
γ-ZT
0
YtdXt=ZT
0
Yt◦dXt+1
2(2γ−1) X
1≤i,j≤dZT
0
Y′
t(i, j) d[Xi, X j]t,
where Y′
t= (Y′
t(i, j))1≤i,j ≤d.
Proof. The first statement follows as Corollary 13 by replacing Theorem 11 with Theorem 23
in the proof. The second statement is proved in analogy to the proof of Theorem 18.
References
[FD06] Denis Feyel and Arnaud De La Pradelle, Curvilinear integrals along enriched paths.,
Electron. J. Probab. 11 (2006), 860–892.
[FDM08] Denis Feyel, Arnaud De La Pradelle, and Gabriel Mokobodzki, A non-commutative
sewing lemma., Electron. Commun. Probab. 13 (2008), 24–34.
[FH13] Peter Friz and Martin Hairer, A short course on rough paths, in preparation (2013).
[F¨ol79] Hans F¨ollmer, Calcul d’Itˆo sans probabilit´es, S´eminaire de Probabilit´es XV 80
(1979), 143–150 (French).
[Fre83] David Freedman, Brownian motion and diffusion, second ed., Springer-Verlag, New
York, 1983. MR 686607 (84c:60121)
[GIP13] Massimiliano Gubinelli, Peter Imkeller, and Nicolas Perkowski, Paracontrolled dis-
tributions and singular PDEs, preprint arXiv:1210.2684v2 (2013).
[GIP14] , A Fourier Approach to pathwise stochastic integration, in preparation,
2014.
[Gub04] Massimiliano Gubinelli, Controlling rough paths, J. Funct. Anal. 216 (2004), no. 1,
86–140.
[LCL07] Terry J. Lyons, Michael Caruana, and Thierry L´evy, Differential equations driven
by rough paths, Lecture Notes in Mathematics, vol. 1908, Springer, Berlin, 2007.
MR 2314753 (2009c:60156)
[Lej09] Antoine Lejay, S´eminaire de Probabilit´es XLII, Lecture Notes in Math. 1979 (2009).
[L´ev40] Paul L´evy, Le mouvement brownien plan, Amer. J. Math. 62 (1940), 487–550
(French). MR 0002734 (2,107g)
17

[Lyo98] Terry J. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoam.
14 (1998), no. 2, 215–310.
[Per14] Nicolas Simon Perkowski, Studies of robustness in stochastic analysis and mathe-
matical finance, Ph.D. thesis, 2014.
[PP13] Nicolas Perkowski and David J. Pr¨omel, Pathwise stochastic integrals for model free
finance, Preprint arXiv:1311.6187 (2013).
[You36] Laurence C. Young, An inequality of the H¨older type, connected with Stieltjes inte-
gration, Acta Math. 67 (1936), no. 1, 251–282.
18