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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

2ZT

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|r1

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

−1sin(2kπs) sin(2lπs)2lπ+ cos(2lπs) cos(2kπs)2kπds

=−

m

X

k,l=1

akal2lπZ1

−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

aksin(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)θr

≤2r−1|Ξs,t|r+ 2r−1C(θ)rω(s, t)rθ ≤2r−1+ 2r−1C(θ)rω(0, T )rθ −1ω(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

2γ-ZT

0

YtdXt+γ-ZT

0

XtdYt+1

2γ-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]∈πhYs+γYs,t , Xs,ti − hXs+γXs,t , Ys,ti,(6)

exists and satisﬁes

AγhX, Y i=AhX, Y i:= lim

|π|→0X

[s,t]∈πhYs′, 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]∈πnhYs, Xs,ti − hXs, Ys,t i−X

[s,t]∈πnhYs′, 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]∈πnhYs,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

2hYT, 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

2hYs+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]∈πhYs+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]∈πnhYs+γYs,t , Xs,ti+hXs+γXs,t , Ys,ti

+1

2X

[s,t]∈πnhYs+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

2hXT, 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+γhYt, Xs,ti+hXt, Ys,t i

=(1 −γ)hYt, Xti − hYs, Xsi+hYs, Xti − hYt, Xti+hXs, Ys,ti

+γhYt, Xti − hYs, Xsi+hYs, Xsi − hYt, Xsi+hXt, Ys,ti

=(1 −γ)hYt, Xti − hYs, Xsi − hXs,t, Ys,ti+γhYt, 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]∈πnhYs+γYs,t , Xs,ti+hXs+γXs,t, Ys,ti

=1

2hYT, 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]∈πhYs′, 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

qω(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

qω(s, t)θ,

with θ:= 2

p+1

p>1. Superadditivity of ωﬁnally gives

X

[s,t]∈πnhYs,s′, Xs,ti − hXs,s′, Ys,ti

≤||X′||∞+ 2||Y′||∞+ω(0, T )

1

qmax

[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.

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