Document [original]

Rough paths, probability and related topics

vorgelegt von

Master of Science

Atul Shekhar

geb. in Darbhanga

Von der Fakult¨at II - Mathematik und Naturwissenschaften

der Technischen Universit¨at Berlin

zur Erlangung des akademischen Grades

Doktor der Naturwissenschaften

Dr.rer.nat.

genehmigte Dissertation

Promotionsausschuss:

Vorsitzende: Prof. Dr. Boris Springborn

Berichter/Gutachter: Prof. Dr. Peter K. Friz

Berichter/Gutachter: Prof. Dr. Terry Lyons

Tag der wissenschaftlichen Aussprache: 09 December 2015

Berlin 2016

To my parents Anandi Das and Lakshman Lal Das

Acknowledgement

Successful completion of my doctorate degree is indebted to many people to whom I

express my sincere gratitude. First and foremost of all, a big thanks goes to my supervisior

Prof. Dr. Peter K. Friz who has constantly supported me during this period. I admire

him for the patience that he had with me. He constantly kept me motivated through

many research discussion sessions, during which i learnt a lot from him. He has shaped

me into a better mathematician and a researcher by giving me great ideas and directing

me into right path. Due to his support, I attended many conferences and summer schools

to deliver a talk, which made me a better teacher. His encouragement indeed has laid

the foundational stones of current work.

I would like to thank Prof. Terry Lyons for being the external examiners of my thesis

which includes careful examining the results in this manuscript. I would also like to

thanks Prof. Boris Springborn for accepting to act as a chairman in my PhD defense.

In June 2014, I had the oppurtunity to visit university of Washington, Seattle, under

the invitation of Prof. Steffen Rohde. He also visited Berlin for 3 months in the winter

semester 2014-15 accompanied by a series of lectures on random geometry. Meeting and

discussing mathematics with him was truly inspirational and I came to learn a lot about

theory of Schramm-Loewner evolutions through him.

In Nov 2014, I was invited as an speaker in the weekly seminar in Oxford-Man institute,

Oxford by Prof. Terry Lyons. I also visited Oxford university during summer semester

2015 for two months as a research visitor, adding to my collaborating research experience.

I thank Prof. Steffen Rohde and Prof. Terry Lyons for their invitation and taking their

valuable time for fruitful discussions we had.

Of course I wouldn’t forget to mention the part of my friends in Berlin for making

my stay so beautiful. Sharing mathematical ideas along with sharing jokes and fun

moments during the coffee breaks leaves you wonder the meaning of stress. A happy

brain is what all you need for mathematics afterall. Thank you Alberto Chiarini, Giovanni

Conforti, Adrian Gonzalez, Giuseppe Cannizzaro, Matti Leimbach, Benedickt Rehle and

Julie Meissner. I also thank the postdoctoral fellows in our research group for exposing

me to different areas of research and to always be there for clearing my doubts and

questions. Thank you Antoine Dalqvist, Romain Allez, Joscha Diehl and Paul Gassiat

from Technical university of Berlin, Horatia Boedihardjo from Oxford university, Huy

Tran from university of Washington and Laure Dumaz from Cambridge university.

I thank Berlin Mathematical School and reseach training group 1845 for providing the

financial support on the one hand and organising wonderful talks and conferences like

BMS fridays, minicourses, RTG summer schools on the other.

Last but not the least, all this couldn’t have been possible without the support of my

family, Anandi Das, Lakshman Lal Das, Leena Rajesh kumari, Uday Shekhar, Rashmi

Kumari and Himanshu Shekhar. Thank you for having belief in me and supporting and

suggesting all the important decisions of my career and life.

Introduction

The contents of this doctoral dissertation consists of results in two different areas of

probability theory and its applications, namely Rough path theory and theory of Loewner

evolutions. In order to describe best, we take a view on the two areas seperately and

then present the results.

Rough path theory

Rough path theory, abbreviated RPT, was introduced by Terry Lyons in a seminal pa-

per [54] in 1998 for a systematic study of continuous paths which are very irregular.

Continuous paths, whether deterministic or random, appear frequently in real world as

a mathematical model for something which evolves in time. It could be describing the

motion of a particle in an external field, it could be some time series data from financial

world, e.g. price of a financial commodity like call options or could it be the trajec-

tory of a script alphabet being written on a touchscreen interactive device like computer

tablets. In many applications, it is very essential to associate parameters to the path

which can be used to derive informations it. One seeks for a set of parameters which is

efficient and easy to implement and also should also contain relevant informations about

the path which will distinguish it from other ones. For example, given a smooth path

X: [0, T]→Rd, length of the path

L(X) = ∫T

0|˙

Xr|dr

is one such parameter measuring the length of the path. Other examples include winding

number of a closed complex valued (or two dimensional) path,

W(X, P) = 1

2πi ∫X

z−Pdz

which measures number of times path Xturns around point P. These examples are very

useful for certain appliations but they do not cover all the aspects of the path. More

generally, in 1957, K. Chen [[7], [8]] introduced signature of a path, which is collection of

its all possible iterated integrals as a set of such parameters which would describe all the

important aspects of the path in an efficient manner.

Definition .0.1. Given a path X: [0, T]→Rd, signature of X, denoted S(X)0,T is

defined as

S(X)0,T =(1,∫0<r1<T

dXr1,∫0<r1<r2<T

dXr1⊗dXr2, ..., ∫0<r1<r2<..<rn<T

dXr1⊗dXr2⊗..⊗dXrn, ...)

given that all the integrals above are well defined, e.g. when Xis a smooth path.

vii

viii

Signature can be considered as the building block of Rough path theory. In 1936,

L.C. Young proved the following remarkable result.

Theorem .0.2 (Young, 1936).Let α, β ∈(0,1] such that α+β > 1. If X∈Cα([0, T],Rd)

and Y∈Cβ([0, T],Rn×d)are two H¨older regular paths. Then

∫T

YrdXr:= lim

|P|→0∑

[s,t]∈P

Ys(Xt−Xs)

is well defined, where Pdenote a partition of [0, T]and |P| its mesh size. Furthermore,

there exist a constant Csuch that for all s<t,

⏐⏐⏐⏐∫t

YrdXr−Ys(Xt−Xs)⏐⏐⏐⏐≤C||X||α||Y||β|t−s|α+β

Young’s result guarantees that iterated integrals in definition of signature are well

defined wheneve Xis α-H¨older path for α > 1

2. But when Xis even more irregular,

it is not possible to make sense of signature of path X. The key idea of T. Lyons

leading to emergence of RPT was abstractifying the concept of signature. When the

path Xis not regular enough, the integrals in the definition of signature are not well

defined. But, if one could construct an objects which mimics signature of Xupto some

algebraic and analytical properties, then one could use that object in the place of signature

for practically all relevant purposes. RPT also comes with a extension theorem which

gurantees that it is only required to construct an objects which mimics signature only

upto some finitely many levels and all the higher order iterated integrals can be naturally

constructed. Let TN

1(Rd) be the truncated tensor algebra defined by TN

1(Rd) := 1⊕Rd⊕

{Rd}⊗2⊕.. ⊕{Rd}⊗Nwith the multiplication g⊗hsuch that

πk(g⊗h) =

∑

i=0

πi(g)⊗πk−i(h)

where πkis the projection map πk:TN

1→ {Rd}⊗k.GN(Rd) denote the step Nnilpotent

group.

Definition .0.3. Let X∈Cα([0, T],Rd)be a path and let N= [ 1

α]. A rough path

associated to Xis a path X: [0, T]→TN

1(Rd)such that π1(X) = Xand for 1≤k≤N,

||πk(X)||kα := sup

s=t

|πk(Xs,t)|

|t−s|kα <∞

where Xs,t := X−1

s⊗Xt. A rough path is called a geometric rough path if it takes value

in GN(Rd)⊂TN

1(Rd).

The rough path Xindeed mimics the signature and it can be proved that when Xis

smooth,

πk(X) = ∫0<r1<r2<..<rk<T

dXr1⊗dXr2⊗.. ⊗dXrk

An another strategy for styding properties of a path is to construct calculus based on

that path, i.e. make sense of integration

∫YrdXr

for appropriate paths Y. This strategy has proved to be very effective in many situations,

e.g. many non-trivial properties of Brownian motion sample path can be derived using

Ito stochastic calculus. The rough path Xassociated to Xcan also be used develop an

integration theory called “rough integration”. In particular, rough integration can be

used to make sense of higher order iterated integrals

∫0<r1<r2<..<rn<T

dXr1⊗dXr2⊗.. ⊗dXrn

for n > N, justying the fact that it is enough to construct the object Xwhich mimics

the signature upto N levels. The key idea behind rough integration is that since Xis not

regular enough to define integration, we enhance the path Xvia X, adding some extra

information to it and Xcan be used to define the (rough) integration. For example in

the case when 1

3< α ≤1

2,N= 2 and integration of one forms f(X) can be defined as

follows.

Theorem .0.4 ( Lyons, Gubinelli).For 1

3< α ≤1

2, rough path X= (1, X, X), and one

forms f(X)with fcontinuously twice differentiable,

∫T

f(X)dX:= lim

|P|→0∑

[s,t]∈P

f(Xs)Xs,t +f′(Xs)Xs,t

exists. Moreover,

⏐⏐⏐⏐∫t

f(X)dX−f(Xs)Xs,t −f′(Xs)Xs,t⏐⏐⏐⏐≤C(||X||3

α+||X||α||X||2α)|t−s|3α

where the constant Conly depends on f.

In the above theorem, the pair of 1-forms (f(X), f′(X)) can be replaced by any pair

(Y, Y ′) such that Y, Y ′are α-H¨older path and Rs,t := Ys,t −Y′

s(Xs,t) satisfy

||R||2α:= sup

s=t

|Rs,t|

|t−s|2α<∞

Such a pair (Y, Y ′) are called controlled paths and rough integration

∫T

Y dX

can be defined same as above for controlled paths.

Rough path theory paves way to give meaning to differential equations which were

previously ill posed. For a irregular path X, the equation

dYt=f(Yt)dXt

can be now given the meaning

Yt=Y0+∫t

f(Yr)dXr

and solution Ycan be looked for in the space of controlled paths. These equations

are so called “rough differential equations” (RDEs) and existence and uniqueness of the

solutions Y:= If(X) can be established using modified picard iteration schemes, as

carried out in [[26], [25]]. One of the most important results which follows from the

analysis of RDE’s is the “Universal limit theorem” which guarantees that the solution

map X→Y:= If(X) is continuous in the appropriate rough path metric, a metric

defined on the space of rough paths. Given that path Xis very irregular, such a result is

very rare and is of great impact and it finds its applications in many areas of mathematics.

One of the most important applications of RPT is in stochastic calculus. When X

is a stochastic process with semimartingale structure, e.g. Brownian motion, there is an

another probabilistic way of making sense of

∫T

Y dX (1)

This theory was developed by K. Ito in mid 20th century and now is known as Ito’s

stochastic calculus named after him. Here, the integrands Yare taken to be adapted left

continuous processes and the stochastic integral

∫T

YrdXr:= lim

|P|→0∑

[s,t]∈P

YsXs,t

where the limit is shown to exist in probability. The Ito’s stochastic calculus is a very

elegant theory and it indeed finds its appliations in many different areas. But it has

some cons too. The limit above is taken in probability which takes into account the

law of Xand thus all the sample paths of X. The stochastic integral so constructed

is only defined almost surely, i.e. on a set of probability one and it is thus impossible

to decide whether the integral is defined for a given particular sample path. In many

application, e.g. mathematical finance, there is only one available time series data and

inherent law of the process is usually unknown. Considering such situations, one seeks

for a an integral calculus which is defined pathwise, i.e. for a particular given sample

path of Xone should be able to define the integral (1). Note that since Brownian motion

Bis only α-H¨older for α < 1

2and no more regular, the Young’s result Theorem .0.2 falls

just short for making sense of (1) in a pathwise sense. Such a situation can perfectly be

handled by RPT as introduced above. We fix an α∈(1

3,1

2) such that Brownian sample

paths are α-H¨older almost surely. If one could construct a rough path Bassociated to

B, then rough integration can be used to give meaning to pathwise integration against

Brownian motion. It can be easily proven that prescribing

Bs,t := ∫t

Bs,r ⊗dBr

indeed defines a rough path B= (1, B, B) reaching us the goal of pathwise integration.

Rough integral in this context is also consistent with the stochastic integral. It can be

shown that rough integral so defined actually matches with the Ito integral. Such a

consistensy is of great impact. What this implies is that stochastic differential equations

(SDEs)

dYt=f(Yt)dBt

can be now viewed as a RDE

dYt=f(Yt)dBt

In constrast to SDEs, RDEs are very well behaved in view of approximations. Note that

the solution map of a SDE Ψ : B→Ydoesn’t have any nice continuity properties w.r.t

to any useful topology put on the path space C([0, T],Rd). On the other hand, “universal

limit theorem” for RDEs gives us continuity of map Y=If(X) in the rough path metric.

Considering SDEs as a RDE, we immediately get the continuity under approximation

type results for SDE, e.g. Wong-Zakai theorem with explicit rates of convergence.

The applications of RPT in stochastic analysis goes beyond the realm of Ito calculus.

Probabilistic semimartingale structure is very crucial for the developement of Ito calcu-

lus. But there are many other stochastic models where one doesn’t have semimartingale

property and the Ito calculus fails to apply, e.g. fractional Brownian motion. On the

other hand, RPT just requires for a construction of an appropriate rough path associated

to the process X. This can be carried out for many Gaussian processes with appropriate

covariance structure. As a result, many interesting results follows, e.g. densities for RDEs

under H¨ormander condition for Gaussian signals, [67].

Relevant to the content of this thesis, we mention one more application of RPT in

approximating the solution of parabolic PDEs. Consider

{∂tu(t, x) = −Lu(t, x)

u(T, x) = f(x)

with the differential operator L=V0+1

2(V2

1+.. +V2

d), where the differential operator

is identified with the corresponding vector fields Vi. Using Feynman-Kac reperesentation

formula, one can can express u(t, x) = E(f(ξT−t,x)), where ξthe solution of Stratonovich

SDE {dξt,x =∑d

i=0 Vi(ξt,x)◦dXi

ξ0,x =x

driven by Xt= (t, Bt) where Btis the d-dimensional Brownian motion. Interested in

fast numerical simulations for the solution u(t, x) of such parabolic PDEs, signature of

Brownian motion can be used very effectively. Stochastic taylor expansion (of order m)

allows one to approximate

u(t, x) = E[f(ξt,x)] ≈∑

(i1,..ik)∈Am

Vi1..Vikf(x)E[∫0<t1<..<tk<t ◦dBi1

t1.. ◦dBik

tk]

where Am:= {(i1, ..ik)∈ {0, .., d}k|k+card{j, ij= 0} ≤ m}. One clearly sees Stratonovich

signature of B(iterated integrals w.r.t. Stratonovich integration) contributing to approx-

imate the solution u(t, x). In fact one only needs the expected value of signature and it is

very desirable to get an closed form formula for the expected signature. One can use the

analysis above to base very efficient approximation technique, e.g. Cubature methods on

wiener space [52].

xii

Theory of Loewner chains

The theory of Loewner chains and Loewner’s differential equation (LDE) was introduced

in early 20th century by C. Loewner in an attempt to solve Bieberbach’s conjecture in

geometric function theory. The conjecture stated that if

f(z) = z+a2z2+a3z3+...

is an univalent conformal map on the unit disk, then for all n≥2

|an| ≤ n.

Bieberbach himself proved the bound |a2| ≤ 2 and Loewner could extend it to |a3| ≤ 3

using LDE. Later in 1986, when De Branges finally resolved the conjecture, LDE was

used as an important component in the proof. A good account on the history and

developements of Bieberbach conjecture can be found in [68].

Loewner’s theory gives a one-to-one correspondence between a family of continuously

growing compact sets in a planar simply connected domain and real valued continuous

curves. For simplicity, we will restrict ourselves to the upper half plane

H={z|z∈C, Im(z)>0}

A bounded subset K⊂His called a compact H-hull if K=H∩¯

Kand H\Kis a simply

connected domain. For each such compact H-hull, there is a unique associated bijective

conformal map gK:H\K→Hsatisfying the so called hydrodynamic normalization

lim

z→∞gK(z)−z= 0

The map gKis called the mapping out function of K. The half plane capacity of K is

defined by

hcap(K) = lim

z→∞z(gK(z)−z)

Definition .0.5. A Loewner chain is a family {Kt}t≥0of compact H-hulls such that

Ks(Ktfor all s<tand satisfying local growth property:

rad(Kt,t+h)→0as h→0 + uniformly on compacts in t

where Ks,t := gKs(Kt\Ks)

Given {Ut}t≥0a continuous real valued curve with U0= 0, for each z∈¯

H\0, let gt(z)

denote the solution of the LDE

˙gt(z) = 2

gt(z)−Ut

, g0(z) = z(2)

The solution exists upto the maximal time T(z)∈(0,∞] and if T(z)<∞,

lim

t→T(z)gt(z)−Ut= 0

Define

Kt={z∈H|T(z)≤t}

xiii

Then the family {Kt}t≥0is a Loewner chain with hcap(Kt)=2tand gtis the mapping

out function of Kt. We call the chain {Kt}t≥0is driven by {Ut}t≥0.

Conversely given a Loewner chain {Kt}t≥0with hcap(Kt) = 2t, then there exist contin-

uous real valued curve Utwith U0= 0 such that mapping out functions gt(z) = gKt(z)

satisfies equation (38) and {Kt}t≥0is driven by {Ut}t≥0. Please refer to [40] and lecture

notes [3] for the details.

In a seminal article by O. Schramm in 1999, [73], correspondence above was utilized

to characterize processes in plane which are conformally invariant and satisfies domain

Markov property. Today these processes are known as Schramm-Loewner evolutions,

SLE(κ), which is a random Loewner chain obtained when Ut=√κBt, where Btis the

one dimensional Brownian motion. SLEs was then proven to arise natuarally as scaling

limit of various discrete lattice models in statistical physics. See results in [76, 74, 73] for

these results.

Interesting examples of Loewner chains inlcude bubble fill of any continuous curve

γ: [0, T]→¯

H(fill in the interior whenever γforms a loop). In this case, the Loewner

chain is actually described by a growing curve rather than just a growing family of

compact sets. But not all Loewner chains fall in this category. There are pathological

examples like logarithmic spiral where Ktis not locally connected. Convergence results

of various statistical mechanics model to SLEs suggests that SLEs are actually random

curves in upper half plane and not just a family of growing compact sets. A natural

question then is that under what conditions the Loewner chain can be described as

bubble fill of a continuous curve in the following sense :

Definition .0.6. A chain {Kt}t≥0is called generated by a curve γ: [0, T]→¯

Hwith

γ0= 0 if for all t≥0,Ht:= H\Ktis the unbounded component of H\γ[0, t].

If a chain is generated by a curve γ, then it is the only such a curve called the trace

of chain. A necessary and sufficient condition for the existence of the trace can be found

in [71]. Denote ft(z) = g−1

t(z).

Theorem .0.7 ([71]).A chain is generated by a curve if and only if

γt:= lim

y→0+ ft(iy +Ut)

exists and is continuous curve. If so, curve γis the trace.

There are many positive results known in this direction. It was proved in [62, 47]

that when the driver Uis 1

2- Holder with ||U||1

2<4, then the trace exist. This is

the best possible known deteministic result and fails to apply for the SLEκcase where

Ut=√κBt, where Btis standard Brownian motion. Nevertheless, proof of existence

of trace of SLE(κ), κ= 8 was carried out in [71] using probabilistic techniques. The

trace also exist for SLE(8), but the proof follows indirectly from convergence of Uniform

spanning tree to SLE(8) and there is no direct proof known so far.

Complete understanding of phenomena of existence of trace is unknown. In the case

of random drivers U, probabilistic techiniques are the only available tool for proving

the existence of trace. Though these techniques works efficiently for SLEs, it doesn’t

give deterministic conditions on the driver and pathwise property of Brownian motion

xiv

responsible for the existence of trace. Such a condition is very desirable to understand

many fine properties of SLE. One immediate corollary to such understanding would be

continuity of the Loewner map Φ : U→γis appropriate metric and Wong-Zakai type

Theorem for SLE. In this thesis, we take a step in this direction and prove some results

on the trace of Loewner chains.

The results of this thesis

In this section, we mention the main results of this thesis. The content is divided into

following research articles.

I. Doob-Meyer Theorem for rough paths

In a recent work [57], Hairer-Pillai introduced the concept of θ-roughness for rough paths,

leading them to a deterministic Norris type lemma. Norris lemma is one of the key

ingredient in H¨ormander theorem for smoothness of probability density of solutions to

Brownian SDEs. Using rough path theory and aforementioned Norris-type lemma, Hairer-

Pillai developed a H¨ormander theorem for rough path driven SDEs. In this article, we

take a step back and prove a deterministic Doob-Meyer type result for rough paths.

In the semimartingale setting, Doob-Meyer Theorem states that it is not possible to

have cancellation between martingale part and bounded variation part and the both

the components are uniquely determined from semimartingale. Intuition behind is that

martingales are nothing but time changed Brownian motion and it is very irregular at

every small scales. We coin the notion of “true roughness” of rough paths which formalises

the above intuition and obtain the Doob-Meyer type results under this assumption. In

the framework of [54], let X: [0, T]→Vis a p-rough path taking value in Banach space

Vand controlled by some control function ω.

Definition .0.8. For fixed s∈[0, T)we call X”rough at time s” if (convention 0/0 := 0)

(∗) : ∀v∗∈V∗\{0}: lim sup

t↓s

|⟨v∗, Xs,t⟩|

ω(s, t)2/p = +∞.

If Xis rough on some dense set of [0, T], we call it truely rough.

Theorem .0.9. (i) Assume Xis rough at time s. Then

∫t

f(X)dX=O(ω(s, t)2/p)as t↓s=⇒f(Xs)=0.

(i’) As a consequence, if Xis truely rough, then

∫·

f(X)dX≡0on [0, T] =⇒f(X·)≡0on [0, T].

(i”) As another consequence, assume |t−s|=O(ω(s, t)2/p), satisfied e.g. when ω(s, t)≍

t−sand p≥2(the ”rough” regime of usual interest) then

∫·

f(X)dX+∫·

g(X)dt ≡0on [0, T] =⇒f(X·), g (X·)≡0on [0, T].

(ii) Assume Z:= X⊕Ylifts to a rough path and set, with ˜

f(z) (x, y) := f(z)x,

∫f(Z)dX:= ∫˜

f(Z)dZ.

Then the conclusions from (i),(i’) and (i”), with g=g(Z), remain valid.

We then verify various stochastic processes to be truly rough. Examples include

Brownian motion, fractional Brownian motion and many other Gaussian processes. In

fact, the results doesn’t require finite dimensionality of state space of rough path and

can be proved in infinite dimension setting. As a result, Q-Wiener processes are also

proven to be truly rough. We close the article by mentioning an application to existence

of density for Non-Markovian systems under H¨ormander condition. This is joint work

with Prof. Peter Friz published in Bulletin of the Institute of Mathematics, Academia

Sinica (New Series) [20].

II. General rough integration, L´evy rough paths and a L´evy–Kintchine type

formula

RPT as introduced above is developed under the basic assumption of Xbeing a con-

tinuous path. The theory fails to work due to some technical reasons if we give up the

continuity and just assume that Xis a c´adl´ag (right continuous with left limits) path.

Needless to say that c´adl´ag paths appear frequently in many situations, e.g. price of

financial commodity undergoing sudden change can be modelled using c´adl´ag processes

like L´evy processes. In this article, we develope RPT for c´adl´ag paths.

We first coin the definition of c´adl´ag rough paths. Since we lack continuity, we choose

to measure the roughness of a path in p-variation metric which a priopri doesn’t imply

continuity rather than H¨older metric. We prove a variant of Lyon’s extension Theorem

in this setting and make sense of signature of a (geometric) c´adl´ag rough path.

Theorem .0.10. Let 1≤p < ∞and N∋n>m:= [p]. A cadlag geometric rough path

X(m)admits an extension to a path X(n)of with values G(n)⊂T(n), unique in the class

of G(n)-valued path starting from 1and of finite p-variation with respect to CC metric on

G(n)subject to the additional constraint

log(n)∆X(n)

t= log(m)∆X(m)

t.(3)

Signature of X(m)is defined as

S(X(m))0,T := (1, π1(X(1)), π2(X(2)), .., πn(X(n)), ...)

The construction of an appropriate extension above uses an adaptation of an idea

due to Marcus. Marcus argued that a jump is nothing but an idealisation a very fast

change happening in a very short time. Thus at each jump time, a fictious time interval

is introduced during which the jump is traversed appropriately, giving us a continuous

path. One can then use the machinery available in continuous RPT followed by removal

of the fictious time intervals in the end to obtain results about c´adl´ag rough paths.

We next make sense of rough integration against such rough paths. For simplicity, we

stick to the case of p∈[2,3).

xvi

Theorem .0.11. Let X= (1, X, X)be a c´adl´ag rough path and (Y, Y ′)a controlled rough

path, then

∫T

Yr−dXr:= lim

|P|→0∑

[s,t]∈P

YsXs,t +Y′

sXs,t.

where the limit exist in refinement Riemann sum (RRS/net convergence) sense.

The c´adl´ag rough integration can be used to give pathwise interpretation of gerenal

stochastic integration. For example, if Xis a L´evy process, it can be naturally lifted to

a rough path by defining

Xs,t := ∫t

Xs,r−⊗dXr

or a geometric rough path by

s,t := ∫t

Xs,r−⊗dXr

where dXrstands for Marcus integration. We verify that such specification actually

constructs a rough path associated to Xand that the rough integration so obtained

matches almost surely with stochastic integration. The solutions to SDEs driven by L´evy

processes can be now seen as corresponding RDE solution. We also introduce rough path

variant of Marcus differential equations as introduced in [39]. In fact, we show that the

signature St=S(X)0,t is the solution of a Marcus type RDE

dSt=St⊗dXt

given the meaning

St= 1 + ∫t

Sr−⊗dXr+∑

0<s≤t

Ss−⊗{exp(log(2) ∆Xs)−∆Xs}

This allows us to compute explicitly the expected value of signature of a L´evy process.

Theorem .0.12. Let Xis a L´evy process associated to L´evy triple (a, b, K)with co-

varinace matrix a, drift band jump measure K. If the measure K1|y|≥1has all finite

moments, then

E[S(X)0,T ] = exp [T(b+1

2a+∫(exp(y)−1−y1|y|<1)K(dy))]

The above formula resembles the well known L´evy-Khintchine formula for character-

stic function of L´evy processes and thus we name it L´evy-Khintchine formula for rough

paths. It is an generalisation of the formula obtained in [13] for the expected signature of

Brownian motion. We give further generalisation of such formula for L´evy rough paths,

defined as Lie group G2(Rd) valued L´evy processes with appropriate variational regular-

ity. We also produce a partial integral differential equation (PIDE) for computing the

expected signature for Markov jump diffusions. Variants of such equations for the case

of Brownian diffusions was carried out in [66].

The contents of this work is collected together in a paper [18](submitted) jointly with

Prof. Peter Friz

xvii

III. Loewner chains driven by semimartingales

In this article, we consider the Loewner chains driven by semimartingales in the con-

text of existence of trace. Apart from producing more interesting examples of Loewner

chains, there is another motivation for considering such models. The deep insight of Oded

Schramm leading to construction of SLE was that domain Markov property (DMP) to-

gether with conformal invariance (CI) forces the driver to have independent and stationary

increment, i.e. Brownian motion with some speed. It is possible to canonically produce

some models which fails to have DMP and CI globally, but do possess these properties

on a local scale. See construction of SLEκ,ρ in [41] for example. Such processes is of great

importance to study the symmetries like duality and reversibility of SLE. Heuristically

speaking, having DMP and CI on a local scale forces the driver to have independent and

stationary increment locally, i.e. diffusions. The main result of this article is the following

deterministic inequality well suited to obtain the existence of trace for random drivers

beyond Brownian motion. To state it, we assume that driver that U: [0, T]→Rhas

finite quadratic-variation in sense of F¨ollmer if (along some fixed sequence of partitions

π= (πn) of [0, T], with mesh-size going to zero)

∃lim

n→∞ ∑

[r,s]∈πn

(Us∧t−Ur∧t)2=: [U]π

and defines a continuous map t↦→ [U]π

t≡[U]t. For each fixed t > 0, define βs=

Ut−Ut−s, s ∈[0, t] and [β]πbe the corresponding quadratic variation.

Theorem .0.13. Let d[β]π

s/ds ≤κ < 2. Then there exist a path C1path Gsuch that ˙

is F¨ollmer integrable against βand

|f′

t(iy +Ut)| ≤ exp[Mπ

t−∫t

rd(r+1

2[β]r)]

where ∫t

0˙

Grdπβr=Mπ

tis the F¨ollmer type integral.

As a result, we are able to prove the following theorem on the existence of trace for

Loewner chains driven by semimartingales.

Theorem .0.14. For each κ < 2, there exist a constant α0(κ)depending only on κsuch

that following holds :

Let Utis a continuous process such that for each t∈[0, T],βs=Ut−Ut−sis a semi-

martingale w.r.t. some filtration with canonical decomposition

βs=Ns+As

with local martingale part Nand bounded variation part A. Assume for all s≤t≤T,

|d[N]s

ds | ≤ κ

and

sup

t∈[0,T]

E[exp(α∫t

rdr)]<∞

for some α > α0(κ). Then the Loewner chain generated by Uis generated by a trace.

xviii

In a special case when Uis a (deterministic) Cameron-Martin path, our method allows

us to prove the following uniform bound:

Theorem .0.15. If Uis a Cameron-Martin path, then

|f′

t(iy +Ut)| ≤ exp[1

4||U||2

H](4)

where ||U||H={∫T

0˙

rdr}1

is the Cameron-Martin norm.

Note that for Cameron-Martin paths U, by Cauchy-Schwarz inequality

|Ut−Us|=|∫t

Urdr| ≤ √t−s√∫t

rdr

which implies

inf

ϵ>0sup

|t−s|<ϵ

|Ut−Us|

√t−s= 0 <4

and the existence of trace follows from results in [62], but it can also be seen directly

from inequality 4. Additionally, as remarked in [33], the Holder regularity of the trace γ

improves as 1

2-Holder norm of the driver gets smaller. Since Cameron-Martin paths are

of vanishing 1

2Holder norm, we can expect the trace to be as regular as possible. Note

that when U≡0, γt= 2i√t, which is at best 1

2-Holder on [0, T], Lipchitz on time interval

[ϵ, T] for ϵ > 0 and is of bounded variation. Bounds like 4 allows us to prove,

Theorem .0.16. If Uis Cameron-Martin path, then ||γ||1

2,[0,T]<∞and for any ϵ > 0,

||γ||1,[ϵ,T]<∞. In fact under suitable time reparametrization φ,||γ◦φ||1<∞and thus

γis a bounded variation path.

Stability under approximation type results follow as corollary:

Theorem .0.17. If Uis Cameron-Martin and Unis a sequence of piecewise linear ap-

proximation to U, then for any α < 1

||γn−γ||α→0as n→ ∞

The contents of this article is collected together in a paper [19] (upcoming) jointly

with Prof. Peter Friz.

IV. Slow points and trace of Loewner chains

Precise pathwise property of the driver responsible for the existence of trace is still un-

known. In this article, we seek for deterministic conditions on the Loewner driver Uwhich

will guarantee the existence of trace and which will also relate to the case Ut=√κBt.

We relate the existence of slow points for the driver to the existence of the trace for

correponding Loewner chain.

xix

Definition .0.18. Given a curve Uand a > 0, point t > 0is called a a-slow point (from

below ) if

lim sup

h→0+

|Ut−h−Ut|

√h≤a

The main result of this article is the following Theorem.

Theorem .0.19. Let Ube bounded variation path. For t > 0and s∈[0, t], define

βs=βt

s:= Ut−Ut−sand ||β||s:= ||β||1−var,[0,s]. Assume for all t > 0,

lim sup

s→0+

||β||s

√s<2 (5)

and ∫t

√rd||β||r<∞(6)

Then the Loewner chain driven by Uis generated by a simple curve.

The slow points are known to exist for Brownian motion and have been extensively

studied in the literature. We refer the following well known result due to Davis, Perkins

and Greenwood.

Theorem .0.20 (Davis, Perkins and Greenwood).For standard Brownian motion B,

almost surely

inf

t∈[0,1] lim sup

h→0+

|Bt+h−Bt|

√h= 1

In particular, a-slow points exists almost surely for a > 1.

Theorem .0.19 and .0.20 gives strong evidence of connection between slow points and

the trace of Loewner chains. But as required in Theorem .0.19, not all the times tare

slow points of Brownian motion. There do exist points where the Brownian motion has

unusually large growth in infinitely many small scales (such points are called fast points)

and Brownian motion falls short to satisfy conditions of Theorem .0.19. We propose

another simple direct approach to the trace which only relies on a weaker condition that

Uis psuedo (a, 1

2)-Holder for some a < 4 according to the following definition.

Definition .0.21. A continuous curve Uis called (a, 1

2)- psuedo Holder if

sup

lim inf

h→0+ |Ut+h−Ut|

√h≤a

We recall a result due to B. Davis [11] suggesting that Brownian motion paths are

psuedo Holder.

Theorem .0.22 (B. Davis).For Brownian motion B, almost surely,

sup

lim inf

h→0+

Bt+h−Bt

√h= 1

Except for the missing modulus, Theorem .0.22 suggest that Brownian motion paths

are almost surely psuedo Holder. Also note that for a fixed time t, since zero set of

Brownian motion has no isolated points, almost surely,

lim inf

h→0+ |Bt+h−Bt|

√h= 0

Conjecture 1. Brownian motion paths are almost surely (1,1

2)- psuedo Holder.

For SLE(κ) process, the driver Ut=√κBt. Thus our approach works well for κ < 16

and potentially answers the role of Brownian motion in the existence of trace for SLE(κ),

including the unresolved case κ= 8.

Contents

I Doob-Meyer Theorem for rough paths 1

I.1 Introduction.................................. 3

I.2 Truely ”rough” paths and a deterministic Doob-Meyer result . . . . . . . 3

I.3 True roughness of stochastic processes . . . . . . . . . . . . . . . . . . . 5

I.3.1 Gaussian processes . . . . . . . . . . . . . . . . . . . . . . . . . . 6

I.4 Anapplication ................................ 8

II General rough integration, L´evy rough paths and a L´evy–

Kintchine type formula 9

II.1 Motivation and contribution of this paper . . . . . . . . . . . . . . . . . 11

II.2 Preliminaries ................................. 12

II.2.1 “General” Young integration . . . . . . . . . . . . . . . . . . . . . 12

II.2.2 “General” Itˆo stochastic integration . . . . . . . . . . . . . . . . . 13

II.2.3 Marcus canonical integration . . . . . . . . . . . . . . . . . . . . . 13

II.2.4 “Continuous” rough integration . . . . . . . . . . . . . . . . . . . 14

II.2.5 Geometric rough paths and signatures . . . . . . . . . . . . . . . 15

II.2.6 Checking p-variation ......................... 17

II.2.7 Expected signatures . . . . . . . . . . . . . . . . . . . . . . . . . 18

II.2.8 L´evyProcesses ............................ 19

II.2.9 The work of D. Williams . . . . . . . . . . . . . . . . . . . . . . . 19

II.3 General rough paths: definition and first examples . . . . . . . . . . . . . 20

II.4 The minimal jump extension of cadlag rough paths . . . . . . . . . . . . 24

II.5 Rough integration with jumps . . . . . . . . . . . . . . . . . . . . . . . . 29

II.6 Rough differential equations with jumps . . . . . . . . . . . . . . . . . . 35

II.7 Roughpathstability ............................. 38

II.8 Rough versus stochastic integration . . . . . . . . . . . . . . . . . . . . . 38

II.9 L´evy processes and expected signature . . . . . . . . . . . . . . . . . . . 40

II.9.1 A L´evy–Khintchine formula and rough path regularity . . . . . . 40

II.9.2 L´evyroughpaths........................... 44

II.9.3 Expected signatures for L´evy rough paths . . . . . . . . . . . . . 48

II.9.4 The moment problem for random signatures . . . . . . . . . . . . 50

II.10 Further classes of stochastic processes . . . . . . . . . . . . . . . . . . . . 51

II.10.1 Markov jump diffusions . . . . . . . . . . . . . . . . . . . . . . . . 51

II.10.2 Semimartingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

II.10.3 Gaussian processes . . . . . . . . . . . . . . . . . . . . . . . . . . 54

CONTENTS

III Loewner chains driven by semimartingales 57

III.1Introduction.................................. 59

III.2 Proof of Theorem III.1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

III.3 Reverse Brownian filtration and diffusion driven Loewner chains . . . . . 69

III.4 Regularity and stability under approximation of the trace . . . . . . . . . 71

IV Slow points and the trace of Loewner chains 75

IV.1Introduction.................................. 77

IV.2 Proof of Theorem IV.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

IV.2.1 Existence of the limit 53 . . . . . . . . . . . . . . . . . . . . . . . 78

IV.2.2 Continuity of map t→γt....................... 82

IV.3Furtherdiscussions .............................. 82

Part I

Doob-Meyer Theorem for rough

paths

I.1. INTRODUCTION 3

Contents of this article is collected together in a published paper [20] appearing in

Bulletin of the Institute of Mathematics, Academia Sinica (New Series), 2012.

I.1 Introduction

Hairer–Pillai [57] proposed the notion of θ-roughness of a path which leads to a deter-

ministic Norris lemma, i.e. some sort of quantitative Doob-Meyer decomposition, for

(level-2, H¨older) rough integrals in the sense of Gubinelli. It is possible to check that

this roughness condition holds for fractional Brownian motion (fBm); indeed in [57] the

author show θ-roughness for any θ > H where Hdenotes the Hurst parameter. (Recall

that Brownian motion corresponds to H= 1/2; in comparison, the regime H < 1/2

should be thought of as ”rougher”.) All this turns out to be a key ingredient in their

H¨ormander type result for stochastic differential equations driven by fBm, any H > 1/3,

solutions of which are in general non-Markovian.

In the present note we take a step back and propose a natural ”roughness” condition

relative to a given p-rough path (of arbitrary level [p]=1,2, ..) in the sense of Lyons; the

aim being a Doob-Meyer result for (general) rough integrals in the sense of Lyons. The

interest in our (weaker) condition is that it is immediately verified for large classes of

Gaussian processes, also in infinite dimensions. (In essence one only needs a Khintchine

law of iterated logarithms for 1-dimensional projections.)

We conclude with an application to non-Markovian systems under H¨ormander’s con-

dition, in the spirit of [23].

I.2 Truely ”rough” paths and a deterministic Doob-

Meyer result

Let Vbe a Banach-space. Let p≥1. Assume f∈Lipγ(V, L (V, W)), γ > p −1, and

X: [0, T]→Vto be a p-rough path in the sense of T. Lyons [48, 50] controlled by ω.

Recall that such a rough path consists of a underlying path X: [0, T]→V, together

with higher order information which somewhat prescribes the iterated integrals ∫·

0dXt1⊗

... ⊗dXtkfor 1 < k ≤[p].

Definition I.2.1. For fixed s∈[0, T)we call X”rough at time s” if (convention 0/0 :=

(∗) : ∀v∗∈V∗\{0}: lim sup

t↓s

|⟨v∗, Xs,t⟩|

ω(s, t)2/p = +∞.

If Xis rough on some dense set of [0, T], we call it truely rough.

Theorem I.2.2. (i) Assume Xis rough at time s. Then

∫t

f(X)dX=O(ω(s, t)2/p)as t↓s=⇒f(Xs)=0.

(i’) As a consequence, if Xis truely rough, then

∫·

f(X)dX≡0on [0, T] =⇒f(X·)≡0on [0, T].

(i”) As another consequence, assume g∈C(V, W)and |t−s|=O(ω(s, t)2/p), satisfied

e.g. when ω(s, t)≍t−sand p≥2(the ”rough” regime of usual interest) then

∫·

f(X)dX+∫·

g(X)dt ≡0on [0, T] =⇒f(X·), g (X·)≡0on [0, T].

(ii) Assume Z:= X⊕Ylifts to a rough path and set, with ˜

f(z) (x, y) := f(z)x,

∫f(Z)dX:= ∫˜

f(Z)dZ.

Then the conclusions from (i),(i’) and (i”), with g=g(Z), remain valid.

Remark I.2.3. Solutions of rough differential equations dY =V(Y)dXin the sense of

Lyons are understood in the integral sense, based on the integral defined in (ii) above.

This is our interest in this (immediate) extension of part (i).

Proof. (i) A basic estimate (e.g. [10]) for the W-valued rough integral is

∫t

f(X)dX =f(Xs)Xs,t +O(ω(s, t)2/p).

By assumption, for fixed s∈[0, T), we have

0 = f(Xs)Xs,t

ω(s, t)2/p +O(1) as t↓s

and thus, for any w∗∈W∗,

|⟨v∗, Xs,t⟩|

ω(s, t)2/p := ⏐⏐⏐⏐⏐⟨w∗,f(Xs)Xs,t

ω(s, t)2/p ⟩⏐⏐⏐⏐⏐

=O(1) as t↓s;

where v∗∈V∗is given by V∋v↦→ ⟨w∗, f (Xs)v⟩recalling that f(Xs)∈L(V, W).

Unless v∗= 0, the assumption (∗) implies that, along some sequence tn↓s, we have

the divergent behaviour |⟨v∗, Xs,tn⟩|/ω (s, tn)2/p → ∞, which contradicts that the same

expression is O(1) as tn↓s. We thus conclude that v∗= 0. In other words,

∀w∗∈W∗, v ∈V:⟨w∗, f (Xs)v⟩= 0.

and this clearly implies f(Xs) = 0. (Indeed, assume otherwise i.e. ∃v:w:= f(Xs)v=

0. Then define ⟨w∗, λw⟩:= λand extend, using Hahn-Banach if necessary, w∗from

span(w)⊂Wto the entire space, such as to obtain the contradiction ⟨w∗, f (Xs)v⟩= 1.)

(i”) From the assumptions, ∫t

sg(Xr)dr ≤ |g|∞|t−s|=O(ω(s, t)2/p).We may thus

use (i) to conclude f(Xs) = 0 on s∈[0, T). It follows that ∫·

0g(Xr)dr ≡0 and by

differentiation, g(X·)≡0 on [0, T].

(ii) By definition of ∫f(Z)dXand ˜

frespectively,

∫t

f(Z)dX:= ∫t

f(Z)dZ

=˜

f(Zs)Zs,t +O(ω(s, t)2/p)

=f(Zs)Xs,t +O(ω(s, t)2/p)

and the identical proof (for (i’), then (i”)) goes through, concluding f(Zs) = 0.

I.3. TRUE ROUGHNESS OF STOCHASTIC PROCESSES 5

Remark I.2.4. The reader may wonder about the restriction to p≥2in (i”) for H¨older

type controls ω(s, t)≍t−s. Typically, when p < 2, one uses Young theory, thereby

avoiding the full body of rough path theory. That said, one can always view a path of finite

p-variation, p < 2, as rough path of finite 2-variation (iterated integrals are well-defined as

Young integrals). Moreover, by a basic consistency result, the respective integrals (Young,

rough) coincide. In the context of fBM with Hurst parameter H∈(1/2,1) ,for instance,

we can take p= 2 and note that in this setting fBM is truely rough (cf. below for a general

argument based on the law of iterated logarithm). By the afore-mentioned consistency,

the Doob–Meyer decomposition of (i”) then becomes a statement about Young integrals.

Such a decomposition was previously used in [6].

Remark I.2.5. The argument is immediately adapted to the Gubinelli setting of ”con-

trolled” paths and would (in that context) yield uniqueness of the derivative process.

Remark I.2.6. In definition I.2.1, one could replace the denominator ω(s, t)2/p by

ω(s, t)θ, say for 1/p < θ ≤2/p. Unlike [57], where 2/p −θaffects the quantitative

estimates, there seems to be no benefit of such a stronger condition in the present con-

text.

I.3 True roughness of stochastic processes

Fix ρ∈[1,2) and p∈(2ρ, 4). We assume that the V-valued stochastic process Xlifts to

a random p-rough path. We assume V∗separable which implies separability of the unit

sphere in V∗and also (by a standard theorem) separability of V. (Separability of the

dual unit sphere in the weak-∗topology, guaranteed when Vis assumed to be separable,

seems not enough for our argument below.)

The following 2 conditions should be thought of as a weak form of a LIL lower bound,

and a fairely robust form of a LIL upper bound. As will be explained below, they are

easily checked for large classes of Gaussian processes, also in infinite dimensions.

Condition 1. Set ψ(h) = h1

2ρ(ln ln 1/h)1/2. Assume (i) there exists c > 0such that for

every fixed dual unit vector ϕ∈V∗and s∈[0, T)

P[lim sup

t↓s|ϕ(Xs,t)|/ψ (t−s)≥c]= 1

and (ii) for every fixed s∈[0, T),

P[lim sup

t↓s

|Xs,t|V

ψ(t−s)<∞]= 1

Theorem I.3.1. Assume Xsatisfies the above condition. Then Xis a.s. truely rough.

Proof. Take a dense, countable set of dual unit vectors, say K⊂V∗. Since Kis countable,

the set on which condition (i) holds simultanously for all ϕ∈Khas full measure,

P[∀ϕ∈K: lim sup

t↓s|ϕ(Xs,t)|/ψ (t−s)≥c]= 1

On the other hand, every unit dual vector ϕ∈V∗is the limit of some (ϕn)⊂K. Then

|⟨ϕn, Xs,t⟩|

ψ(t−s)≤|⟨ϕ, Xs,t⟩|

ψ(t−s)+|ϕn−ϕ|V∗|Xs,t|V

ψ(t−s)

so that, using lim (|a|+|b|)≤lim (|a|) + lim (|b|), and restricting to the above set of full

measure,

c≤lim

t↓s|⟨ϕn, Xs,t⟩|

ψ(t−s)≤lim

t↓s|⟨ϕ, Xs,t⟩|

ψ(t−s)+|ϕn−ϕ|V∗lim

t↓s|Xs,t|V

ψ(t−s).

Sending n→ ∞ gives, with probability one,

c≤lim

t↓s|⟨ϕ, Xs,t⟩|

ψ(t−s).

Hence, for a.e. sampe X=X(ω) we can pick a sequence (tn) converging to ssuch that

|⟨ϕ, Xs,tn⟩|/ψ (tn−s)≥c−1/n. On the other hand, for any θ≥1/(2ρ)

|⟨ϕ, Xs,tn(ω)⟩|

|tn−s|θ=|⟨ϕ, Xs,tn(ω)⟩|

ψ(tn−s)

|tn−s|θ

≥(c−1/n)|tn−s|1

2ρ−θL(tn−s)

→ ∞

since c > 0 and θ≥1/(2ρ) and slowly varying L(τ) := (ln ln 1/τ)1/2(in the extreme

case θ= 1/(2ρ) the divergence is due to the (very slow) divergence L(τ)→ ∞ as

τ=tn−s→0 .)

I.3.1 Gaussian processes

The conditions put forward here are typical for Gaussian process (so that the pairing

⟨ϕ, X⟩is automatically a scalar Gaussian process). Sufficient conditions for (i), in fact,

a law of iterated logarithm, with equality and c= 1 are e.g. found in [59, Thm 7.2.15].

These conditions cover immediately - and from general principles - many Gaussian (rough

paths) examples, including fractional Brownian motion (ρ= 1/(2H), lifted to a rough

path [10, 21]) and the stationary solution to the stochastic heat equation on the torus,

viewed as as Gaussian processes parametrized by x∈[0,2π]; here ρ= 1, the fruitful lift

to a ”spatial” Gaussian rough path is due to Hairer [30].

As for condition (ii), it holds under a very general condition [21, Thm A.22]

∃η > 0 : sup

0≤s,t≤T

Eexp (η|Xs,t|2

|t−s|1/ρ )<∞.

In presence of some scaling, this condition is immediately verfied by Fernique’s theorem.

Example 1. d-dimensional fBM is a.s. truely rough (in fact, H-rough)

In order to apply this in the context of (random) rough integration, we need to

intersect the class of truely rough Gaussian processes with the classes of Gaussian pro-

cesses which amit a rough path lift. To this end, we recall the following standard setup

[26]. Consider a continuous d-dimensional Gaussian process, say X, realized as coordi-

nate process on the (not-too abstract) Wiener space (E, H, µ) where E=C([0, T],Rd)

I.3. TRUE ROUGHNESS OF STOCHASTIC PROCESSES 7

equipped with µis a Gaussian measure s.t. Xhas zero-mean, independent components

and that Vρ-var (R, [0, T]2), the ρ-variation in 2D sense of the covariance Rof X, is finite

for ρ∈[1,2). (In the fBM case, this condition translates to H > 1/4). From [21, The-

orem 15.33] it follows that we can lift the sample paths of Xto p-rough paths for any

p > 2ρand we denote this process by X, called the enhanced Gaussian process. In this

context, modulo a deterministic time-change, condition (ii) will always be satisfied (with

the same ρ). The non-degeneracy condition (i), of course, cannot be expect to hold true

in this generality; but, as already noted, conditions are readily available [59].

Example 2. Q-Wiener processes are a.s. truely rough. More precisely, consider a sepa-

rable Hilbert space Hwith ONB (ek),(λk)∈l1,λk>0for all k, and a countable sequence

(βk)of independent standard Brownians. Then the limit

Xt:= ∞

∑

k=1

λ1/2

kβk

tek

exists a.s. and in L2, uniformly on compacts and defines a Q-Wiener process, where

Q=∑λk⟨ek,·⟩ is symmetric, non-negative and trace-class. (Conversely, any such

operator Qon Hcan be written in this form and thus gives rise to a Q-Wiener process.)

By Brownian scaling and Fernique, condition (ii) is obvious. As for condition (i), let ϕ

be an arbitrary unit dual vector and note that ϕ(X·)/σϕis standard Brownian provided

we set

σ2

ϕ:= ∑λk⟨ϕ, ek⟩2>0.

By Khintchine’s law of iterated logarithms for standard Brownian motion, for fixed ϕand

s, with probability one,

lim sup

t↓s|ϕ(Xs,t)|/ψ (t−s)≥√2σϕ.

Since ϕ↦→ σ2

ϕis weakly continuous (this follows from (λ)∈l1and dominated convergence)

and compactness of the unit sphere in the weak topology, c:= inf σϕ>0, and so condition

(ii) is verified.

Let us quickly note that Q-Wiener processes can be naturally enhanced to rough

paths. Indeed, it suffices to define the H⊗H-valued ”second level” increments as

(s, t)↦→ Xs,t := ∑

i,j

λ1/2

iλ1/2

j∫t

βi

s,·◦dβjei⊗ej.

which essentially reduces the construction of the ”area-process” to the L´evy area of

a 2-dimensional standard Brownian motion. (Alternatively, one could use integration

against Q-Wiener processes.) Rough path regularity, |Xs,t|H⊗H=O(|t−s|2α)for some

α∈(1/3,1/2] (in fact: any α < 1/2), is immediate from a suitable Kolmogrov-type or

GRR criterion (e.g. [16, 21]).

Variations of the scheme are possible of course, it is rather immediate to define Q-

Gaussian processes in which (βk)are replaced by (Xk), a sequence of independent Gaus-

sian processes, continuous each with covariance uniformly of finite ρ-variation, ρ < 2.

Let us insist that the (random) rough integration against Brownian, or Q-Wiener

processes) is well-known to be consistent with Stratonovich stochastic integration (e.g.

[50, 21, 16]). In fact, one can also construct a rough path lift via Itˆo-integration, in this

case (random) rough integration (now against a ”non-geometric” rough path) coincides

with Itˆo-integration.

I.4 An application

Let Xbe a continuous d-dimensional Gaussian process which admits a rough path lift

in the sense described at the end of the previous section. Assume in addition that

the Cameron-Martin space Hhas complementary Young regularity in the sense that H

embeds continuously in Cq-var ([0, T],Rd)with 1

p+1

q>1. Note q≤pfor µis supported

on the paths of finite p-variation. This is true in great generality with q=ρwhenever

ρ < 3/2 and also for fBM (and variations thereof) for all H > 1/4. Complementary

Young regularity of the Cameron-Martin space is a natural condition, in particular in

the context of Malliavin calculus and has been the basis of non-Markovian H¨ormander

theory, the best results up to date were obtained in [23] (existence of density only, no

drift, general non-degenerate Gaussian driving noise) and then [57] (existence of a smooth

density, with drift, fBM H > 1/3). We give a quick proof of existence of density, with

drift, with general non-degenerate Gaussian driving noise (including fBM H > 1/4). To

this end, consider the rough differential equation

dY =V0(Y)dt +V(Y)dX

subject to a weak H¨ormander condition at the starting point. (Vector fields, on Re, say

are assumed to be bounded, with bounded derivatives of all orders.) In the drift free

case, V0= 0, conditions on the Gaussian driving signal Xwhere given in [23] which

guarantee existence of a density. The proof (by contradiction) follows a classical pat-

tern which involves a deterministic, non-zero vector zs.t. zTJX(ω)

0←· (Vk(Y·(ω))) ≡0 on

[0,Θ (ω)),every k∈ {1, .., d}for some a.s. positive random time Θ. (This follows from

a global non-degeneracy condition, which, for instance, rules out Brownian bridge type

behaviour, and a 0-1 law, see conditions 3,4 in [23]). From this

∫·

zTJX

0←t([V, Vk] (Yt)) dX+∫·

zTJX

0←t([V0, Vk] (Yt)) dt ≡0

on [0,Θ (ω)); here V= (V1, ..Vd) and V0denote smooth vector fields on Realong which

the RDEs under consideration do not explode. Now we assume the driving (rough) path

to be truely rough, at least on a positive neighbourhood of 0. Since Z:= (X, Y, J) can be

constructed simultanously as rough path, say Z, we conclude with Theorem I.2.2, (iii):

zTJX

0←· ([Vl, Vk] (Y·)) ≡0≡zTJX

0←· ([V0, Vk] (Yt)) .

Usual iteration of this argument shows that zis orthogonal to V1, .., Vdand then all Lie-

brackets (also allowing V0), always at y0. Since the weak-H¨ormander condition asserts

precisely that all these vector fields span the tangent space (at starting point y0) we

then find z= 0 which is the desired contradiction. We note that the true roughness

condition on the driving (rough) path replaces the support type condition put forward in

[23]. Let us also note that this argument allows a painfree handling of a drift vector field

(not including in [23]); examples include immediately fBM with H > 1/4 but we have

explained above that far more general driving signals can be treated. In fact, it transpires

true roughness of Q-Wiener processes (and then, suitables generalizations to Q-Gaussian

processes) on a seperable Hilbert space Hallows to obtain a H¨ormander type result where

the Q-process ”drives” countably many vectorfields given by V:Re→Lin (H,Re).

The Norris type lemma put forward in [57] suggests that that the argument can

be made quantitative, at least in finite dimensions, thus allowing for a H¨ormander type

theory (existence of smooth densities) for RDE driven by general non-degenerate Gaussian

signals. (In [57] the authors obtain this result for fBM, H > 1/3 .)

Part II

General rough integration, L´evy

rough paths and a L´evy–Kintchine

type formula

II.1. MOTIVATION AND CONTRIBUTION OF THIS PAPER 11

Contents of this article is collected together in an upcoming paper [18].

II.1 Motivation and contribution of this paper

An important aspect of “general” theory of stochastic processes [35, 70, 69] is its ability to

deal with jumps. On the other hand, the (deterministic) theory of rough paths [55, 51, 29,

56, 22, 17] has been very successful in dealing with continuous stochastic processes (and

more recently random fields arising from SPDEs [31, 17]). It is a natural question to what

extent there is a “general” rough path theory which can handle jumps and ultimately

offers a (rough)pathwise view on stochastic integration against c´adl´ag processes. In the

spirit of Marcus canonical equations (e.g. [39, 1]) related questions were first raised by

Williams [80] and we will comment in more detail in Section II.2.9 on his work and the

relation to ours. We can also mention the “pathwise” works of Mikosch–Norvaiˇsa [63] and

Simon [75], although their works assumes Young regularity of sample paths (q-variation,

q < 2) and thereby does not cover the “rough” regime of interest for general processes.

Postponing the exact definition of “general” (by convention: c´adl´ag) rough path, let

us start with a list of desirable properties and natural questions.

•An analogue of Lyons’ fundamental extension theorem (Section II.2.5 below for a

recall) should hold true. That is, any general geometric p-rough path Xshould

admit canonically defined higher iterated integrals, thereby yielding a group-like

element (the “signature” of X).

•A general rough path Xshould allow the integration of 1-forms, and more general

suitable “controlled rough paths” Yin the sense of Gubinelli [17], leading to rough

integrals of the form

∫f(X−)dXand ∫Y−dX.

•Every semimartingale X=X(ω) with (rough path) Itˆo-lift XI=XI(ω), should

give rise to a (random) rough integral that coincides under reasonable assumptions

with the Itˆo-integral, so that a.s.

(Itˆo) ∫f(X−)dX =∫f(X)dXI.

•As model case for both semi-martingales and jump Markov process, what is the

precisely rough path nature of L´evy processes? In particular, it would be desirable

to have a class of “L´evy rough paths” that captures natural (but “non-canoncial”)

examples such as the pure area Poisson process or the Brownian rough path in a

magnetic field?

•To what extent can we compute the expected signature of such processes? And

what do we get from it?

In essence, we will give reasonable answers to all these points. We have not tried to

push for maximal generality. For instance, in the spirit of Friz–Hairer [17, Chapter 3-5], we

develop general rough integration only in the level 2-setting, which is what matters most

for probability. But that said, the required algebraic and geometric picture to handle the

level N-case is still needed in this paper, notably when we discuss the extension theorem

and signatures. For the most, we have chosen to work with (both “canonically” and

“non-canonically” lifted) L´evy processes as model case for random c´adl´ag rough paths,

this choice being similar to choosing Brownian motion over continuous semimartingales.

In the final chapter we give discuss some extensions, notably to Markov jump diffusions

and some simple Gaussian examples.

In his landmark paper [55, p.220], Lyons gave a long and visionary list of advantages

(to a probabilist) of constructing stochastic objects in a pathwise fashion: stochastic

flows, differential equations with boundary conditions, Stroock–Varadhan support theo-

rem, stochastic anlysis for non-semimartingales, numerical algorithms for SDEs, robust

stochastic filtering, stochastic PDE with spatial roughness. Many other applications have

been added to this list since. (We do not attempt to give references; an up-to-date bib-

liography with many applications of the (continuous) rough path theory can be found

e.g. in [17].) The present work lays in particular the foundation to revisit many of these

problems, but not allowing for systematic treatment of jumps. We also note that inte-

gration against general rough paths can be considered as a generalization of the F¨ollmer

integral [14] and, to some extent, Karandikar [38], (see also Soner et al. [77]1), but free

of implicit semimartingale features.

II.2 Preliminaries

II.2.1 “General” Young integration

[81, 12]

We briefly review Young’s integration theory. Consider a path X: [0, T]→Rdof

finite p-variation, that is

∥X∥p-var;[0,T ]:= ⎛

⎝sup

P∑

[s,t]∈P |Xs,t|p⎞

⎠

1/p

<∞

with Xs,t =Xt−Xsand sup (here and later on) taken over all for finite partitions P

of [0, T]. As is well-known tsuch paths are regulated in the sense of admitting left- and

right-limits. In particular, X−

t:= lims↑tXsis c´agl´ad and X+

t:= lims↓tXsc´adl´ag (by

convention: X−

0≡X0, X+

T≡XT). Let us write X∈Wp([0, T]) for the space of c´adl´ag

path of finite p-variation. A generic c´agl´ad path of finite q-variation is then given by Y−

for Y∈Wq([0, T]). Any such pair (X, Y −) has no common points of discontinuity on the

same side of a point and the Young integral of Y−against X,

∫T

Y−dX ≡∫T

Y−

rdXr≡∫T

Ys−dXs,

is well-defined (see below) provided 1/p + 1/q > 1 (or p < 2, in case p=q). We need

Definition II.2.1. Assume S=S(P)is defined on the partitions of [0, T]and takes

values in some normed space.

(i) Convergence in Refinement Riemann–Stieltjes (RRS) sense: we say (RRS) lim|P|→0S(P) =

Lif for every ε > 0there exists P0such that for every “refinement” P ⊃ P0one has

1There is much renewed interest in this theories from a model independent finance point of view.

II.2. PRELIMINARIES 13

|S(P)−L|< ε.

(ii) Convergence in Mesh Riemann–Stieltjes (MRS) sense: we say (MRS) lim|P|→0S(P) =

Lif for every ε > 0there exists δ > 0s.t. ∀P with mesh |P| < δ, one has |S(P)−L|< ε.

Theorem II.2.2 (Young).If X∈Wpand Y∈Wqwith 1

p+1

q>1, then the Young

integral is given by

∫T

Y−dX := lim

|P|→0∑

[s,t]∈P

Y−

sXs,t = lim

|P|→0∑

[s,t]∈P

YsXs,t (7)

where both limit exist in (RRS) sense. Moreover, Young’s inequality holds in either form

⏐⏐⏐⏐∫t

Y−dX −Y−

sXs,t⏐⏐⏐⏐

.

Y−

q-var;[s,t]∥X∥p-var;[s,t],(8)

⏐⏐⏐⏐∫t

Y−dX −YsXs,t⏐⏐⏐⏐

.∥Y∥q-var;[s,t]∥X∥p-var;[s,t].(9)

At last, if X, Y are continuous (so that in particular Y−≡Y), the defining limit of the

Young integral exists in (MRS) sense.

Everything is well-known here, although we could not find the equality of the limits in

(7) pointed out explicitly in the literature. The reader can find the proof in Proposition

II.5.1 below.

II.2.2 “General” Itˆo stochastic integration

[35, 70, 69]

Subject to the usual conditions, any semimartingale X=X(ω) may (and will) be

taken with c´adl´ag sample paths. A classical result of Monroe allows to write any (real-

valued) martingale as a time-change of Brownian motion. As an easy consequence, semi-

martingales inherit a.s. finite 2+variation of sample paths from Brownian sample paths.

See [45] for much more in this direction, notably a quantification of ∥X∥p-var;[0,T ]for any

p > 2 in terms of a BDG inequality. Let now Ybe another (c´adl´ag) semimartingale, so

that Y−is previsible. The Itˆo integral of Y−against Xis then well-defined, and one has

the following classical Riemann–Stieltjes type description,

Theorem II.2.3 (Itˆo).The Itˆo integral of Y−against Xhas the presentation, with

i=i T

2n,

∫T

Y−dX = lim

n∑

Y−

i−1Xtn

i−1,tn

i= lim

n∑

Ytn

i−1Xtn

i−1,tn

i(10)

where the limits exists in probability, uniformly in Tover compacts.

Again, this is well-known but perhaps the equality of the limits in (10) which the

reader can find in Protter [69, Chapter 2, Theorem 21].

II.2.3 Marcus canonical integration

[60, 61, 39, 1] Real (classical) particles do not jump, but may move at extreme speed.

In this spirit, transform X∈Wp([0, T]) into ˜

X∈Cp-var([0,˜

T]), by ”streching” time

whenever

Xt−Xt−≡∆tX= 0,

followed by replacing the jump by a straight line connecting Xs−with Xs, say

[0,1] ∋θ↦→ Xt−+θ∆tX.

Implemented in a (c´adl´ag) semimartingale context , this leads to Marcus integration

∫T

f(X)dX := ∫T

f(Xt−)dXt+1

2∫T

Df (Xt−)d[X, X]c

+∑

t∈(0,T]

∆tX{∫1

f(Xt−+θ∆tX)−f(Xt−)}dθ.

(A Young canonical integral, providied p < 2 and f∈C1, is defined similarly, it suffices

to omit the continuous quadratic variation term.) A useful consequence, for f∈C3(Rd),

say, is the chain-rule

∫t

∂if(X)Xi=f(Xt)−f(X0).

It is also possible to implement this idea in the context of SDE’s,

dZt=f(Zt)dXt(11)

for f:Rd→Rd×kwhere Xis a semi-martingale, [39]. The precise meaning of this

Marcus canoncial equation is given by

Zt=Z0+∫t

f(Zs−)dXs+1

2∫t

f′f(Zs)d[X, X]c

+∑

0<s≤t{φ(f∆Xs, Zs−)−Zs−−f(Zs−)∆Xs}

=Z0+∫t

f(Zs−)dXs+1

2∫t

f′f(Zs)d[X, X]s

+∑

0<s≤t{φ(f∆Xs, Zs−)−Zs−−f(Zs−)∆Xs−f′f(Zs)1

2(∆Xs)⊗2}

where φ(g, x) is the time 1 solution to ˙y=g(y), y(0) = x. As one would expected fromt

the afore-mentioned (first order) chain-rule, such SDEs respect the geometry.

Theorem II.2.4 ([39]).If Xis a c´adl´ag semi-martingale and fand f′fare globally

Lipchitz, then solution to the Marcus canoncial SDE (11) exists uniquely and it is a

c´adl´ag semimartingale. Also, if Mis manifold without boundary embedded in Rdand

{fi(x) : x∈M}1≤i≤kare vector fields on M, then

P(Z0∈M) = 1 =⇒P(Zt∈M∀t≥0) = 1.

II.2.4 “Continuous” rough integration

[55, 29, 17]

Young integration of (continuous) paths has been the inspiration for the (continuous)

rough integration, elements of which we now recall. Consider p∈[2,3) and X= (X, X)∈

Cp-var([0, T]) which in notation of [17] means validity of Chen’s relation

Xs,u =Xs,t +Xt,u +Xs,t ⊗Xt,u (12)

II.2. PRELIMINARIES 15

and ∥X∥p-var := ∥X∥p-var +∥X∥1/2

p/2-var <∞, where

∥X∥p/2-var := ⎛

⎝sup

P∑

[s,t]∈P |Xs,t|p/2⎞

⎠

2/p

For nice enough F(e.g. F∈C2), both Ys:= F(Xs) and Y′:= DF(Xs) are in Cp-var

and we have

∥R∥p/2-var =⎛

⎝sup

P∑

[s,t]∈P |Rs,t|p/2⎞

⎠

2/p

<∞where Rs,t := Ys,t −Y′

sXs,t.(13)

Theorem II.2.5 (Lyons, Gubinelli).Write Pfor finite partitions of [0, T]. Then

∃lim

|P|→0∑

[s,t]∈P

YsXs,t +Y′

sXs,t =: ∫T

Y dX

where the limit exists in (MRS) sense, cf. Definition II.2.1.

Rough integration extends immediately to the integration of so-called controlled rough

paths, that is, pairs (Y, Y ′) subject to (13). This gives meaning to a rough differential

equation (RDE)

dY =f(Y)dX

provided f∈C2, say: A solution is simply as a path Ysuch that (Y, Y ′) := (Y, f(Y))

satisfies (13) and such that the above RDE is satisfied in the (well-defined!) integral

sense, i.e. for all t∈[0, T],

Yt−Y0=∫t

f(Y)dX.

II.2.5 Geometric rough paths and signatures

[55, 51, 56, 22]

Ageometric rough path X= (X, X) is rough path with Sym (Xs,t) = 1

2Xs,t ⊗Xs,t ; and

we write X= (X, X)∈ Cp-var

g([0, T]) accordingly. We work with generalized increments of

the form Xs,t = (Xs,t,Xs,t) where we write Xs,t =Xt−Xsfor the path increment, while

second order increments Xs,t are determined from (X0,t) by Chen’s relation

X0,s +X0,s ⊗Xs,t +Xs,t =X0,t.

Behind all this is the picture that X0,t := (1, X0,t,X0,t) takes values in a Lie group

T(2)

1(Rd)≡ {1}⊕Rd⊕Rd×d, embedded in the (truncated) tensor algebra T(2) (Rd), and

Xs,t =X−1

0,s ⊗X0,t. From the usual power series in this tensor algebra one defines, for

a+b∈Rd⊕Rd×d,

log (1 + a+b) = a+b−1

2a⊗a,

exp (a+b) = 1 + a+b+1

2a⊗a.

The linear space g(2)(Rd) = Rd⊕so (d) is a Lie algebra under

[a+b, a′+b′] = a⊗a′−a′⊗a;

its exponential image G(2)(Rd) := exp (g(2)(Rd))is then a Lie (sub) group under

(1, a, b)⊗(1, a′, b′) = (1, a +a′, b +a⊗a′+b′).

At last we recall that G(2)(Rd) admits a so-called Carnot–Caratheodory norm (abbrevi-

ated as CC norm henceforth), with infimum taken over all curves γ: [0,1] →Rdof finite

length L,

∥1 + a+b∥CC : = inf (L(γ) : γ1−γ0=a, ∫1

(γt−γ0)⊗dγt=b)

≍ |a|+|b|1/2

≍ |a|+|Anti (b)|1/2.

A left-invariant distance is induced by the group structure,

dCC (g, h) = 

g−1⊗h

CC

which turns G(2)(Rd) into a Polish space. Geometric rough paths with roughness param-

eter p∈[2,3) are precisely classical paths of finite p-variation with values in this metric

space.

Proposition II.2.6. X = (X, X)∈ Cp-var

g([0, T]) iff X= (1, X, X)∈Cp-var ([0, T], G(2)(Rd)).

Moreover,

∥X∥p-var ≍⎛

⎝sup

P∑

[s,t]∈P∥Xs,t∥p

CC⎞

⎠

1/p

The theory of geometric rough paths extends to all p≥1, and a geometric p-rough

path is a path with values in G([p]) (Rd), the step-[p] nilpotent Lie group with dgenerators,

embedded in T([p]) (Rd), where

T(m)(Rd)=⊕m

k=0 (Rd)⊗k⊂ ⊕∞

k=0 (Rd)⊗k⊂T((Rd))

(the last inclusion is strict, think polyomials versus power-series) and again of finite p-

variation with respect to the Carnot–Caratheodory distance (now defined on G([p])).

Theorem II.2.7 (Lyons’ extension).Let 1≤m:= [p]≤p≤N < ∞. A (continuous)

geometric rough path X(m)∈Cp-var ([0, T], G(m))admits an extension to a path X(N)with

values G(N)⊂T(N), unique in the class of G(N)-valued path starting from 1and of finite

p-variation with respect to CC metric on G(N). In fact,



X(N)

p-var;[s,t].

X(m)

p-var;[s,t].

Remark II.2.8. In view of this theorem, any X∈Cp-var ([0, T], G(m))may be regarded

as X∈Cp-var ([0, T], G(N)), any N≥m, and there is no ambiguity in this notation.

II.2. PRELIMINARIES 17

Definition II.2.9. Write π(N)resp. πMfor the projection T((Rd)) →T(N)(Rd)resp.

(Rd)⊗M. Call g∈T((Rd)) group-like, if π(N)(g)∈G(N)for all N. Consider a geometric

rough path X∈Cp-var ([0, T], G[p]). Then, thanks to the extension theorem,

S(X)0,T := (1, π1(X), .., πm(X), πm+1 (X), ..)∈T((Rd))

defines a group-like element, called the signature of X.

The signature solves a rough differential equation (RDE, ODE if p= 1) in the tensor-

algebra,

dS =S⊗dX, S0= 1.(14)

To a significant extent, the signature determines the underlying path X, if of bounded

variation, cf. [32]. (The rough path case was recently obtained in [5]). A basic, yet

immensely useful fact is that multiplication in T((Rd)), if restricted to group-like elements,

can be linearized.

Proposition II.2.10. (Shuffle product formula) Consider two multiindices v= (i1, .., im), w =

(j1, .., jn)

XvXw=∑Xz

where the (finite) sum runs over all shuffles zof v, w.

II.2.6 Checking p-variation

[43, 27, 58]

(i) As in whenever γ > p −1>0, with tn

k=k2−nT, one has

∥X∥p

p-var;[0,T ].∞

∑

n=1

nγ

∑

k=1 ⏐⏐⏐Xtn

k−Xtn

k−1⏐⏐⏐

p.(15)

This estimate has been used [43] to verify finite (sample path) p-variation, simply by

taking expectation, e.g. for the case of Brownian motion by using that E[|Bs,t|p] =

|t−s|1+ϵfor ϵ > 0, provided p > 2. Unfortunately, this argument does not work for jump

processes. Even for the standard Poisson process one only has E[|Ns,t|p]∼Cp|t−s|

as t−s→0, so that the expected value of the right-hande side of (15) is infinity. An

extension of (15) to rough path is

∥X∥p

p-var;[0,T ].∞

∑

n=1

nγ

∑

k=1 {⏐⏐⏐Xtn

k−1,tn

k⏐⏐⏐

p+⏐⏐⏐Xtn

k−1,tn

k⏐⏐⏐

p/2}

and we note that for a geometric rough path X= (X, X), i.e. when Sym (Xs,t) =

2Xs,t ⊗Xs,t, we may replace Xon the right-hand side by the area A= Anti (X). This

has been used in [51], again by taking expecations, to show that Brownian motion B

enhanced with Bs,t := ∫t

sBs,r ⊗◦dBrconstitutes a.s. an element in the rough path space

Cp-var([0, T], G(2)), for p∈(2,3).

(ii) In [27] an embedding result Wδ,q ↪→Cp-var is shown, more precisely

∥X∥q

p-var;[0,T ].∫T

0∫T

|Xt−Xs|q

|t−s|1+δq dsdt,

provided 1 < p < q < ∞with δ= 1/p ∈(0,1). The extension to rough paths reads

∥X∥q

p-var;[0,T ].∫T

0∫T

0{|Xs,t|q

|t−s|1+δq +|Xs,t|q/2

|t−s|1+δq }dsdt.

Since elements in Wδ,q are also α-H¨older, with α=δ−1/q > 0, these embeddings are

not suitable for non-continuous paths.

(iii) In case of a strong Markov process Xwith values in some Polish space (E, d), a

powerful criterion has been established by Manstavicius [58]. Define

α(h, a) := sup {P(d(Xs,x

t, x)≥a)}

with sup taken over all x∈E, and s<tin[0, T] with t−s≤h. Under the assumption

α(h, a).hβ

aγ,

uniformly for h, a in a right neighbourhood of zero, the process Xhas finite p-variation for

any p > γ/β. In the above Poisson example, noting E[|Ns,t|] = O(h) whenever t−s≤h,

Chebychev inequality immediately gives α(h, a)≤h/a, and we find finite p-variation,

any p > 1. (Of course p= 1 here, but one should not expect this borderline case from a

general criterion.) The Manstavicius criterion will play an important role for us.

II.2.7 Expected signatures

[13, 32, 42, 9]

Recall that for a smooth path X: [0, T]→Rd, its signature S=S(X) is given by

the group-like element

(1,∫0<t1<T

dXr1,∫0<t1<t2<T

dXr1⊗dXr2, ..)∈T((Rd)).

The signature solves an ODE in the tensor-algebra,

dS=S⊗dX, S0=1.(16)

Generalizations to semimartingales are immediate, by interpretation of (16) as Itˆo, Stratonovich

or Marcus stochastic differential equation. In the same spirit Xcan be replaced by a

generic (continuous) geometric rough path with the according interpretation of (16) as

(linear) rough differential equation.

Whenever X=X(ω), or X=X(ω) is granted sufficient integrability, we may con-

sider the expected signature, that is

EST∈T((Rd))

defined in the obvious componentwise fashion. To a significant extent, this object behaves

like a moment generating function. In a recent work [9], it is shown that under some mild

condition, the expected signature determines the law of the ST(ω).

II.2. PRELIMINARIES 19

II.2.8 L´evy Processes

[72, 4, 1, 34]

Recall that a d-dimensional L´evy process (Xt) is a stochastically continuous process

such that (i) for all 0 <s<t<∞, the law of Xt−Xsdepends only on t−s; (ii) for

all t1, .., tksuch that 0 < t1< .. < tkthe random variables Xti+1 −Xtiare independent.

L´evy process can (and will) be taken with c´adl´ag sample paths and are characterized by

the L´evy triplet (a, b, K), where a= (ai,j) is a positive semidefinite symmetric matrix,

b= (bi) a vector and K(dx) a L´evy measure on Rd(no mass at 0, integrates min(|x|2,1))

so that

E[ei⟨u,Xt⟩]= exp (−1

2⟨u, au⟩+i⟨u, b⟩+∫Rd

(eiu y −1−iu y1{|y|<1})K(dy)).(17)

The Itˆo–L´evy decomposition asserts that any such L´evy process may be written as,

Xt=σBt+bt +∫(0,t]×{|y|<1}

y˜

N(ds, dy) + ∫(0,t]×{|y|≥1}

yN (ds, dy) (18)

where Bis a d-dimensional Browbian motion, σσT=a, and N(resp. ˜

N) is the Pois-

son random meausre (resp. compensated PRM) with intensity ds K (dy). A Markovian

description of a L´evy process is given in terms of its generator

(Lf) (x) = 1

∑

i,j=1

ai,j∂i∂jf+

∑

i=1

bi∂if+∫Rd(f(x+y)−f(x)−1{|y|<1}

∑

i=1

yi∂if)K(dy).

(19)

By a classical result of Hunt [34], this characterization extends to L´evy process with

values in a Lie group G, defined as above, but with Xt−Xsreplaced by X−1

sXt. Let

{u1, .., um}be a basis of the Lie algebra g, thought of a left-invariant first order differential

operators. In the special case of exponential Lie groups, meaning that exp : g→Gis an

analytical diffeomorphism (so that g= exp (xiui) for all g∈G, with canonical coordinates

xi=xi(g) of the first kind) the generator reads

(Lf) (x) = 1

∑

v,w=1

av,wuvuwf+

∑

v=1

bvuvf+∫G(f(xy)−f(x)−1{|y|<1}

∑

v=1

yvuvf)K(dy).

(20)

As before the L´evy triplet (a, b, K) consists of (av,w) positive semidefinite symmetric,

b= (bv) and K(dx) a L´evy measure on G(no mass at the unit element, integrates

min(|x|2,1), with |x|2:= ∑m

i=v(xv)2.)

II.2.9 The work of D. Williams

[80]

Williams first considers the Young regime p∈[1,2) and shows that every X∈

Wp([0, T]) may be turned into ˜

X∈Cp-var([0,˜

T]), by replacing jumps by segments

of straight lines (in the spirit of Marcus canonical equations, via some time change

[0, T]→[0,˜

T]) Crucially, this can be done with a uniform estimate || ˜

X||p-var .||X||p-var.

In the rough regime p≥2, Williams considers a generic d-dimensional L´evy process X

enhanced with stochastic area

As,t := Anti ∫(s,t]

(X−

r−Xs)⊗dXr

where the stochastic integration is understood in Itˆo-sense. On a technical level his main

results [80, p310-320] are summarized in

Theorem II.2.11 (Williams).Assume Xis a d-dimensional L´evy process Xwith triplet

(a, b, K).

(i) Assume Khas compact support. Then

E[|As,t|2].|t−s|2.

(ii) For any p > 2, with sup taken over all partitions of [0, T],

sup

P∑

[s,t]∈P |As,t|p/2<∞a.s.

Clearly, (X, A) (ω) is all the information one needs to have a (in our terminology)

cadlag geometric p-rough path X=X(ω), any p∈(2,3). However, Williams does not

discuss rough integration, nor does he give meaning (in the sense of an integral equation)

to a rough differential equations driven by cadlag p-rough paths. Instead he constructs,

again in the spirit of Marcus, ˜

X∈ Cp-var([0,˜

T]), and then goes on to define a solution Y

to an RDE driven by X(ω) as reverse-time change of a (classical) RDE solution driven

by the (continuous) geometric p-rough path ˜

X. While this construction is of appealing

simplicity, the time-change depends in a complicated way on the jumps of X(ω) and the

absence of quantitative estimates, makes any local analysis of so-defined RDE solution

difficult (starting with the identification of Yas solution to the corresponding Marcus

canonical equation). We shall not rely on any of Williams’ result, although his ideas will

be visible at various places in this paper. A simplified proof of Theorem II.2.11 will be

given below.

II.3 General rough paths: definition and first exam-

ples

The following definitions are fundamental.

Definition II.3.1. Fix p∈[2,3). We say that X= (X, X)is a general (c´adl´ag) rough

path over Rdif

(i) Chen’s relation holds, i.e. for all s≤u≤t,Xs,t −Xs,u −Xu,t =Xs,u ⊗Xu,t;

(ii) the following map

[0, T]∋t↦→ X0,t = (X0,t,X0,t)∈Rd⊕Rd×d

is c´adl´ag;

(iii) p-variation regularity in rough path sense holds, that is

∥X∥p-var;[0,T]+∥X∥1/2

p/2-var;[0,T]<∞.

We then write

X∈ Wp=Wp([0, T],Rd).

II.3. GENERAL ROUGH PATHS: DEFINITION AND FIRST EXAMPLES 21

Definition II.3.2. We call X∈ Wpgeometric if it takes values in G(2)(Rd), in symbols

X∈ Wp

g. If, in addition,

(∆tX, ∆tA) := log ∆tX∈Rd⊕{0} ⊂ g(2)(Rd)

we call XMarcus-like, in symbols X∈ Wp

As in the case of (continuous) rough paths, cf. Section II.2.5,

g:= Wp

g([0, T],Rd)=Wp([0, T], G(2)(Rd))

so that general geometric p-rough paths are precisely paths of finite p-variation in G(2)(Rd)

equipped with CC metric. We can generalize the definition to general p∈[1,∞) at the

price of working in the step-[p] free nilpotent group,

g=Wp([0, T], G([p])).

As a special case of Lyons’ extension theorem (Theorem II.2.7), for a given continuous

path X∈Wpfor p∈[1,2), there is a unique rough path X= (X, X)∈ Wp. (Uniqueness

is lost when p≥2, as seen by the perturbation ¯

Xs,t =Xs,t +a(t−s), for some matrix

a.)

The situation is different in presence of jumps and Lyons’ First Theorem fails, even

when p= 1. Essentially, this is due to the fact that there are non-trivial pure jump paths

of finite q-variation with q < 1.

Proposition II.3.3 (Canonical lifts of paths in Young regime).Let X∈Wp([0, T],Rd)

be a c´adl´ag path of finite p-variation for p∈[1,2). (i) It is lifted to a (in general,

non-geometric) rough path X= (X, X)∈ Wpby enhancing Xwith

Xs,t = (Young) ∫(s,t]

Xs,r−⊗dXr

(ii) It is lifted to a Marcus-like c´adl´ag rough path XM=(X, XM)∈ Wp

Mby enhancing X

with

s,t =Xs,t +1

2∑

r∈(s,t]

(∆rX)⊗(∆rX).

Proof. As an application of Young’s inequality, it is easy to see that

|Xs,t|.||X||2

p-var;[s,t]

Note that ω(s, t) := ||X||p

p-var;[s,t]is superadditive, i.e. for all s<u<t,ω(s, u)+ω(u, t)≤

ω(s, t), so that

∑

[s,t]∈P |Xs,t|p

2.∑

[s,t]∈P ||X||p

p-var;[s,t].||X||p

p-var;[0,T ]

Taking sup over P,Xhas p

2variation. We then note that

⎛

⎝∑

r∈(s,t]

(∆rX)⊗(∆rX)⎞

⎠

≤⎛

⎝∑

r∈(s,t]|∆rX|2⎞

⎠

≤∑

r∈(s,t]|∆rX|p

Since the jumps of Xare p-summable, we immediately conclude that XMalso is of

finite p

2variation.

Also, from “integration by parts formula for sums”, it can be easily checked that

Sym(XM

s,t) = 1

2Xs,t ⊗Xs,t. The fact that (X, XM)forms a Marcus-like rough path comes

from the underlying idea of the Marcus integral replaces jumps by straight lines which

do not create area. Precisely,

lim

s↑tXM

s,t =: ∆tXM=1

2(∆tX)⊗2

which is symmetric. Thus ∆tA=Anti(∆tXM) = 0.

Clearly, in the continuous case every geometric rough path is Marcus-like and so there

is need to distinguish them. The situation is different with jumps and there are large

classes of Marcus-like as well as non-Marcus-like geometric rough paths. We give some

examples.

Example 3 (Pure area jump rough path).Consider a so (d)-valued path (At)of

finite 1-variation, started at A0= 0. Then

X0,t := exp (At)

defines a geometric rough path, for any p≥2, i.e. X(ω)∈ Wp

gbut, unless Ais continu-

ous,

X(ω)/∈Wp

It is not hard to randomize the above non-Marcus Mrough path example.

Example 4 (Pure area Poisson process).Consider an i.i.d. sequence of a so (d)-

valued r.v. (an(ω)) and a standard Poisson process Ntwith rate λ > 0. Then, with

probability one,

X0,t (ω) := exp (Nt

∑

n=1

an(ω))

yields a geometric, non-Marcus like c´adl´ag rough path for any p≥2.

It is instructive to compare the last examples with the following two classical examples

from (continuous) rough path theory.

Example 5 (Pure area rough path).Fix a∈so (d). Then

X0,t := exp (at),

yields a geometric rough path, X∈ Cp

g([0, T],Rd), above the trivial path X≡0, for any

p∈[2,3).

Example 6 (Brownian rough path in magnetic field).Write

s,t =(Bs,t,∫t

Bs,r ⊗◦dBr)

II.3. GENERAL ROUGH PATHS: DEFINITION AND FIRST EXAMPLES 23

for the Brownian rough path based on iterated Stratonovich integration. If one considers

the (zero-mass) limit of a physical Brownian particle, with non-zero charge, in a constant

magnetic field [24] one finds the (non-canonical) Brownian rough path

0,t := BS0,t + (0,at),

for some a∈so(d). This yields a continuous, non-canonical geometric rough path lift of

Brownian motion. More precisely, Bm∈ Cp

g([0, T],Rd)a.s, for any p∈(2,3).

As is well-known in rough path theory, it is not trivial to construct suitable Xgiven

some (irregular) path X, and most interesting constructions are of stochastic nature. At

the same time, Xdoes not determine X, as was seen in the above examples. That said,

once in possession of a (c´adl´ag) rough path, there are immediate ways to obtain further

rough paths, of which we mention in particular perturbation of Xby increments of some

p/2-variation path, and, secondly, subordination of (X, X) by some increasing (c´adl´ag)

path. For instance, in a stochastic setting, any time change of the (canonical) Brownian

rough path, by some L´evy subordinator for instance, will yield a general random rough

path, corresponding to the (c´adl´ag) rough path associated to a specific semimartingale.

For Brownian motion, as for (general) semimartingales, there are two “canoncial”

candidates for X, obtained by Itˆo- and Marcus canonical (=Stratonovich in absence of

jumps) integration, respectively. We have

Proposition II.3.4. Consider a d-dimensional (c´adl´ag) semimartingale Xand let p∈

(2,3). Then the following three statements are equivalent.

(i) XI(ω)∈ Wpa.s where XI= (X, XI)and

s,t := ∫t

Xs,r−⊗dXr(Itˆo)

(ii) XM(ω)∈ Wp

M(⊂ Wp

g)a.s. where XM= (X, XM)and

s,t := ∫t

Xs,r−⊗dXr(Marcus).

(iii) The stochastic area (identical for both Itˆo- and Marcus lift)

As,t := Anti (XI

s,t)= Anti (XM

s,t)

has a.s. finite p/2-variation.

Proof. Clearly

Sym(XM

s,t) =´1

2Xs,t ⊗Xs,t

is of finite p/2-variation, a consequence of X∈Wpa.s., for any p > 2. Note that XM−XI

is symmetric,

(XM

s,t)i,j −(XI

s,t)i,j =1

2[Xi, Xj]c

s,t +1

2∑

r∈(s,t]

∆rXi∆rXj.

and is of finite p

2variation as [Xi, Xj]cis of bounded variation, while

⏐⏐⏐⏐⏐⏐∑

r∈(s,t]

∆rXi∆rXj⏐⏐⏐⏐⏐⏐

≤⏐⏐⏐⏐⏐⏐

2∑

r∈(s,t]|∆rX|2⏐⏐⏐⏐⏐⏐

.∑

r∈(s,t]|∆rX|p<∞a.s.

because jumps of semimartingale is square summable and thus p≥2 summable.

We now given an elegant criterion which allows to check finite 2+-variation of G(2)-

valued processes.

Proposition II.3.5. Consider a G(2)(Rd)-valued strong Markov process Xs,t := X−1

s⊗

Xt= exp (Xs,t,As,t). Assume

E|Xs,t|2.|t−s|,

E|As,t|2.|t−s|2,

uniformly in s, t ∈[0, T]. Then, for any p > 2,

∥X∥p-var +∥A∥p/2-var <∞a.s.

Equivalently, ||X||p-var <∞a.s.

Proof. Consider s, t ∈[0, T] with |t−s| ≤ h. Then

P(|Xs,t| ≥ a)≤1

a2E|Xs,t|2.h

a2,

P(|As,t|1/2≥a)=P(|As,t| ≥ a2)≤1

a2E|As,t|

≤1

a2(E|As,t|2)1/2=h

a2.

From properties of the Carnot–Caratheodory metric dCC(Xs,Xt)≍ |Xs,t|+|As,t|1/2and

the above estimates yield

P(d(Xs,Xt)≥a).h

a2.

Applying the result of Manstavicius (cf. Section II.2.6) with β= 1, γ = 2 we obtain

a.s. finite p-variation of X, any p > γ/β = 2, with respect to dCC and the statement

follows.

As will be detailed in Section II.9.1 below, this criterion, combined with the ex-

pected signature of a d-dimensional L´evy process, provides an immediate way to recover

Williams’ rough path regularity result on L´evy process (Theorem II.2.11) and then sig-

nificantly larger classes of jump diffusions. With the confidence that there exists large

classes of random c´adl´ag rough paths, we continue to developt the deterministic theory.

II.4 The minimal jump extension of cadlag rough

paths

In view of Theorem II.2.7, it is natural to ask for such extension theorem for c´adl´ag

rough paths. (For continuous paths in Young regime, extension is uniquely given by

n-fold iterated young integrals.) However, in presence of jumps the uniqueness part of

Lyons’ extension theorem fails, as already seen by elementary examples of finite variation

paths.

II.4. THE MINIMAL JUMP EXTENSION OF CADLAG ROUGH PATHS 25

Example 7. Let p= 1, N = 2 and consider the trivial path X≡0∈W1([0,1],Rd),

identified with X≡(1,0) ∈W1([0,1], G(1)). Consider a non-trivial so (d)-valued cadlag

path a(t), of pure finite jump type, i.e.

a0,t =∑

s∈(0,t]

(finite)

∆as.

Then two possible lifts of Xare given by

X(2) ≡(1,0,0) ,˜

X(2)

t≡(1,0, at−a0)∈ W1-var

g=W1([0,1], G(2)).

We can generalize this example as follows.

Example 8. Again p= 1, N = 2 and consider X∈W1-var. Then

X(2)

t:= (1, Xt,XM

t)∈ W1-var

and another choice is given by

X(2)

t≡(1, Xt,XM

t+at−a0)∈ W1-var

whenever, at∈so(d)is piecewise constant, with finitely many jumps ∆at= 0.

Note that, among all such lifts ˜

X(2)

t, the X(2)

tis minimal in the sense that log(2) ∆X(2)

has no 2-tensor component, and in fact,

log(2) ∆X(2)

t= ∆Xt.

We have the following far-reaching extension of this example. Note that we consider

gn⊃gmin the obvious way whenever n≥m.

Theorem II.4.1 (Minimal jump extension).Let 1≤p < ∞and N∋n>m:= [p]. A

cadlag rough path X(m)∈ Wp

g=Wp([0, T], G(m))admits an extension to a path X(n)of

with values G(n)⊂T(n), unique in the class of G(n)-valued path starting from 1and of

finite p-variation with respect to CC metric on G(n)subject to the additional constraint

log(n)∆X(n)

t= log(m)∆X(m)

t.(21)

For the proof, we will adopt the Marcus / Willliams idea of introducing an artificial

additional time interval at each jump times of Xm, during which the jump will be suitably

traversed. Since Xmhas countably infinite many jumps, we number the jumps as follows.

Let t1is such that

||∆t1X(m)||CC = sup

t∈[0,T]{||∆tX(m)||CC}

Similarly, define t2with

||∆t2X(m)||CC = sup

t∈[0,T],t=t1{||∆tX(m)||CC}

and so on. Note that the suprema are always attained and if ||∆tX(m)||CC = 0, then

t=tkfor some k. Indeed, it readily follows from the c´adl´ag (or regulated) property that

for any ϵ > 0, there are only finitely many jumps with ||∆tX(m)||CC > ϵ.

Choose any sequence δk>0 such that ∑kδk<∞. Starting from t1, we recursively

introduce an interval of length δkat tn, during which the jump ∆tkX(m)is traversed

suitably, to get a continuous curve ˜

X(m)on the (finite) interval [0,˜

T] where

T=T+∞

∑

k=1

δk<∞.

Taking motivation from simple examples, in order to get minimal jump extensions, we

choose the “best possible” curve traversing the jump, so that it doesn’t create additional

terms in log(n)∆X(n)

t. If [a, a +δk]⊂[0,˜

T], is the jump segment corresponding to the kth

jump, define

γk

t= exp(m)(a+δk−t

δk

log(m)X(m)

tk−+t−a

δk

log(m)X(m)

tk)

Lemma II.4.2. γk: [a, a +δk]→G(m)is a continuous path of finite pvariation w.r.t.

the CC metric and we have the bound

||γk||p

p-var;[a,a+δk].||∆tkX(m)||p.(22)

Proof. Omit k. W.l.o.g. we can assume that γt= exp(m)((1 −t) log(m)x+tlog(m)y)

for t∈[0,1] for some x, y ∈G(m). Also, as an application of Campbell-Baker-Hausdorff

formula,

exp(m)(log(m)x)⊗exp(m)(tlog(m)(x−1⊗y)) = exp(m)((1 −t) log(m)x+tlog(m)y)

so that we can assume x= 1. At this point, we have

γs,t = exp(m)((t−s) log(m)y)

Also, since p≥m, it is easy to check that for z∈gmand λ∈[0,1],

||exp(m)(λx)||p.λ||exp(m)(z)||p

So,

||γs,t||p.(t−s)||y||p

which finishes the claim.

Lemma II.4.3. The curve ˜

X(m): [0,˜

T]→G(m)constructed as above from X(m)∈

Wp([0, T], G(m))is a continuous path of finite pvariation w.r.t. the CC metric and we

have the bound

||˜

X(m)||p-var;[0,˜

T].||X(m)||p-var;[0,T](23)

Proof. For simpler notation, omit mand write ˜

X,X. The curve ˜

Xis continuous by

construction. To see the estimate, introduce ω(s, t) = ||X||p

p−var,[s,t]with the notation

ω(s, t−) := ||X||p

p-var,[s,t). Note that ω(s, t) is superadditive. Call t′the preimage of

t∈[0,˜

T] under the time change from [0, T]→[0,˜

T]. Note that [0,˜

T] contains (possibly

countably many) jump segements Inof the form [a, a +δk). Let us agree that point in

these jump segments are “red” and all remaining points are “blue”. Note that jump

segements correspond to one point in the pre-image. For 0 ≤s < t ≤˜

T, there are

following possiblities;

II.4. THE MINIMAL JUMP EXTENSION OF CADLAG ROUGH PATHS 27

•Both s, t are blue, in which case ||˜

Xs,t||p=||Xs′,t′||p≤ ||X||p

p-var,[s′,t′]=ω(s′, t′)

•Both s, t are red and in same jump segment [a, a +δk), in which case

||˜

Xs,t||p≤ ||γ||p

p-var;[a,a+δk]

•Both s, t are red but in different jump segment s∈[a, a +δk) and t∈[b, b +δl), in

which case s′= (a+δk)′,t′= (b+δl)′and



Xs,t



p≤3p−1(



Xs,a+δk



p+



Xa+δk,b



p+



Xb,t



p).

≤3p−1(||γk||p

p-var;[a,a+δk]+||Xs′,t′−||p+||γl||p

p-var,[b,b+δl])

≤3p−1(||γk||p

p-var;[a,a+δk]+ω(s′, t′−) + ||γl||p

p-var,[b,b+δl])

•sis blue and t∈[a, a +δk) is red, in which case



Xs,t



p≤2p−1(



Xs,a



p+



Xa,t



≤2p−1(ω(s′, t′−) + ||γk||p

p-var;[a,a+δk])

•s∈[a, a +δk) is red and tis blue, then



Xs,t



p≤2p−1(



Xs,a+δk



p+



Xa+δk,t



≤2p−1(||γk||p

p-var;[a,a+δk]+ω(s′, t′))

In any case, by using Lemma II.4.2, we see that



Xs,t



p.ω(s′, t′) + ||∆s′X||p+||∆t′X||p

which implies for any partition Pof [0,˜

T],

∑

[s,t]∈P 



Xs,t



p.∑

[s′,t′]

ω(s′, t′) + ||∆s′X||p+||∆t′X||p

.ω(0, T) + ∑

0<s≤T||∆sX||p

Finally, note that

∑

0<s≤T||∆sX||p≤ ||X||p

p-var;[0,T ]

which proves the claim.

Proof of Theorem II.4.1. Since ˜

X(m)is continuous path of finite p-variation on [0,˜

T],

from Theorem II.2.7, it admits an extension ˜

X(n)taking values in G(n)starting from 1

for all n>m. We emphasize that S=˜

X(n)can be obtained as linear RDE solution to

dS =S⊗dX(m), S0= 1 ∈T(n).(24)

We claim that for each jump segment [a, a +δk],

X(n)

a,a+δk= exp(n)(log(m)(∆tkX(m)))

which amounts to proving that if γt= exp(m)((1 −t) log(m)x+tlog(m)y) for t∈[0,1] for

some x, y ∈G(m), then its extension γ(n)to G(n)satisfies

γ(n)

0,1= exp(n)(log(m)(x−1⊗y))

By Campbell-Baker-Hausdorff formula,

exp(m)(log(m)x)⊗exp(m)(tlog(m)(x−1⊗y)) = exp(m)((1 −t) log(m)x+tlog(m)y)

Thus,

γs,t = expm((t−s) logm(x−1⊗y))

Here we have used our crucial construction that log(m)γtis linear in t. Now by guessing

and uniqueness of Theorem II.2.7,

γ(n)

s,t = exp(n)((t−s) log(m)(x−1⊗y))

which proves that claim and defining X(n)

t′=˜

X(n)

tfinishes the existence part of Theorem

II.4.1.

For uniqueness, w.l.o.g., assume n=m+1. Let Z(n)

t=X(m)

t+Mtand Y(n)

t=X(m)

t+Nt

are two extension of X(m)

tas prescribed of Theorem II.4.1, where Mt, Nt∈(Rd)⊗n.

Consider

St=Z(n)

t⊗{Y(n)

t}−1= (X(m)

t+Mt)⊗(X(m)

t+Nt)−1= 1 + Mt−Nt

where the last equality is due to truncation in the (truncated) tensor product. This in

particular implies Stis in centre of the group G(n)(actually group Tn

1) and thus so is

S−1

s⊗St. So, by using symmetry and subadditivity of CC norm,

||S−1

s⊗St|| =||Y(n)

s⊗Z(n)

s,t ⊗{Y(n)

t}−1|| =||Z(n)

s,t ⊗{Y(n)

s,t }−1|| ≤ ||Z(n)

s,t ||+||Y(n)

s,t ||

which implies Stis of finite p-variation. Also,

∆tS=Y(n)

t−⊗∆tZ(n)⊗Y(n)

Since ∆tZ(n)= ∆tY(n), we see that log(n)∆tS= 0, i.e. Stis continuous. Thus, M−Nis a

continuous path in (Rd)⊗nwith finite p

n<1 variation, which implies Mt=Ntconcluding

the proof.

II.5. ROUGH INTEGRATION WITH JUMPS 29

Remark II.4.4. In the proof of uniqueness of minimal jump extension, we didn’t use the

structure of group G(n). The fact that the minimal jump extension takes value in G(n)

follows by construction. That said, if Z(n)and Y(n)are two extensions of X(m)taking

values in T(n)(Rd), of finite p-variation w.r.t. norm

||1 + g|| := |g1|+|g2|1

2+.. +|gn|1

and

∆tZ(n)= ∆tY(n)= exp(n)(log(m)(∆tX(m)))

then same argument asw above implies

Z(n)

t=Y(n)

Definition II.4.5 (Signature of a cadlag rough path).Given X∈Wp([0, T], G([p]))call

X(n)constructed above the step-nsignature of X. The T((Rd))-valued projective limit of

X(n)

0,T as n→ ∞ is called signature of Xover [0, T].

II.5 Rough integration with jumps

In this section, we will define rough integration for c´adl´ag rough paths in the spirit of

[81, 29] and apply this for pathwise understanding of stochastic integral. We restrict

ourselves to case p < 3. For p∈[1,2), Young integration theory is well established and

interesting case is for p∈[2,3). Recall the meaning of convergence in (RRS) sense, cf.

Definition II.2.1. In order to cause no confusion between following two choices of Riemann

sum approximation

S(P) := ∑

[s,t]∈P

YsXs,t

and

S′(P) := ∑

[s,t]∈P

Ys−Xs,t

we add that, if Xand Yare regulated paths of finite p-variation for p < 2, then

C:= (RRS) lim

|P|→0∑

[s,t]∈P

YsXs,t

exist if either Yis c´adl´ag or Yis c´agl´ad (left continuous with right limit) and Xis c´adl´ag.

This can be easily verified by carefully reviewing the proof of existence of Young

integral as in [12]. Note that we have restricted ourselves to left point evaluation in

Riemann sums. Thus if Yis a c´adl´ag path then,

C1:= (RRS) lim

|P|→0∑

[s,t]∈P

YsXs,t

and

C2:= (RRS) lim

|P|→0∑

[s,t]∈P

Ys−Xs,t

both exists. But it doesn’t cause any ambiguity because in fact they are equal.

Proposition II.5.1. If Xand Yare c´adl´ag paths of finite p-variation for p < 2, then

C1=C2.

Proof. For each ϵ > 0,

S(P) := ∑

[s,t]∈P

∆YsXs,t =∑

[s,t]∈P

∆Ys1|∆Ys|>ϵXs,t +∑

[s,t]∈P

∆Ys1|∆Ys|≤ϵXs,t

Since there are finitely many jumps of size bigger than ϵand Xis right continuous,

lim

|P|→0∑

[s,t]∈P

∆Ys1|∆Ys|>ϵXs,t = 0

On the other hand,

⏐⏐⏐⏐⏐⏐∑

[s,t]∈P

∆Ys1|∆Ys|≤ϵXs,t⏐⏐⏐⏐⏐⏐

≤∑

[s,t]∈P

(|∆Ys|21|∆Ys|≤ϵ)∑

[s,t]∈P |Xs,t|2

≤ε2−p∑

[s,t]∈P |∆Ys|p∑

[s,t]∈P |Xs,t|2

≤ε2−p∥Y∥p

p-var ∥X∥2

2-var

where we used p < 2 in the step. It thus follows that

lim

ϵ→0lim

|P|→0∑

[s,t]∈P

∆Ys1|∆Ys|≤ϵXs,t = 0

which proves the claim.

One fundamental difference between continuous and c´adl´ag cases is absence of uniform

continuity which implies small oscillation of a path in small time interval. This becomes

crucial in the construction of integral, as also can be seen in construction of Young

integral (see [12]) when the integrator and integrand are assumed to have no common

discontinuity on the same side of a point. This guarantees at least one of them to have

small oscillation on small time intervals.

Definition II.5.2. A pair of functions (Xs,t, Ys,t)defined for {0≤s≤t≤T}is called

compatible if for all ϵ > 0, there exist a partition τ={0 = t0< t1··· < tn=T}such

that for all 0≤i≤n−1,

Osc(X, [ti, ti+1]) ≤ϵOR Osc(Y, [ti, ti+1]) ≤ϵ

where Osc(Z, [s, t]) := sup{|Zu,v|⏐⏐s≤u≤v≤t}.

Proposition II.5.3. If Xis a c´adl´ag path and Yis c´agl´ad path, then (X, Y )is a com-

patible pair.

Proof. See [12]

II.5. ROUGH INTEGRATION WITH JUMPS 31

Ler X= (X, X) be c´adl´ag rough path in the sense of Definition II.3.1. For the purpose

of rough integration we will use a different enhancement

Xs,t =Xs,t + ∆sX⊗Xs,t

Note clearly that ˜

Xis also of finite p

2variation, ˜

X0,t is c´adl´ag path and for s≤u≤t,

Xs,t −˜

Xs,u −˜

Xu,t =X−

s,u ⊗Xu,t (25)

where X−

t:= Xt−,X−

0=X0= 0.

Lemma II.5.4. For any ϵ > 0, there exist a partition τ={0 = t0< t1··· < tn=T}

such that for all 0≤i≤n−1,

Osc(˜

X,(ti, ti+1)) ≤ϵ

Proof. Since ˜

X0,t is c´adl´ag, from (25), it follows that for each y∈(0, T), there exist a

δy>0 such that

Osc(˜

X,(y−δy, y)) ≤ϵand Osc(˜

X,(y, y +δy)) ≤ϵ

Similarly there exist δ0and δTsuch that Osc(˜

X,(0, δ0)) ≤ϵand Osc(˜

X,(T−δT, T)) ≤ϵ.

Now family of open sets

[0, δ0),(y−δy, y +δy), ..., (T−δT, T]

form a open cover of interval [0, T], so it has a finite subcover [0, δ0),(y1−δy1, y1+

δy1), .., (yn−δyn, yn+δyn),(T−δT, T]. Without loss of generality, we can assume that

each interval in the finite subcover is the first interval that intersects its previous one and

the claim follows by choosing

t0= 0, t1∈(y1−δy1, δ0), t2=y1, t3∈(y2−δy2, y1+δy1), ..., t2n+1 =T

Lemma II.5.5. For any c´agl´ad path Y, the pair (Y, ˜

X)is a compatible pair.

Proof. Choose a partition τsuch that for all [s, t]∈τ,

Osc(Z, (s, t)) ≤ϵ

for Z=Y, X, X−,˜

X. We refine the partition τby adding a common continuity point of

Yt,Xt,X−

tand ˜

X0,t in each interval (s, t). Note that such common continuity points will

exist because a regulated paths can have only countably many discontinuities. With this

choice of partition, we observe that on every odd numbered [s, t]∈τ,

Osc(˜

X,[s, t]) ≤ϵ

and on every even numbered [s, t]∈τ,

Osc(Y, [s, t]) ≤ϵ

Definition II.5.6. Given X∈Wp, a pair of c´adl´ag paths (Y, Y ′)of finite p-variation is

called controlled rough path if Rs,t =Ys,t −Y′

sXs,t has finite p

2-variation, in the sense

||R||p

2:= sup

P{∑

[s,t]∈P |Rs,t|p

2}2

p<∞.

It is easy to see that 1-forms (Yt, Y ′

t) := (f(Xt), f′(Xt)) for f∈C2is a controlled

rough path. Also ˜

Rs,t := Y−

s,t −Y′

s−X−

s,t

is also of finite p

2-variation and pair ( ˜

R, X) is a compatible pair.

Theorem II.5.7. Let X= (X, X)be a c´adl´ag rough path and (Y, Y ′)a controlled rough

path, then

∫T

Yr−dXr:= lim

|P|→0S(P) = lim

|P|→0S′(P)

where both limits exist in (RRS) sense, as introduced in Definition II.2.1 and

S(P) := ∑

[s,t]∈P

Ys−Xs,t +Y′

s−˜

Xs,t =∑

[s,t]∈P

Ys−Xs,t +Y′

s−(Xs,t + ∆sX⊗Xs,t)

S′(P) = ∑

[s,t]∈P

YsXs,t +Y′

sXs,t.

Furthermore, we have the following rough path estimates: there exist a constant Cde-

pending only on psuch that

⏐⏐⏐⏐∫t

Yr−dXr−Ys−Xs,t −Y′

s−˜

Xs,t⏐⏐⏐⏐≤C(||˜

R||p

2,[s,t]||X||p,[s,t]+||Y′−||p,[s,t]||˜

X||p

2,[s,t])

(26)

⏐⏐⏐⏐∫t

Yr−dXr−YsXs,t −Y′

sXs,t⏐⏐⏐⏐≤C(||R||p

2,[s,t]||X||p,[s,t]+||Y′||p,[s,t]||X||p

2,[s,t]).(27)

Proof. We first consider the approximations given by S(P). We first note that if ωis a

superadditive function defined on intervals, i.e. for all s≤u≤t

ω[s, u] + ω[u, t]≤ω[s, t]

then, for any partition Pof [s, t] into r≥2 intervals, there exist intervals [u−, u] and

[u, u+] such that

ω[u−, u+]≤2

r−1ω[s, t] (28)

Also, we can immediately verify that for Zof finite p-variation,

ω[s, t] := ||Z||p

p,[s,t]

defines and superadditive function and if ω1and ω2are two positive superaddtive func-

tions, then for α, β ≥0, α +β≥1,

ω:= ωα

1ωβ

II.5. ROUGH INTEGRATION WITH JUMPS 33

is also a superadditive function.

Now, it is enough to prove that for any ϵ > 0, there exist a partition τ(to be chosen

properly) such that for all refinement partition Pof τ,

|S(P)−S(τ)| ≤ ϵ

Choose p<p′<3 and let [s, t]∈τand Ps,t be the partition of [s, t] by refinement points

of P. Note that

ω[s, t] := ||˜

R||

p′

2,[s,t]||X||

p′

p′,[s,t]+||Y′−||

p′

p′,[s,t]||˜

X||

p′

2,[s,t]

is a superadditive and there exist u−< u < u+∈ Ps,t such that (28) holds. Using (25)

|S(Ps,t)−S(Ps,t \u)|=|˜

Ru−,uXu,u++Y′−

u−,u ˜

Xu,u+|

≤ ||˜

R||p′

2,[u−,u+]||X||p′,[u−,u+]+||Y′−||p′,[u−,u+]||˜

X||p′

2,[u−,u+]

≤(||˜

R||

p′

2,[u−,u+]||X||

p′

p′,[u−,u+]+||Y′−||

p′

p′,[u−,u+]||˜

X||

p′

2,[u−,u+])3

p′

≤C

(r−1) 3

p′

(||˜

R||

p′

2,[s,t]||X||

p′

p′,[s,t]+||Y′−||

p′

p′,[s,t]||˜

X||

p′

2,[s,t])3

p′

≤C

(r−1) 3

p′

(||˜

R||p′

2,[s,t]||X||p′,[s,t]+||Y′−||p′,[s,t]||˜

X||p′

2,[s,t])

where Cis a generic constant. Iterating this, since p′<3, we get that

|S(Ps,t)−Ys−Xs,t +Y′

s−˜

Xs,t| ≤ C(||˜

R||p′

2,[s,t]||X||p′,[s,t]+||Y′−||p′,[s,t]||˜

X||p′

2,[s,t])

Thus,

|S(P)−S(τ)| ≤ C∑

[s,t]∈τ||˜

R||p′

2,[s,t]||X||p′,[s,t]+||Y′−||p′,[s,t]||˜

X||p′

2,[s,t]

Note that ( ˜

R, X) and (Y′−,˜

X) are compatible pairs. Properly choosing τ,

|S(P)−S(τ)| ≤ Cϵ ∑

[s,t]∈τ||˜

R||

p′

2,[s,t]||X||

p′

p,[s,t]+||Y′−||

p′

p,[s,t]||˜

X||

p′

2,[s,t]

Finally, the term under summation sign is superadditive which thereby implies

|S(P)−S(τ)| ≤ Cϵ

Also, the estimate (26) follows immediately as a by product of the analysis above.

At last, let us deal with the case of Riemann sum approximations

S′(P) = ∑

[s,t]∈P

YsXs,t +Y′

sXs,t.

It suffices to consider the difference

S′(P)−S(P) = ∑

[s,t]∈P

Rs−,sXs,t + ∆Y′

sXs,t

and then use arguments similar as those in the proof of proposition II.5.1 to see tgat

(RRS) lim

|P|→0(S′(P)−S(P)) = 0.

The rest is then clear.

As an immediate corollary of (26) and (27), we have

Corollary II.5.8. For a controlled rough path (Y, Y ′),

(Zt, Z′

t) := (∫t

Yr−dXr, Yt)

is also a controlled rough path.

Corollary II.5.9. If (Y, Y ′)is a controlled rough path and Zt=∫t

0Yr−dXr, then

∆tZ= lim

s↑t∫t

Yr−dXr=Yt−∆tX+Y′

t−∆tX

where ∆tX= lim

s↑tXs,t.

Though we avoid to write down the long expression for the bounds of ||Z||p,||Z′||p

and ||RZ||p

2, it can be easily derived from (27). The important point here is that we can

again, for Ztaking value in suitable spaces, readily define

∫t

Zr−dXr

The rough integral defined above is also compatible with Young integral. If Xis a finite

p-variation path for p < 2, we can construct c´adl´ag rough path Xby

Xs,t := ∫t

(Xr−−Xs)⊗dXr

where right hand side is understood as a Young integral.

Proposition II.5.10. If X, Y are c´adl´ag path of finite pand qvariation respectively with

p+1

q>1, then for any θ > 0with 1

p+1

q≥1

θ,

Zs,t := ∫t

(Yr−−Ys)dXr

has finite θvariation. In particular, Xhas finite p

2variation.

Proof. From Young’s inequality,

|Zs,t|θ≤C||X||θ

p,[s,t]||Y||θ

q,[s,t]

If 1

p+1

q≥1

θ, right hand side is superaddtive, which implies ||Z||θ<∞

Theorem II.5.11. If X, Y are c´adl´ag paths of of finite p-variation for p < 2, then

∫t

Yr−dXr=∫t

Yr−dXr

II.6. ROUGH DIFFERENTIAL EQUATIONS WITH JUMPS 35

Proof. The difference between Riemann sum approximation of corresponding integrals

can be written as

S(P) = ∑

[s,t]∈P

Y′

s−˜

Xs,t

Choose p<p′<2. From Young’s inequality,

|˜

Xs,t|=|∫t

(Xr−−Xs−)⊗dXr| ≤ C||X−||p′,[s,t]||X||p′,[s,t]

(X−, X) is a compatible pair, which implies for properly chosen P,

|S(P)| ≤ Cϵ ∑

[s,t]∈P ||X−||

p′

p,[s,t]||X||

p′

p,[s,t]

Noting again that the term under summation sign is superadditive,

(RRS) lim

|P|→0S(P) = 0

II.6 Rough differential equations with jumps

In the case of continuous RDEs the difference between non-geometric (Itˆo-type) and

geometric situations, is entirely captured in one’s choice of the second order information

X, so that both cases are handled with the same notion of (continuous) RDE solution.

In the jump setting, the situation is different and a geometric notion of RDE solution

requires additional terms in the equation in the spirit of Marcus’ canonical (stochastic)

equations [60, 61, 39, 1]. We now define both solution concepts for RDEs with jumps, or

course they coincide in absence of jumps, (∆Xs,∆Xs)≡(0,0).

Definition II.6.1. (i) For suitable fand a c´adl´ag geometric p-rough path X= (X, X)∈

g, call a path Z(or better: controlled rough path (Z, f(Z))) solution to the rough

canonical equation

dZt=f(Zt)dXt

if, by definition,

Zt=Z0+∫t

f(Zs−)dXs+∑

0<s≤t

φ(f∆Xs+1

2[f, f] ∆Xs;Zs−)−Zs−−f(Zs−)∆Xs−f′f(Zs−) ∆Xs}

where, as in Section II.2.3, φ(g, x)is the time 1solution to ˙y=g(y), y(0) = x. When

Xis Marcus like, i.e. X∈ Wp

Mso that ∆Xs= (∆Xs)⊗2/2, this becomes

Zt=Z0+∫t

f(Zs−)dXs+∑

0<s≤t{φ(f∆Xs, Zs−)−Zs−−f(Zs−)∆Xs−f′f(Zs−)1

2(∆Xs)⊗2}.

(ii) For suitable fand a c´adl´ag p-rough path call a path Z(or better: controlled rough

path (Z, f(Z))) solution to the (general) rough differential equation

dZt=f(Zt−)dXt

if, by definition,

Zt=Z0+∫t

f(Zs−)dXs.

We shall not consider the solution type (ii) further here.

Theorem II.6.2. Fix initial data Z0. Then Zis a solution to dZt=f(Zt)dXtif and

only if ˜

Zis a solution to the (continuous) RDE

d˜

Zt=f(˜

Zt)d˜

where ˜

X∈ Cp

gis constructed from X∈ Wp

gas in Theorem II.4.1.

Proof. We illustrate the idea by considering Xof finite 1-variation, with one jump at

τ∈[0, T]. This jump time becomes an interval ˜

I= [a, a +δ]⊂[0,˜

T] = [0, T +δ] in the

stretched time scale. Now ˜

Z0,˜

T≈∑

[s,t]∈P

f(˜

Zs)˜

Xs,t

in the sense of (MRS) convergence, as |P| → 0. In particular, noting that ˜

Xs,t =

(t−s)

δ∆Xτwhenever [s, t]⊂[a, a +δ]

Za,a+δ= lim

|˜

P|→0∑

[s,t]∈˜

f(˜

Zs)˜

Xs,t =1

δ∫a+δ

f(˜

Zr)∆Xτdr

=⇒˜

Za,a+δ=φ(f∆Xτ,˜

Za)−˜

On the other hand, by refinement of P, we may insist that the end-point of ˜

Iare contained

in Pwhich thus has the form

P=P1∪˜

P ∪P2

and so

Z0,˜

T≈∑

[s,t]∈P1

f(˜

Zs)˜

Xs,t +∑

[s,t]∈˜

f(˜

Zs)˜

Xs,t +∑

[s,t]∈P2

f(˜

Zs)˜

Xs,t

from which we learn, by sending ⏐⏐⏐

P⏐⏐⏐→0, that

Z0,˜

T≈∑

[s,t]∈P1

f(˜

Zs)˜

Xs,t +φ(f∆Xτ,˜

Za)−˜

Za+∑

[s,t]∈P2

f(˜

Zs)˜

Xs,t.

We now switch back to the original time scale. Of course, Z≡˜

Zon [0, τ) while Zt=˜

Zt+δ

on [τ, T] and in particular

Z0,T =˜

Z0,˜

Zτ−=˜

Zτ=˜

Za+δ.

But then, with P′

1and P′

2partitions of [0, τ] and [τ, T], respectively,

Z0,T ≈∑

[s′,t′]∈P′

t′<τ

f(Zs′)Xs′,t′+∑

[s′,t′]∈P′

f(Zs′)Xs′,t′+φ(f∆Xτ, Zτ−)−Zτ−

≈∑

[s′,t′]∈P′

f(Zs′)Xs′,t′+φ(f∆Xτ,˜

Za) + {φ(f∆Xτ, Zτ−)−Zτ−−f(Zτ−) ∆Xτ}

II.6. ROUGH DIFFERENTIAL EQUATIONS WITH JUMPS 37

since f(Zs′)Xs′,τ →f(Zτ−) ∆Xτas |P′| → 0, with [s′, τ]∈ P′. By passing to the (RRS)

limit, find

Z0,T =∫T

f(Z−

s)dX +{φ(f∆Xτ, Zτ−)−Zτ−−f(Zτ−) ∆Xτ}.

This argument extends to countable many jumps. We want to show that

ZT=Z0+∫T

f(Zs−)dXs+∑

0<s≤T{...}

=Z0+ (RRS) lim

|P|→0∑

[s,t]∈P

f(Zs)Xs,t + lim

η↓0∑

s∈(0,t]:

|∆Xs|>η

{...}.

What we know is (MRS)-convergence of the time-changed problem. That is, given ε > 0,

there exists δs.t. |P| < δ implies

Z0,˜

T≈ε∑

[s,t]∈P

f(˜

Zs)˜

Xs,t

where a≈εbmeans |a−b| ≤ ε. For fixed η > 0, include all (but only finitely many, say

N) points s∈(0, t] : |∆Xs|> η in P, giving rise to (˜

Pj: 1 ≤j≤N). Sending the mesh

of these to zero gives, as before,

Z0,T ≈ε∑

[s′,t′]∈P′

f(Zs′)Xs′,t′+∑

s∈(0,t]:

|∆Xs|>η

{φ(f∆Xs, Zs−)−Zs−−f(Zs−) ∆Xs}

In fact, due to summability of ∑s∈(0,t]{...}, we can pick η > 0 such that

Z0,T ≈2ε∑

[s′,t′]∈P′

f(Zs′)Xs′,t′+∑

s∈(0,t]: {φ(f∆Xs, Zs−)−Zs−−f(Zs−) ∆Xs}

and this is good enough to take the (RRS) lim as |P′| → 0. Going from finite variation

Xto the rough case, is just more notational effort. After all, the rough integral is a sort

of (abstract) Riemann integral.

We will need to the following corollary.

Corollary II.6.3. For a c´adl´ag rough path X= 1 + X+X= exp (X+A)∈ Wp

for p∈[2,3), the minimal jump extenstion X(n)taking values in G(n)(Rd)satisfies the

Marcus-type differential equation

X(n)

t= 1 + ∫t

X(n)

r−⊗dXr+∑

0<s≤t

X(n)

s−⊗{exp(n)(log(2) ∆Xs)−∆Xs}(29)

where the integral is understood as a rough integral and summation term is well defined

as absolutely summable series.

Proof. This follows from (24).

II.7 Rough path stability

We briefly discuss stability of rough integration and rough differential equations. In the

context of cadlag rough integration, Section II.5, it is a natural to estimate Z1−Z2, in

p-variation norm, where

Zi=∫0

YidXifor i= 1,2.

Now, the analysis presented in Section II.5 adapts without difficulties to this situation.

For instance, when Yi=F(Xi), one easily find

⏐⏐Z1−Z2⏐⏐p-var ≤CF,M (⏐⏐X1

0−X2

0⏐⏐+

X1−X2

p-var +

X1−X2

p-var)

provided F∈C2and |Xi

0|+∥Xi∥p-var +∥Xi∥p-var ≤M. (The situation can be compared

with [17, Sec 4.4] where analogous estimate in the α-H¨older setting.)

The situation is somewhat different in the case of Marcus type RDEs, dY i

t=f(Yi

t)

dXi

t. The observation here, quite possibly already contained implicitly in the works of

Williams [80], is simply that the difference Y1−Y2, in p-variation norm, is controlled,

as above uniformly on bounded sets, by



X1−˜

X2



p-var +



X1−˜

X2



p-var

where ˜

Xi= ( ˜

Xi,˜

Xi)∈ Cp

gis constructed from Xi∈ Wp

gas in Theorem II.4.1. (It should

be possible to rederive the convergence results of [39] within this framework.)

II.8 Rough versus stochastic integration

Consider a d-dimensiona L´evy process Xtenhanced with

Xs,t := (Itˆo) ∫(s,t]

(Xr−−Xs)⊗dXr.

We show that rough integration against the Itˆo lift actually yields standard stochastic

integral in Itˆo sense. An immediate benefit, say when taking Y=f(X) with f∈C2,

is the universality of the resulting stochastic integral, defined on a set of full measure

simultanously for all such integrands.

Theorem II.8.1. Let Xbe a d-dimensional L´evy process, and consider adapted processes

Yand Y′such that (Y, Y ′)is controlled rough path. Then Itˆo- and rough integral coincide,

∫(0,T]

Ys−dXs=∫T

Ys−dXsa.s.

Proof of Theorem II.8.1. By Theorem II.5.7, there exist partitions Pnwith

|S(Pn)−∫T

Ys−dXs| ≤ 1

where

S(Pn) := ∑

[s,t]∈Pn

YsXs,t +Y′

sXs,t.

II.8. ROUGH VERSUS STOCHASTIC INTEGRATION 39

Let Xt=Mt+Vtbe the L´evy Ito decomposition with martingale Mand bounded

variation part V. Since (V−, V ), (V−, M), (M−, V ) are compatible pairs we can choose

the corresponding τnfor ϵ=1

nfrom their compatibility. W.l.o.g., we can assume Pn−1∪

τn∪Dn⊂ Pn, where Dnis the n-th dyadic partition. We know from general stochastic

integration theory that, possibly along some subsequence, almost surely,

S′(Pn) = ∑

[s,t]∈Pn

YsXs,t →∫(0,T]

Ys−dXsas n→ ∞

Thus it suffices to prove that almost surely, along some subsequence,

S′′ (Pn) = ∑

[s,t]∈Pn

Y′

sXs,t →0

Now,

Xs,t =Ms,t +Vs,t +∫(s,t]

(Mr−−Ms)⊗dVr+∫(s,t]

(Vr−−Vs)⊗dMr

Using a similar argument as in Theorem II.5.11,

∑

[s,t]∈Pn

Y′

s(Vs,t +∫(s,t]

(Mr−−Ms)⊗dVr+∫(s,t]

(Vr−−Vs)⊗dMr)→0

We are left to show that ∑

[s,t]∈Pn

Y′

sMs,t →0

By the very nature of Itˆo lift

Sym (Ms,t) = 1

2Ms,t ⊗Ms,t −1

2[M, M]s,t

and it follows from standard (convergence to) quadratic variation results for semimartin-

gale (due to F¨ollmer [14]) that one is left with

∑

[s,t]∈Pn

Y′

sAs,t →0

where As,t = Anti (Ms,t). At this point, let us first assume that |Y′|∞≤Kuniformly in

ω. We know from Theorem II.2.11 (or Corollary II.9.2 below)

E[|As,t|2]≤C|t−s|2(30)

and using standard martingale argument (orthogonal increment property),

E[|∑

[s,t]∈Pn

Y′

sAs,t|2] = ∑

[s,t]∈Pn

E[|Y′

sAs,t|2]≤K2C∑

[s,t]∈Pn|t−s|2=O(|Pn|)

which implies, along some subsequence, almost surely,

∑

[s,t]∈Pn

Y′

sAs,t →0 (31)

Finally, for unbounded Y′, introduce stopping times

TK= inf{t∈[0, T] : sup

s∈[0,t]|Y′

s| ≥ K}

Similarly as in the previous case,

E[|∑

[s,t]∈Pn,s≤TK

Y′

sAs,t|2] = O(Pn)

Thus almost surely on the event {TK> T}

∑

[s,t]∈Pn

Y′

sAs,t →0

and sending K→ ∞ concludes the proof.

We remark that the identification of rough with stochastic integrals is by no means

restricted to L´evy processes, and the method of proof here obviously applies to semi-

martingale situation. As a preliminary remark, one can always drop the bounded vari-

ation part (and thereby gain integrability). Then, with finite p-variation rough path

regularity, for some p < 3, of (Itˆo -, by Proposition II.3.4 equivalently: Stratonovich) lift,

see Section II.10.2, the proof proceeds along the same lines until the moment where one

shows (31). For the argument then to go through, one only needs

∑

[s,t]∈P

E[|As,t|2]→0 as |P| → 0,

which follows from (30), an estimate which will be extended to general classes of Markov

jump processes in Section II.10.1. That said, we note that much less than (30) is necessary

and clearly this has to exploited in a general semimartingale context.

II.9 L´evy processes and expected signature

II.9.1 A L´evy–Khintchine formula and rough path regularity

In this section, we assume (Xt) is a d-dimensional L´evy process with triplet (a, b, K).

The main insight of section is that the expected signature is well-suited to study rough

path regularity. More precisely, we consider the Marcus canoncial signature S=S(X),

given as solution to

dS =S⊗dX

S0= (1,0,0..)∈T((Rd)).

With Ss,t =S−1

s⊗Stas usual this givens random group-like elements

Ss,t =(1,X1

s,t,X2

s,t, ..)=(1, Xs,t,XM

s,t, ..)

and then the step-nsignature of X|[s,t]by projection,

X(n)

s,t =(1,X1

s,t, .., Xn

s,t)∈G(N)(Rd).

II.9. L´

EVY PROCESSES AND EXPECTED SIGNATURE 41

The expected signature is obtained by taking component-wise expectation and exists

under a natural assumption on the tail behaviour of the L´evy measure K=K(dy). In

fact, it takes “L´evy–Kintchine” form as detailed in the followig theorem. We stress that

fact that expected signature contains significant information about the process (Xt: 0 ≤

t≤T), where a classical moment generating function of XTonly carries information

about the random variable XT.

Theorem II.9.1 (L´evy–Kintchine formula).If the measure K1|y|≥1has moments up to

order N, then

E[X(N)

0,T ] = exp(CT)

with tensor algebra valued exponent

C=(0, b +∫|y|≥1

yK (dy),a

2+∫y⊗2

2! K(dy), .., ∫y⊗N

N!K(dy))∈T(N)(Rd).

In particular, if KI|y|≥1has finite moments of all orders, the expected signature is given

E[S(X)0,T ] = exp [T(b+1

2a+∫(exp(y)−1−y1|y|<1)K(dy))]∈T((Rd)).

The proof is based on the Marcus SDE dS =S⊗dX in T(N)(Rd), so that X(n)

0,T =S

and will be given in detail below. We note that Fawcett’s formula [13, 53, 2] for the

expected value of iterated Stratonovich integrals of of d-dimensional Brownian motion

(with covariance matrix a=Iin the afore-mentioned references)

E[S(B)0,T ] = E[(1,∫0<s<T ◦dB, ∫0<s<t<T ◦dB ⊗◦dB, ..)]= exp [T

2a]

is a special case of the above formula. Let us in fact give a (novel) elementary argument for

the validity of Fawcett’s formula. The form E[S(B)0,T ] = exp (TC) for some C∈T((Rd))

is actually an easy consequence of independent increments of Brownian motion. But

Brownian scaling implies the kth tensor level of S(B)0,T scales as Tk/2, which alreay

implies that Cmust be a pure 2-tensor. The identification C=a/2 is then an immediate

computation. Another instructive case which allows for an elementary proof is the case of

when If Xis a compound Poisson process, i.e. Xt=∑Nt

i=1 Jifor some i.i.d. d-dimensional

random variables Jiand Nta Poisson process with intensity λ. In L´evy terminology, one

has triplet (0,0, K) where Kis λtimes the law of Ji. Since jumps are to be traversed

along straight lines, Chen’s rule implies

E[SN(X)0,1|N1=n] = E[exp(J1)⊗.. ⊗exp(Jn)] = E[exp(J1)]⊗n

and thus

E[SN(X)0,1] = exp[(λ(Eexp(J)−1)]

which gives, with all integrations over Rd,

C=∫(exp(y)−1)K(dy).

Before turning to the proof of Theorem II.9.1 we give the following application. It

relies on the fact that the expected signature allows to extract easily information about

stochastic area.

Corollary II.9.2. Let Xbe a d-dimensional L´evy process. Then, for any p > 2, a.s.

(X, XM)∈ Wp

M([0, T],Rd)a.s.

We call the resulting Marcus like (geometric) rough path the Marcus lift of X.

Proof. W.l.o.g. all jumps have size less than 1. (This amounts to drop a bounded

variation term in the Itˆo-L´evy decomposition. This does not affect the p-variation sample

path properties of X, nor - in view of basic Young (actually Riemann–Stieltjes) estimates -

those of XM). We establish the desired rough path regularity as application of Proposition

II.3.5 which requires as to show

E|Xs,t|2.|t−s|

E|As,t|2.|t−s|2.

While the first estimate is immediate from the L2-isometry of stochastic integrals against

Poisson random measures (drift and Brownian component obviously pose no problem),

the second one is more subtle in nature and indeed fails - in presence of jumps - when A

is replaced by the full second level XM. (To see this take, d= 1 so that XM

s,t =X2

s,t/2 and

note that even for standard Poisson process E|Xs,t|4.|t−s|but not |t−s|2.)

It is clearly enough to consider Ai,j

s,t for indices i=j. It is enough to work with

S4(X) =: X. Using the geometric nature of X, by using shuffle product formula,

(Ai,j

s,t)2=1

4(Xi,j

s,t −Xj,i

s,t)(Xi,j

s,t −Xj,i

s,t)

=Xiijj

s,t −Xijji

s,t −Xjiij

s,t +Xjjii

s,t

On the other hand,

EXs,t = exp [(t−s)C] = 1 + (t−s)C+O(t−s)2

so that it is enough to check that Ciijj −Cijji −Cjiij +Cjjii = 0. But this is obvious

from the symmetry of

π4C=1

4! ∫y⊗4K(dy).

We now given the proof of the L´evy–Kintchine formula for the expected signature of

L´evy–processes. We first state some lemmas required.

The following lemma, a generalization of [69, Ch. 1, Thm. 38], is surely well-known

but since we could not find a precise reference we include the short proof.

Lemma II.9.3. Let Fsbe a c´agl´ad adapted process with sup0<s≤tE[|Fs|]<∞and gbe a

measurable function with |g(x)| ≤ C|x|kfor some C > 0, k ≥2and g∈L1(K). Then

E[∑

0<s≤t

Fsg(∆Xs)] = ∫t

E[Fs]ds ∫Rd

g(x)K(dx)

II.9. L´

EVY PROCESSES AND EXPECTED SIGNATURE 43

Proof. At first we prove the following,

E[∑

0<s≤t|Fs||g(∆Xs)|]≤t||g||1sup

0<s≤t

E[|Fs|] (32)

To this end, w.l.o.g, we can assume gvanishes in a neighbourhood of zero. The general

case follows by an application of Fatou’s lemma. Also, it is easy to check the inequality

when Fsis a simple predictable process. For general Fs, we choose a sequence of simple

predictable process Fn

s→Fspointwise. Since there are only finitely many jumps away

from zero, we see that

∑

0<s≤t|Fn

s||g(∆Xs)| → ∑

0<s≤t|Fs||g(∆Xs)|a.s.

and the claim follows again by Fatou’s lemma.

Now, define ¯g=∫Rd\0g(x)K(dx) and Mt=∑0<s≤tg(∆Xs)−t¯g. Then it is easy to

check that Mtis a martingale. Also,

Nt:= ∫(0,t]

FsdMs=∑

0<s≤t

Fsg(∆Xs)−¯g∫t

Fsds

is a local martingale. From (32), E[sup0<s≤t|Ns|]<∞. So, Ntis a martingale, which

thereby implies that E[Nt] = 0 finishing the proof.

Lemma II.9.4. If the measure KI|y|≥1has moments upto order Nthen with St=

SN(X)0,t,

E[ sup

0<s≤t|Ss|]<∞

Proof. We will prove it by induction on N. For N= 1, St=1+Xt, and the claim

follows from the classical result that E[sup0<s≤t|Xs|]<∞iff KI|y|≥1has finite first

moment. Now, note that

St= 1+∫t

πN,N−1(Sr−)⊗dXr+1

2∫t

πN,N−1(Sr)⊗adr+∑

0<s≤t

πN,N−1(Sr−)⊗{e∆Xs−∆Xs−1}

where πN,N−1:TN

1(Rd)→TN−1

1(Rd) is the projection map. From induction hypothesis

and lemma (II.9.3), last two terms on right hand side has finite expectation in supremum

norm. Using L´evy-Ito decomposition,

∫t

πN,N−1(Sr−)⊗dXr=∫t

πN,N−1(Sr−)⊗dMr+∫t

πN,N−1(Sr−)⊗bdr

+∑

0<s≤t

πN,N−1(Sr−)⊗∆Xs1|∆Xs|≥1

where Mis the martingale. Again by induction hypothesis and Lemma II.9.3, last

two terms are of finite expectation in supremum norm. Finally,

Lt=∫t

πN,N−1(Sr−)⊗dMr

is a local martingale. By Burkholder-Davis-Gundy inequality and noting that

[M]t=at +∑

0<s≤t

(∆Xs)21|∆Xs|<1

we see that

E[ sup

0<s≤t|Ls|].E[{∫t

0|πN,N−1(Sr−)|2d[M]r}1

]

.E[{∫t

0|πN,N−1(Sr−)|2dr}1

] + E[{∑

0<r≤t|πN,N−1(Sr−)|2|∆Xr|21|∆Xr|<1}1

]

.E[sup

r≤t|πN,N−1(Sr−)|] + E[{sup

r≤t|πN,N−1(Sr−|}1

2{∑

0<r≤t|πN,N−1(Sr−)||∆Xr|21|∆Xr|<1}1

]

.E[sup

r≤t|πN,N−1(Sr−)|] + E[sup

r≤t|πN,N−1(Sr−|] + E[∑

0<r≤t|πN,N−1(Sr−)||∆Xr|21|∆Xr|<1]

where in the last line, we have used √ab .a+b. Again by induction hypothesis and

(II.9.3), we conclude that

E[ sup

0<s≤t|Ls|]<∞

finishing the proof.

Proof. (Theorem II.9.1) As before,

St= 1 + ∫t

Sr−⊗dMr+∫t

Sr⊗(b+a

2)dr +∑

0<s≤t

Ss−⊗{e∆Xs−∆Xs1|∆Xs|<1−1}

By Lemma II.9.4 below ∫t

0Sr−⊗dMris indeed a martingale. Also note that Sthas a

jump iff Xthas a jump, so that almost surely St−=St. Thanks to Lemma II.9.3 below

ESt= 1 + ∫t

ESr⊗(b+a

2)dr +∫t

ESrdr ⊗∫(ey−y1|y|<1−1)K(dy)

and solving this linear ODE in TN

1(Rd) completes the proof.

II.9.2 L´evy rough paths

Corollary II.9.2 tells us that the Marcus lift of some d-dimensional L´evy process Xhas

sample paths of finite p-variation with respect to the CC norm on G(2), that is

XM:= (1, X, XM)∈Wp

g([0, T], G(2) (Rd)).

It is clear from the nature of Marcus integration that XM

s,t is σ(Xr:r < s ≤t)-measurable.

It easily follows that XMis a Lie group valued L´evy process, with values in the Lie group

G(2) (Rd), and in fact a L´evy rough path in the following sense

Definition II.9.5. Let p∈[2,3). A G(2)(Rd)-valued process (X)with (c´adl´ag) rough

sample paths X(ω)∈Wp

ga.s. (on any finite time horizon) is called L´evy p-rough path

iff it has stationary independent left-increments (given by Xs,t (ω) = X−1

s⊗Xt).

II.9. L´

EVY PROCESSES AND EXPECTED SIGNATURE 45

Not every L´evy rough path arises as Marcus lift of some d-dimensional L´evy process.

For instance, the pure area Poisson process from Example 5and then the non-canonical

Brownian rough path (“Brownian motion in a magnetic field”) from Example 6 plainly

do not arise from iterated Marcus integration.

Given any L´evy rough path X= (1, X, X), it is clear that its projection X=π1(X) is

a classical L´evy process on Rdwhich then admits, thanks to Corollary II.9.2, a L´evy rough

path lift XM. This suggests the following terminology. We say that Xis a canoncial

L´evy rough path if Xand XMare indistinguisable, call Xanon-canonical L´evy

rough path otherwise.

Let us also note that there are G(2)(Rd)-valued L´evy processes which are not L´evy p-

rough path in the sense of the above definition, for the may fail to have finite p-variation

for p∈[2,3) (and thereby missing the in rough path theory crucial link between regularity

and level of nilpotency, [p] = 2.) To wit, area-valued Brownian motion

Xt:= exp(2) (Bt[e1, e2]) ∈G(2)(Rd)

is plainly a G(2)(Rd)-valued L´evy processes, but

sup

P∑

[s,t]∈P ||Xs,t||p

CC ∼sup

P∑

[s,t]∈P |Bs,t|p/2<∞

if and only if p > 4.

Remark II.9.6. One could define G4(Rd)-valued L´evy rough paths, with p-variation reg-

ularity where p∈[4,5), an example of which is given by area-valued Brownian motion.

But then again not every G4(Rd)-valued L´evy process will be a G4(Rd)-valued L´evy rough

path and so on. In what follows we remain in the step-2setting of Definition II.9.5.

We now characterize L´evy rough paths among G(2)(Rd)-valued L´evy processes, them-

selves characterized by Hunt’s theory of Lie group valued L´evy porcesses, cf. Section

II.2.8. To this end, let us recall G(2) (Rd)= exp (g2(Rd)), where

g(2) (Rd)=Rd⊕so (d).

For g∈G(2) (Rd), let |g|be the Euclidean norm of log g∈g(2) (Rd). With respect to

the canoncial basis, any element in g(2) (Rd)can be written as in coordinates as (xv)v∈J

where

J:= {i: 1 ≤i≤d}∪{jk : 1 ≤j < k ≤d}.

Write also

I:= {i: 1 ≤i≤d}.

Theorem II.9.7. Every G(2) (Rd)-valued L´evy process (X)is characterized by a triplet

(a,b,K)with

a= (av,w :v, w ∈J),

b= (bv:v∈J),

K∈ M(G(2) (Rd)):∫G(2)(Rd)(|g|2∧1)K(dg).

The projection X:= π1(X)is a standard d-dimensional L´evy process, with triplet

(a, b, K) := ((ai,j :i, j ∈I),(bk:k∈I),(π1)∗K)

where Kis the pushforward of Kunder the projection map. Call (a,b,K)enhanced

L´evy triplet, and Xenhanced L´evy process.

Proof. This is really a special case of Hunt’s theory. Let us detail, however, an explicit

construction which we will be useful later on: every G(2) (Rd)-valued L´evy process X

(started at 1) can be written as in terms of a g(2) (Rd)-valued (standard) L´evy process

(X, Z), started at 0, as

Xt= exp (Xt,At+Zt)

where At=A0,t is the stochastic area associated to X. Indeed, for v, w ∈J, write

x= (xv) for a generic element in g(2) and then

((av,w),(bv),K)

for the L´evy-triplet of (X, Z). Of course, Xand Zare also (Rd- and so (d)-valued) L´evy

process with triplets

((ai,j),(bi), K)and ((ajk,lm),(bjk),K),

respectively, where Kand Kare the image measures of Kunder the obvious projection

maps, onto Rdand so (d), respectively. Define also the image measure under exp, that

is K= exp∗K. It is then easy to see that Xis a L´evy process in the sense of Hunt

(cf. Section II.2.8) with triplet (a,b,K). Conversely, given (a,b,K), one constructs a

g(2) (Rd)-valued L´evy process (X, Z) with triplet ((av,w),(bv),log∗K) and easily checks

that the exp (X, A+Z) is the desired G(2)-valued L´evy process.

Recall that definition of the Carnot–Caratheodory (short: CC) norm on G(2) (Rd)from

Section II.2.5. The definition below should be compared with the classical definition of

Blumenthal–Getoor (short: BG) index.

Definition II.9.8. Given a L´evy measure Kon the Lie group G(2) (Rd), call

β:= inf {q > 0 : ∫G(2)(Rd)

(||g||q

CC ∧1 ) K(dg)}

the Carnot–Caratheodory Blumenthal–Getoor (short: CCBG) index.

Unlike the classical BG index, the CCBG index is not restricted to [0,2].

Lemma II.9.9. The CCBG index takes values in [0,4].

Proof. Set log (g) = x+a∈Rd⊕so (d). Then

||g||q

CC ≍∑

i⏐⏐xi⏐⏐

q+∑

j<k ⏐⏐ajk⏐⏐

q/2.

By the very nature of K, it integrates |xi|2and ⏐⏐ajk⏐⏐

2and hence β≤4. (The definition of

CC Blumenthal–Getoor extends immediately to G(N)(Rd), in which case β≤2N.)

Theorem II.9.10. Consider a G(2) (Rd)-valued L´evy process Xwith enhanced triplet

(a,b,K). Assume

(i) the sub-ellipticity condition

av,w ≡0unless v, w ∈I={i: 1 ≤i≤d};

(ii) the following bound on the CCBG index

β < 3.

Let p∈(2,3). Then a.s. Xis a L´evy p-rough path if p>βand this condition is sharp.

II.9. L´

EVY PROCESSES AND EXPECTED SIGNATURE 47

Proof. Set log (g) = x+a∈Rd⊕so (d). Then

||g||2ρ

CC ≍∑

i⏐⏐xi⏐⏐

2ρ+∑

j<k ⏐⏐ajk⏐⏐

ρ.

Let Kdenote the image measure of Kunder the projection map g↦→ x∈Rd. Let also

Kdenote the image measure under the map g↦→ a∈so (d). Since Kis a L´evy measure

on G(2)(Rd), we know that

∫so(d)

(|a|ρ∧1) K(da)<∞.(33)

whenever β < 2ρ < 3. We now show that Xenjoys p-variation. We have seen in the

proof of Theorem II.9.7 that any such L´evy process can be written as

log X= (X, A+Z)

where Xis a d-dimensional L´evy process with triplet

((ai,j),(bi), K)

with so (d)-valued area A=As,t and a so (d)-valued L´evy process Zwith triplet

(0,(bjk),K).

We know that E[|Xs,t|2].|t−s|and E[|As,t|2].|t−s|2and so, for |t−s| ≤ h,

P(|Xs,t|> a)≤h

P(|As,t|1/2> a)≤h

a2.

On the other hand,

P(|Zs,t|1/2> a)≤1

a2ρE(|Zs,t|ρ)∼h

a2ρ∫so(d)

(|a|ρ∧1) K(da)

and so

P(||Xs,t||CC > a).h

a2ρ∨2.

It then follows from Manstavicius’ criterion, cf. Section II.2.6, applied with β= 1, γ =

2ρ∨2, that Xhas indeed p-variation, for any p > 2ρ∨2, and by taking the infimum, for

all p>β∨2.

It remains to see that the conditions are sharp. Indeed, if the sup-ellipticity condition

is violated, say if av,w = 0 for some v=jk, say, this means (Brownian) diffusity (and

hence finite 2+- but not 2-variation) in direction [ej, ek]∈so (d). As a consequence,

Xhas 4+-variation (but not 4-variation), in particular, it fails to have p-variation for

some p∈[2,3). Similarly, if one considers an α-stable process in direction [ej, ek], with

well-known finite α+- but not α-variation, we see that the condition p>βcannot be

weakened.

II.9.3 Expected signatures for L´evy rough paths

Let us return to the Theorem II.9.1, where we computed, subject to suitable integrability

assumptions of the L´evy measure, the expected signature of a L´evy process, lifted by

means of “Marcus” iterated integrals. There we found that the expected signature over

[0, T] takes L´evy–Kintchine form

E[X0,T ] = exp{T(b+a

2+∫Rd

(exp(y)−1−yI|y|<1)K(dy))}

for some symmetric, positive semidefinite matrix a, a vector band a L´evy measure K,

provided KI|y|≥1has moments of all orders. In absence of a drift band jumps, the formula

degenerate to Fawcett’s form, that is

exp(Ta

for a symmetric 2-tensor a. Let us present two examples of L´evy rough paths, for which

the expected signature is computable and different from the above form.

Example 9. We return to the non-canonical Brownian rough path Bm, the zero-mass

limit of physcial Brownain motion in a magnetic field, as discucssed in Example 6. The

signature S=Smis then given by Lyons’ extension theorem applied to Bm, or equiva-

lently, by solving the following rough differential equation

dSt=St⊗dBm

t(ω), S0= 1

In [24] it was noted that the expected signature takes the Fawcett form,

E[Sm

0,T ] = exp{T˜a

but now for a not necessarily symmetric 2-tensor ˜a, the antisymmetric part of which

depends on the charge of the particle and the strength of the magnetic field.

Example 10. Consider the pure area Poisson process from Example 4. Fix some a∈

so (d)and let (Nt)be standard Poisson process, rate λ > 0. We set

Xt:= ⊗Nt

i=1 exp(2) (a)∈G(2)(Rd);

noting that the underlying path is trivial, X=π1(X)≡0and clearly Xis a non-Marcus

L´evy p-rough path, any p≥2. The signature of Xis by definition the minimal jump

extension of Xas provided by Theorem II.4.1. We leave it as easy exercise to the reader

to see that the signature Sis given by

St=⊗Nt

i=1 exp (a)∈T((Rd)).

With due attention to the fact that computations take places in the (non-commutative)

tensor algebra, we then compute explicitly

EST=∑

k≥0

eake−λT (λT)k/k!

=e−λT ∑

k≥0

(λTea)k/k!

= exp[λT(ea−1)].

Note that the jump is not described by a L´evy-measure on Rdbut rather by a Dirac

measure on G(2), assigning unit mass to exp a∈G(2).

II.9. L´

EVY PROCESSES AND EXPECTED SIGNATURE 49

We now give a general result that covers all these examples. Indeed, Example 9 is

precisely the case of ˜a=a+ 2bwith antisymmetric b= (bj,k)= 0, and symmetric

a= (ai,j). As for example (ii), everything is trivial but K, which assigns unit mass to

the element exp a.)

Theorem II.9.11. Consider a L´evy rough path Xwith enhanced triplet (a,b,K). As-

sume that K1{|g|>1}integrates all powers of |g|:= |log g|Rd⊕so(d). Them the signature of

X, by definition the minimal jump extension of Xas provided by Theorem II.4.1, is given

ES0,T = exp [T(1

∑

i,j=1

ai,jei⊗ej+

∑

i=1

biei+∑

j<k

bj,k [ej, ek] + ∫G(2) {exp(log(2) g)−g1{|g|<1}}K(dg))]

(34)

Proof. We saw in Corollary II.6.3 that Ssolves

St= 1 + ∫t

Ss−⊗dXs+∑

0<s≤t

Ss−⊗{exp(log(2) ∆Xs)−∆Xs}.

With notation as in the proof of Theorem II.9.10,

Xs,t =π2exp (Xs,t +As,t +Zt−Zs) = 1

2Xs,t ⊗Xs,t +As,t +Zs,t

s,t =1

2(Xs,t ⊗Xs,t −[X, X]s,t)+Zs,t

where we recall that (X, Z) is a Rd⊕so (d) valued L´evy process. With Zs,t =Zt−Zs,

we note additivity of Ξ := X−XIgiven by

Ξs,t : = 1

2[X, X]s,t +Zs,t

2a(t−s) + 1

2∑

r∈(s,t]|∆Xr|⊗2+Zs,t.

But then ∫t

Ss−⊗dXs=∫t

Ss−⊗dXI

s+∫t

Ss−⊗dΞ

and so, thanks to Theorem II.8.1 on consistency of Itˆo- with rough integration, we can

express Sas solution to a proper Itˆo integral equation,

St= 1 + ∫t

Ss−⊗dXs+∫t

Ss−⊗dΞ + ∑

0<s≤t

Ss−⊗{exp(log(2) ∆Xs)−∆Xs}

≡1 + (1) + (2) + (3).

Let MXbe the martingale part in the Itˆo–L´evy decomposition of X, write also NKfor

the Poisson random measure with intensity dsK(dy). Then, with b≡∑j<k bjk [ej, ek],

Xt=MX

t+bt +∫(0,t]×{|y|+|a|≥1}

yNK(ds, d (y, a)) ∈Rd

Zt=MZ

t+bt+∫(0,t]×{|y|+|a|≥1}

aNK(ds, d (y, a)) ∈so (d)

Ξt=1

2at +1

2∫(0,t]×{|y|≥1}

y⊗2NK(ds, d (y, a)) + Zt∈(Rd)⊗2.

Check (inductively) integrability of Stand note that ∫Ss−dMshas zero mean, for either

martingale choice. It follows that

Φt= 1 + ∫t

Φs⊗(C1+C2+C3)ds where

C1=b+∫g2(Rd)

y1{|y|+|a|>1}K(y, a),

C2=1

2a+1

2∫g2(Rd)

y⊗21{|y|+|a|>1}K(y, a) + b+∫g2(Rd)

a1{|y|+|a|>1}K(y, a),

C3=∫G(2)(Rd){exp(log(2)g)−g}K(dg).

Recall K= log(2)

∗Kso that the sum of the three integrals over g2(Rd) is exactly

∫G(2)

g1{|g|≥1}}K(dg)

where |g|=|log g|=|y|+|a|. And it follows that

C1+C2+C3=1

2a+b+b+∫G(2)(Rd){exp(log(2) g)−g1{|g|<1}}K(dg)

which concludes our proof.

II.9.4 The moment problem for random signatures

Any L´evy rough path X(ω) over some fixed time horizon [0, T] determines, via minimal

jump exension theorem, a random group-like element, say S0,T (ω)∈T((Rd)). What

information does the expected signature really carry? This was first investigated by

Fawcett [13], and more recently by Chevyrev [9]. Using his criterion we can show

Theorem II.9.12. The law of S0,T (ω)is uniquely determined from its expected signature

whenever

∀λ > 0 : ∫y∈G(2):|y|>1

exp (λ|y|)K(dy)<∞.

Proof. As in [9], we need to show that exp (C), equivalently C= (C0, C1, C2, ...)∈

T((Rd)), has sufficiently fast decay as the tensor levels grow. In particular, only the the

jumps matter. More precisely, by a criterion put forward in [9] we need to show that

∑λmCm<∞

where (for m≥3),

Cm=πm(∫G(2) (elog(2) g

(n)−g)K(dg))∈(Rd)⊗m.

We leave it as elementary exercise to see that this is implied by the exponential moment

condition on K.

II.10. FURTHER CLASSES OF STOCHASTIC PROCESSES 51

II.10 Further classes of stochastic processes

II.10.1 Markov jump diffusions

Consider a d-dimensional strong Markov with generator

(Lf) (x) = 1

2∑

i,j∈I

ai,j (x)∂i∂jf+∑

i∈I

bi(x)∂if

+∫Rd{f(x+y)−f(x)−1{y≤1}∑

i∈I

yi∂if}K(x, dy).

Throughout, assume a=σσTand σ, b bounded Lipschitz, K(x, ·) a L´evy measure, with

uniformly integrable tails. Such a process can be constructed as jump diffusion [35], the

martingale problem is discussed in Stroock [78]. As was seen, even in the L´evy case,

with (constant) L´evy triplet (a, b, K), showing finite p-variation in rough path sense is

non-trivial, the difficulty of course being the stochastic area

As,t (ω) = Anti ∫(s,t]

(X−

r−Xs)⊗dX ∈so (d) ;

where stochastic integration is understood in Itˆo sense. In this section we will prove

Theorem II.10.1. With probability one, X(ω)lifts to a G(2)-valued path, with incre-

ments given by

Xs,t := exp(2) (Xs,t +As,t) = X−1

s⊗Xt

and Xis a c´adl´ag Marcus like, geometric p-rough path, for any p > 2.

Note the immediate consequences of this theorem: the minimal jump extension of

the geometeric rough (X, XM)can be identified with the Marcus lift, stochastic integrals

and differential equations driven by Xcan be understood deterministically as function of

X(ω) and are identified with corresponding rough integrals and canonical equations. As

in the L´evy case discussed earlier, we base the proof on the expected signature and point

out some Markovian aspects of independent interest. Namely, we exhibit the step-N

Marcus lift as G(N)-valued Markov process and compute its generator. To this end, recall

(e.g. [22, Remark 7.43]) the generating vector fields Ui(g) = g⊗eion G(N), with the

property that

Lie (U1, .., Ud)|g=TgG(N).

Proposition II.10.2. Consider a d-dimensional Markov process (X)with generator as

above and the Marcus canonical equation dS =S⊗dX, started from

1≡(1,0, .., 0) ∈G(N)(Rd)⊂T(N)(Rd).

Then Stakes values in G(N)(Rd)and is Markov with generator, for f∈C2

(Lf)(x) = (L(N)f) (x) = 1

2∑

i,j∈I

ai,j (π1(x)) UiUjf+∑

i∈I

bi(π1(x)) Uif

+∫Rd{f(x⊗Y)−f(x)−1{y≤1}∑

i∈I

yiUif}K(x, dy),

with Y≡exp(n)(y).

Proof. (Sketch) Similar to the proof of Theorem II.9.1. Write X=M+Vfor the

semimartingale decomposition of X. We have

dS =S⊗dX =∑

i∈I

Ui(S)dXi

and easily deduce an evolution equation for f(St) = f(1). Taking the expected value

leads to the form (Lf).

Since Nwas arbitrary, this leads to the expected signature. We note that in the

(L´evy) case of x-independent characteristics, Φ does not depend on xin which case

the PIDE reduces to the ODE ∂tΦ = C⊗Φ which leads to the L´evy–Kinthchine form

Φ(t) = exp(Ct) obtained previously. We also that the solution Φ = (1,Φ1,Φ2, ..) to the

PIDE system given in the next theorem can be iteratively constructed. In absence of

jumps this systems reduces to a system of PDEs derived by Ni Hao [66, 49].

Theorem II.10.3 (PIDE for expected signature).Assume uniformly bounded jumps, σ, b

bounded and Lipschitz, a=σσT, the expected signautre Φ (x, t) = ExS0,t exists. Set

C(x) := ∑

i∈I

bi(x)ei+1

2∑

i,j∈I

ai,j (x)ei⊗ej+∫Rd(Y−1−I{y≤1}∑

i∈I

yiei)K(x, dy)

with Y= exp (y)∈T((Rd)).

Then Φ (x, t)solves

⎧

⎨

⎩

∂tΦ = C⊗Φ + LΦ + ∑i,j∈Iai,j (∂jΦ) (x)ei

+∫Rd(Y−1) ⊗(Φ (x⊗Y)−Φ (x)) K(x, dy)

Φ (x, 0) = 1.

Proof. It is enough to establish this in T(N)(Rd), for arbitrary integer N. We can see

that

ExX(N)

t=: u(x, t),

for x∈G(N)(Rd)⊂T(N)(Rd)is well-defined, in view of the boundedness assumptions

made on the coefficients, and then a (vector-valued, unique linear growth) solution to the

backward equation

∂tu=Lu,

u(x, 0) = x∈T(N)(Rd).

It is then clear that

ExX(N)

0,t =x−1⊗u(x, t) =: Φ (x, t)

also satisfies a PDE. Indeed, noting the product rule for second order partial-integro

operators,

(L[fg]) (x) = ((L[f]) g) (x)+(fL[g]) (x) + Γ(f, g),

Γ(f, g) = ∑

i,j∈I

ai,j (UifUjg) (x) + ∫G(2)

(f(x⊗Y)−f(x)) (g(x⊗Y)−g(x)) K(dy)

II.10. FURTHER CLASSES OF STOCHASTIC PROCESSES 53

and also noting the action of Uvon f(x)≡x, namely Uif=x⊗ev, we have

Lx=x⊗C:= x⊗{∑

v∈J

bv⊗ev+1

2∑

i,j∈I

ai,jei⊗ej+∫G(2) (Y−1−I{y≤1}∑

v∈J

Yv⊗ev)K(dy)},

Γ(x, g) = x⊗{∑

i,j∈I

ai,j (Ujg) (x)ei+∫G(2)

(Y−1) (g(x⊗Y)−g(x)) K(dy)}.

As a consequence,

x⊗∂tΦ = ∂tu=Lu=L(x⊗Φ) = (Lx)⊗Φ + x⊗L[Φ] + Γ(x, Φ)

and hence

∂tΦ = C⊗Φ+{L[Φ] + ∑

i,j∈I

ai,j (UjΦ) (x)ei+∫G(2)

(Y−1) (Φ (x⊗Y)−Φ (x)) K(dy)}.

(35)

We can now show rough path regularity for general jump diffusions.

Proof. (Theorem II.10.1) Only p-variation statement requires a proof. The key remark

is that the above PIDE implies

Φt= 1 + (∂t|t=0φ)t+O(t2)= 1 + Ct +O(t2)

where our assumptions on a, b, K guarantee uniformity of the O-term in x. We can then

argue exactly as in the proof of Corollary II.9.2.

II.10.2 Semimartingales

In [45] L´epingle established finite p-variation of general semimartingales, any p > 2,

together with powerful Burkholder–Davis–Gundy type estimates. For continuous semi-

martingales the extension to the (Stratonovich=Marcus) rough path lift was obtained in

[28], see also [22, Chapter 14], but so far the general (discontinuous) case eluded us. (By

Proposition II.3.4 it does not matter if one establishes finite p-variation in rough path

sense for the Itˆo- or Marcus lift.)

As it is easy to explain, let us just point to the difficulty in extending L´epingle in the

first place: he crucially relies on Monroe’s result [64], stating that every (scalar!) c´adl´ag

semimartingale can be written as a time-changed scalar Brownian motion for a (c´adl´ag)

family of stopping times (on a suitably extended probability space). This, however, fails

to hold true in higher dimensions and not every (Marcus or Itˆo) lifted general semimartin-

gale2will be a (c´adl´ag) time-change of some enhanced Brownian motion [22, Chapter 13],

in which case the finite p-variation would be an immediate consequence of known facts

about the enhanced Brownian motion (a.k.a. Brownian rough path) and invariance of

p-variation under reparametrization.

A large class of general semimartingales for which finite p-variation (in rough path

sense, any p > 2) can easily be seen, consists of those with summable jumps. Following

2... and certainly not every Markov jump diffusion as considered in the last section ....

Kurtz et al. [39, p. 368], the Marcus version” of such a s semimartingale, i.e. with

jump replaced by straight lines over stretched time, may be interpreted as continuous

semimartingale. One can then apply [28, 22] and again appeal to invariance of p-variation

under reparametrization, to see that such (enhanced) semimartingales have a.s. p-rough

sample paths, any p > 2.

Another class of general semimartingales for which finite p-variation can easily be seen,

consists of time-changed L´evy processes (a popular class of processes used in mathematical

finance). Indeed, appealing once more to invariance of p-variation under reparametriza-

tion, the statement readily follows from the corresponding p-variation regularity of L´evy

rough paths.

II.10.3 Gaussian processes

We start with a brief review of some aspects of the work of Jain–Monrad [36]. Given a

(for the moment, scalar) zero-mean, separable Gaussian process on [0, T], set σ2(s, t) =

EX2

s,t =|Xt−Xs|2

L2.We regard the process Xas Banach space valued path [0, T]→H=

L2(P) and assume finite 2ρ-variation, in the sense of Jain–Monrad’s condition

F(T) := sup

P∑

[u,v]∈P |σ2(u, v)|ρ= sup

P∑

[u,v]∈P |Xt−Xs|2ρ

L2<∞(36)

with partitions Pof [0, T]. It is elementary to see that p-variation paths can always be

written as time-changed H¨older continuous paths with exponent 1/p (see e.g. Lemma

4.3. in [12]). Applied to our setting, with α∗= 1/(2ρ), ˜

X∈Cα∗-H¨ol ([0, F(T)], H) so

that ˜

X◦F=X∈W2ρ([0, T], H).

Now in view of the classical Kolmogorov criterion, and equivalence of moments for Gaus-

sian random variables, knowing

⏐⏐⏐

Xt−˜

Xs⏐⏐⏐L2≤C|t−s|α∗

implies that ˜

X(or a modification thereof) has a.s. α-H¨older samples paths, any α < α∗.

But then, trivialy, ˜

Xhas a.s. finite p-variation sampe paths, any p > 1/α = 2ρ, and

so does Xby invariance of p-variation under reparametrization. (I should be noted that

such Xhas only discontinuities at deterministic times, inherited from the jumps of F.)

In a nutshell, this is one of the main results of Jain–Monrad [36], as summarized in by

Dudley–Norvaiˇsa in [12, Thm 5.3]. We have the following extension to Gaussian rough

paths.

Theorem II.10.4. Consider a d-dimensional zero-mean, separabale Gaussian process

(X)with independent components. Let ρ∈[1,3/2) and assume

sup

P,P′∑

[s,t]∈P

[u,v]∈P′

|E(Xs,t ⊗Xu,v)|ρ<∞(37)

Then Xhas a c´adl´ag modification, denoted by the same letter, which lifts a.s. to a random

geometric c´adl´ag rough path, with A= Anti (X)given as L2-limit of Riemann–Stieltjes

approximations.

II.10. FURTHER CLASSES OF STOCHASTIC PROCESSES 55

Proof. In a setting of continuous Gaussian processes, condition (37), i.e. finite ρ-variation

of the covariance, is well-known [22, 17]. It plainly implies the Jain–Monrad condition

(36), for each component (Xi). With F(t) := ∑d

i=1 Fi(t) we can then write

X◦F=X

for some d-dimensional, zero mean, (by Kolmogorov criterion: continuous) Gaussian

process ˜

X, whose covariance also enjoys finite ρ-variation. We can now emply standard

(continuous) Gaussian rough path theory [22, 17] and construct a canoncial geometric

rough path lift of ˜

X. That is,

X= ( ˜

X, ˜

X)∈ Cρ

with probability 1. The desired geometric c´adl´ag rough path lift is then given by.

(X, X) = X:= ˜

X◦F.

The statement about L2-convergence of Riemann–Stieltjes approximations follows imme-

diately for the corresponding statements for Anti(˜

X), as found in [17, Ch. 10.2].

Part III

Loewner chains driven by

semimartingales

III.1. INTRODUCTION 59

Contents of this article is collected together in an upcoming paper [19].

III.1 Introduction

The theory of Loewner chains and Loewner’s differential equation (LDE) was introduced

in early 20th century by C. Loewner in an attempt to solve Bieberbach’s conjecture in

geometric function theory. The conjecture stated that if

f(z) = z+a2z2+a3z3+...

is an univalent conformal map on the unit disk, then for all n≥2

|an| ≤ n.

Bieberbach himself proved the bound |a2| ≤ 2 and Loewner could extend it to |a3| ≤ 3

using LDE. Later in 1986, when De Branges finally resolved the conjecture, LDE was

used as an important component in the proof. A good account on the history and

developements of Bieberbach conjecture can be found in [68].

Loewner’s theory gives a one-to-one correspondence between a family of continuously

growing compact sets in a planar simply connected domain and continuous curves running

on the boundary of the domain. For simplicity, we will restrict ourselves to the upper

half plane

H={z|z∈C, Im(z)>0}

A bounded subset K⊂His called a compact H-hull if K=H∩¯

Kand H\Kis a simply

connected domain. For each such compact H-hull, there is a unique associated bijective

conformal map gK:H\K→Hsatisfying the so called hydrodynamic normalization

lim

z→∞gK(z)−z= 0

The map gKis called the mapping out function of K. The half plane capacity of K is

defined by

hcap(K) = lim

z→∞z(gK(z)−z)

Definition III.1.1. A Loewner chain is a family {Kt}t≥0of compact H-hulls such that

Ks(Ktfor all s<tand satisfying local growth property:

rad(Kt,t+h)→0as h→0 + uniformly on compacts in t

where Ks,t := gKs(Kt\Ks)

Given {Ut}t≥0a continuous real valued curve with U0= 0, for each z∈¯

H\0, let gt(z)

denote the solution of the LDE

˙gt(z) = 2

gt(z)−Ut

, g0(z) = z(38)

The solution exists upto the maximal time T(z)∈(0,∞] and if T(z)<∞,

lim

t→T(z)gt(z)−Ut= 0

Define

Kt={z∈H|T(z)≤t}

Then the family {Kt}t≥0is a Loewner chain with hcap(Kt)=2tand gtis the mapping

out function of Kt. We call the chain {Kt}t≥0is driven by {Ut}t≥0.

Conversely given a Loewner chain {Kt}t≥0with hcap(Kt) = 2t, then there exist contin-

uous real valued curve Utwith U0= 0 such that mapping out functions gt(z) = gKt(z)

satisfies equation (38) and {Kt}t≥0is driven by {Ut}t≥0. Please refer to [40] and lecture

notes [3] for the details.

In a seminal paper by O. Schramm in 1999, [73], the above correspondence between

Loewner chains and real valued curves was utilized to characterize processes in plane

which satisfy conformal invariance and domain Markov property. Today these processes

are known as Schramm-Loewner evolutions, SLE(κ), which is a random Loewner chain

obtained when Ut=√κBt, where Btis the one dimensional Brownian motion. SLE’s

was then proven to arise natuarally as scaling limit of various discrete lattice models in

statistical physics. See [76, 74, 73] for such results.

These convergence results suggests that SLE’s are not only a family of growing com-

pact sets, but also a growing curve. This motivates to find conditions on driver {Ut}t≥0

which guarantees that {K}t≥0is generated by a curve in the following sense:

Definition III.1.2. A chain {Kt}t≥0is called generated by a curve γ: [0, T]→¯

Hwith

γ0= 0 if for all t≥0,Ht:= H\Ktis the unbounded component of H\γ[0, t].

If a chain is generated by a curve γ, then it is the only such curve called the trace of

chain. A necessary and sufficient condition for the existence of the trace can be found in

[71]. Denote ft(z) = g−1

t(z).

Theorem III.1.3 ([71]).A chain is generated by a curve if and only if

γt:= lim

y→0+ ft(iy +Ut)

exists and is continuous curve. If so, curve γis the trace.

It was proved in [62, 47] that when the driver is 1

2- Holder with ||U||1

2<4, then the

trace exist. This is the best possible known deteministic result and fails to apply for

Ut=√κBt. Nevertheless, proof of existence of trace of SLE(κ), κ= 8 was carried out in

[71] using probabilistic techniques. The trace also exists for SLE(8), but the proof follows

indirectly from convergence of Uniform spanning tree to SLE(8) and there is no direct

proof known so far.

Phenomena of existence of trace for Loewner chains is not completely well under-

stood. For random drivers, probabilistic techniques seems to be the only efficient tool

and it doesn’t give understanding of pathwise properties of driver responsible for the

trace. Having more examples to this phenomena will lead us to better understanding and

thus we consider Loewner chains driven by semimartingales. There is another motivation

for considering such models. The deep insight of Oded Schramm leading to construction

of SLE was that domain Markov property (DMP) together with conformal invariance (CI)

forces the driver to have independent and stationary increment, i.e. Brownian motion

III.1. INTRODUCTION 61

with some speed. It is possible to canonically produce some models which fails to have

DMP and CI globally, but do possess these properties on a local scale. See construction of

SLEκ,ρ in [41] for example. Such processes is of great importance to study the symmetries

like duality and reversibility of SLE. Heuristically speaking, having DMP and CI on a local

scale will force the driver to have independent and stationary increment locally, i.e. diffu-

sions. This motivates to consider Loewner chains driven by diffusions/semimartingales.

The main contribution of this article is the following Theorem:

Theorem III.1.4. For each κ < 2, there exist a constant α0(κ)depending only on κ

such that following holds :

Let Utis a continuous process such that for each t∈[0, T],βs=Ut−Ut−sis a semi-

martingale w.r.t. some filtration with canonical decomposition

βs=Ns+As

with local martingale part Nand bounded variation part A. Assume for all s≤t≤T,

|d[N]s

ds | ≤ κ

and

sup

t∈[0,T]

E[exp(α∫t

rdr)]<∞

for some α > α0(κ). Then the Loewner chain driven by Uis generated by a curve.

In a special case when Uis a (deterministic) Cameron-Martin path, our method allows

us to prove the following uniform bound:

Theorem III.1.5. If Uis a Cameron-Martin path, then

|f′

t(iy +Ut)| ≤ exp[1

4||U||2

H](39)

where ||U||H={∫T

0˙

rdr}1

is the Cameron-Martin norm.

Note that for Cameron-Martin paths U, by Cauchy-Schwarz inequality

|Ut−Us|=|∫t

Urdr| ≤ √t−s√∫t

rdr

which implies

inf

ϵ>0sup

|t−s|<ϵ

|Ut−Us|

√t−s= 0 <4

and the existence of trace follows from results in [62]. But it can also be seen directly

from Theorem III.1.5. Additionally, as remarked in [33], the Holder regularity of the trace

γimproves as 1

2-Holder norm of the driver gets smaller. Since Cameron-Martin paths are

of vanishing 1

2Holder norm, we can expect the trace to be as regular as possible. Note

that when U≡0, γt= 2i√t, which is at best 1

2-Holder on [0, T], Lipchitz on time interval

[ϵ, T] for ϵ > 0 and is of bounded variation. Bounds like 39 allows us to prove,

Theorem III.1.6. If Uis Cameron-Martin path, then ||γ||1

2,[0,T]<∞and for any ϵ > 0,

||γ||1,[ϵ,T]<∞. In fact under suitable time reparametrization φ,||γ◦φ||1<∞and thus

γis a bounded variation path.

Stability under approximation type results follow as corollary:

Theorem III.1.7. If Uis Cameron-Martin and Unis a sequence of piecewise linear

approximation to U, then for any α < 1

||γn−γ||α→0as n→ ∞

III.2 Proof of Theorem III.1.4

The proof of the Theorem III.1.4 will be based on Theorem III.1.3. Thus, achieving some

uniform in tupper bound on |f′

t(iy +Ut)|would imply existence and continuity of

γt= lim

y→0+ ft(iy +Ut)

This is made precise in the following lemma.

Lemma III.2.1. Suppose there exist a θ < 1and y0>0such that for all y∈(0, y0]

sup

t∈[0,T]|f′

t(iy +Ut)| ≤ y−θ(40)

then the trace exists.

Proof. Note that for y1< y2< y0,

|ft(iy2+ut)−ft(iy1+Ut)|=|∫y2

f′

t(ir +Ut)dr| ≤ ∫y2

r−θdr =1

1−θ(y1−θ

2−y1−θ

which implies that ft(iy +Ut) is Cauchy in yand thus

γt= lim

y→0+ ft(iy +Ut)

exists. For continuity of γ, observe that

|γt−ft(iy +Ut)| ≤ y1−θ

1−θ

Now,

|γt−γs|≤|γt−ft(iy +Ut)|+|γs−fs(iy +Us)|+|ft(iy +Ut)−fs(iy +Us)|

.y1−θ+|ft(iy +Ut)−fs(iy +Us)|

It is easy to see that for y > 0,

lim

s→t|ft(iy +Ut)−fs(iy +Us)|= 0

and since ywas arbitrary, this concludes the proof.

III.2. PROOF OF THEOREM ?? 63

In the case of random Loewner chains, e.g. SLE(κ) when Ut=√κBt, it is usually

difficult to prove (40) using pathwise techniques. But it becomes feasible via probabilistic

techniques. Before stating next lemma, we recall following definitions,

Definition III.2.2. A subpower function is a continuous non-decreasing function φ:

[0,∞)→(0,∞)such that for all constants c > 0

lim

x→∞x−cφ(x) = 0

Definition III.2.3. We say a curve U: [0, T]→Ris weakly 1

2-Holder if there exist a

subpower function φsuch that

|Ut−Us| ≤ √|t−s|φ(1

|t−s|)

Lemma III.2.4. If Uis weakly 1

2-Holder and there exist constant b > 2,θ < 1and

C < ∞such that for all t∈[0, T]and y > 0

P[|f′

t(iy +Ut)| ≥ y−θ]≤Cyb

then the trace exists.

Proof. By using of Borel-Cantelli lemma, it is easy that almost surely for nlarge enough,

|f′

k2−2n(i2−n+Uk2−2n)| ≤ 2nθ

for all k= 0,1, .., 22n−1. Now applying results in section 3 of [37] ( Lemma 3.7 and

distortion Theorem in particular ) completes the proof.

For proving the conditions in Lemma III.2.1 and III.2.4, we first get a representation

formula of |f′

t(iy +Ut)|.

Lemma III.2.5. For each fixed t≥0and s∈[0, t], define βs=Ut−Ut−s. Then

log |f′

t(z+Ut)|=∫t

2(X2

r−Y2

(X2

r+Y2

r)2dr (41)

where z=x+iy and (Xs, Ys), s ∈[0, t]is the solution of the ODE

dXs=dβs−2Xs

s+Y2

ds, X0=x

dYs=2Ys

s+Y2

ds, Y0=y

Proof. For each z∈H, the path gt−s(ft(z)) joins zto ft(z) as svaries from 0 to t. It is

then easy to see that

ft(z+Ut) = Pt(z) + Ut

where Ps(z) for s∈[0, t] is the solution of ODE

Ps(z) = −2

Ps(z) + βs

, P0(z) = z

Writing in polar form, P′

s=rseiθs, we see that

Re(|P′

P′

∂sP′

s) = Re(e−iθs(eiθs∂srs+irseiθs∂sθs)) = ∂srs

So it follows that,

∂slog|P′

s|=Re(1

P′

∂sP′

Noting that ∂sP′

s= (∂sPs)′,

∂slog|P′

s|=Re(1

P′

(−2

Ps+βs

)′) = 2Re((Ps+βs)−2)

=⇒log|P′

s|= 2 ∫s

Re((Pr+βr)−2)dr

and the claim follows.

A naive approximation to the RHS of (41)

∫t

2(X2

r−Y2

(X2

r+Y2

r)2dr =∫t

(X2

r−Y2

(X2

r+Y2

r)dlog Yr≤log(Yt

Also, it is easy to see that

Ys≤√y2+ 4s

implying

|f′

t(iy +Ut)|.y−1

which as per Lemma III.2.1 is just not enough for existence of trace. We thus need to

improve upon these naive approximation.

The key step in the proof of Theorem III.1.4 is the following deterministic estimate

on |f′

t(iy +Ut)|which eventually can be used to give an estimate of form |f′

t(iy +Ut)| ≤

y−θ, θ < 1. To state it, we would need that there is some calculus based on path βso

that integration of appropriate paths against βis well defined. This can be achieved if β

is α-H¨older for some α∈(1

3,1

2] (or βis of finite p-variation with p∈[2,3)) and βcan be

lifted to a α-rough path (or p-rough path). Since βis one dimensional, one can naturally

associate a geometric rough path to βby assigning βs,t = (1, βt−βs,1

2(βt−βs)2). Please

see [25] for details. Denote Gs=βs−Xs. Note that even if βis very irregular, Gis

always a C1curve and

Gs=2Xs

s+Y2

In fact it can be easily checked that ˙

Gis a controlled path in the sense of Gubinelli with

the Gubinelli derivate ˙

G′=˙

Y−˙

G2and the rough integral

Ms:= ∫s

Grdβr= lim

|π|→0∑

[u,v]∈π

Gu(βv−βu) + 1

2˙

G′

u(βv−βu)2,

III.2. PROOF OF THEOREM ?? 65

where πis a partition of [0, t], is well defined. We will also assume that βhas finite

quadratic variation in the sense of F¨ollmer [15]. We say that β: [0, T]→Rhas finite

quadratic-variation in sense of F¨ollmer if (along some fixed sequence of partitions π= (πn)

of [0, t], with mesh-size going to zero)

∃lim

n→∞ ∑

[r,s]∈πn

(βs∧t−βr∧t)2=: [β]π

and defines a continuous map t↦→ [β]π

t≡[β]t. A function Von [0, t] is called F¨ollmer-Itˆo

integrable (against β, along π) if

∃lim

n→∞ ∑

[r,s]∈πn

Vs(βs∧t−βr∧t) =: ∫t

V dπβ.

(F¨ollmer [15] shows that integrands of gradient form are integrable in this sense and

so defines pathwise integrals of the form ∫∇F(β)dπβ.) If the bracket is furthermore

Lipschitz, in the sense that

sup

0≤s<t≤T

[β]t−[β]s

t−s≤κ < ∞,(42)

write β∈ Qπ,κ

T. Also note that Xnaturally inherits the calculus from βbecause they

differ by a C1path G. Such freedom to be able to define integrals against βallows one to

write log |f′

t(iy +Ut)|as sum of two parts, one which is singular and blows up as y→0+

and another part which remains (more or less) bounded.

Proposition III.2.6. Let β∈Cαwith α∈(1/3,1/2]. With Gas above,

log |f′

t(z+Ut)|=Mt−∫t

rdr + log(Yt

y)−log(X2

t+Y2

x2+y2) (43)

If in addition, βhas continuous finite quadratic-variation in sense of F¨ollmer (along

π) then

log |f′

t(z+Ut)|=Mπ

t+1

2∫t

G′

sd[β]π

s−∫t

rdr + log(Yt

y)−log(X2

t+Y2

x2+y2) (44)

with (deterministic) F¨ollmer–Itˆo integral

Mπ

t= lim

n∑

[u,v]∈πn

Gu(βv−βu) =: ∫t

Gdπβ. (45)

Proof. Consider first the case of βin C1. Then

Grdβr−1

2˙

rdr +YrdYr

r+Y2

r−2XrdXr+ 2YrdYr

r+Y2

=2Xr

r+Y2

dβr−2X2

(X2

r+Y2

r)2dr −2XrdXr

r+Y2

r−YrdYr

r+Y2

=2Xr

r+Y2

d(βr−Xr)−2X2

(X2

r+Y2

r)2dr −2Y2

(X2

r+Y2

r)2dr

=4X2

(X2

r+Y2

r)2dr −2X2

(X2

r+Y2

r)2dr −2Y2

(X2

r+Y2

r)2dr

=2(X2

r−Y2

(X2

r+Y2

r)2dr

Next note that

2˙

rdr +1

2˙

rdr =˙

and

YrdYr

r+Y2

2˙

rdr =˙

dr −1

2˙

rdr

Putting all together, we get

2(X2

r−Y2

(X2

r+Y2

r)2dr =˙

Grdβr−˙

rdr +˙

dr −2XrdXr+ 2YrdYr

r+Y2

and integrating both side, the claim follows with Mt=∫t

0˙

Gsdβs. In the case of rough

driver, meaning βin Cαwith α > 1/3, let βnbe piecewise linear approximations to β

on partition πnwith |πn| → 0 and let Gnbe the one corresponding to βn. As a result of

continuity of rough integrals in rough path metric (universal limit theorem, see [25] for

details), one can observe that

∫s

rdβn

r→(rough)∫s

Grdβr

as n→ ∞ and equation 43 is then evident. Finally if βhas finite quadratic variation,

then

∑

[u,v]∈πn

G′

u(βv−βu)2→∫s

G′

rd[β]π

which finishes the proof.

Let us apply Proposition (III.2.6) to specific situation. We consider first the case when

U(or equivalently β) is a Cameron-Martin path. In this case βis bounded variation path

and thus [β]≡0. We easily get following proposition:

Proposition III.2.7. If Uis a Cameron-Martin path, then

|f′

t(z+Ut)| ≤ y

(1 + x2

y2) exp[1

4∫t

rdr]

In particular,

|f′

t(iy +Ut)| ≤ y

exp[1

4∫t

rdr]≤exp[1

4∫t

rdr]

and the trace exists.

Proof. Note that

Mt=∫t

Grdβr=∫t

Gr˙

βrdr

≤∫t

rdr +1

4∫t

β2

rdr

Thus form Proposition III.2.6,

III.2. PROOF OF THEOREM ?? 67

log |f′

t(z+Ut)|=Mt−∫t

rdr + log(Yt

y)−log(X2

t+Y2

x2+y2)

≤1

4∫t

β2

rdr + log(Yt

y)−log(X2

t+Y2

x2+y2)

≤1

4∫t

β2

rdr + log(Yt

y)−log( Y2

x2+y2)

4∫t

β2

rdr + log( y

) + log(1 + x2

y2)

and the claim follows. Finally the existence of trace follows from Lemma III.2.1

We can also apply Proposition III.2.6 to other situations where [β] is not necessarily

zero.

Proposition III.2.8. In the context of Proposition III.2.6, with continuous finite quadratic-

variation in sense of F¨ollmer so that d[β]π

s/ds ≤κ < 2one has the following estimate

|f′

t(iy +Ut)| ≤ exp[Mπ

t−∫t

rd(r+1

2[β]r)]

where ∫t

0˙

Grdπβr=Mπ

tis the Itˆo-F¨ollmer type integral introduced in (45).

Proof. From (44) and by ˙

G′=˙

Y−˙

G2, taking z=iy (i.e. x= 0),

log |f′

t(iy +Ut)|=∫t

Grdβr−∫t

rdr −1

2∫t

rd[β]r

+ log(Yt

y)−log(X2

t+Y2

y2) + 1

2∫t

d[β]r

Using positivity of ˙

Yr/Yr,

log(Yt

y)−log(X2

t+Y2

y2) + 1

2∫t

d[β]r

≤log(Yt

y)−2 log(Yt

y) + κ

2∫t

= (κ

2−1) log(Yt

≤0.

and the desired estimate follows.

We are now ready to prove Theorem III.1.4. For each fixed t, let βs=Ut−Ut−s

for s∈[0, t] is a semimartingale with respect to some filtration ( and thus w.r.t its

own filtration). Standard martingale argument implies that for any nested sequence of

partition πnwith |πn| → 0, a.s.

lim

n→∞ ∑

[r,s]∈πn

(βs∧t−βr∧t)2= [β]t

where [β] is the quadratic variation process of semimartingale β.

Proof of Theorem III.1.4. Under the assumptions of Theorem III.1.4, it is easy to verify

that Uis weakly 1

2-Holder. Thus by Lemma III.2.4 and Chebyshev’s inequality, it is

enough to exhibit constant b > 2 and C < ∞such that for all tand y > 0,

E[|f′

t(iy +Ut)|b]≤C

The constant bis chosen as follows. Since κ < 2, we can find p, q > 1 with p−1+q−1= 1

and ϵ∈(0,1) small enough such that

b:= 1

p(1 + 2(1 −ϵ)

κ)>2

Now, since βs=Ns+Aswith Aof bounded variation, [β]s= [N]s. Also since βis

semimartingale, the F¨ollmer integral Mπ

tcan be identified with the Ito intgral ∫t

0˙

Grdβr.

Thus by Proposition III.2.8,

log |f′

t(iy +Ut)| ≤ ∫t

GrdNr+∫t

GrdAr−∫t

rd(r+1

2[N]r)

Note firstly

∫t

GrdAr=∫t

Gr˙

Ardr

≤ϵ∫t

rdr +1

4ϵ∫t

rdr

Secondly,

∫t

rdr ≥1

κ∫t

rd[N]r

Putting all together, we get

log |f′

t(iy +Ut)| ≤ ∫t

GrdNr−(1

2+1−ϵ

κ)∫t

rd[N]r+1

4ϵ∫t

rdr

Thus by Holder’s inequality,

E[|f′

t(iy +Ut)|b]≤E[exp(b∫t

GrdNr−pb2

2∫t

rd[N]r)exp(b

4ϵ∫t

rdr)]

≤E[exp(pb ∫t

GrdNr−p2b2

2∫t

rd[N]r)]1

E[exp(qb

4ϵ∫t

rdr)]1

Finally note that ˙

Gis adapted to filtration of N(or β) and

∫s

GrdNr

is a local martingale. Thus

exp(pb ∫s

GrdNr−p2b2

2∫s

rd[N]r)

III.3. REVERSE BROWNIAN FILTRATION AND DIFFUSION DRIVEN LOEWNER CHAINS69

is a positive local martingale. Since positive local martingales are super-martingale, we

conclude that

E[exp(pb ∫t

GrdNr−p2b2

2∫t

rd[N]r)]≤1

implying

E[|f′

t(iy +Ut)|b]≤E[exp(qb

4ϵ∫t

rdr)]1

Note that for αlarge enough (only depending on κ), it possible to find such p, ϵ with

4ϵ=α

and thus

sup

y>0

sup

t∈[0,T]

E[|f′

t(iy +Ut)|b]<∞

which concludes the proof.

Remark III.2.9. We cannot make a semimartingale asumption for the Loewner driver

Usince the time-reversal of a semimartingales can fail to be a semimartingale. That said,

time-reversal of diffusion was studied by a number of authors including Millet, Nualart,

Sanz, Pardoux ... and sufficient conditions on “diffusion Loewner drivers” could be given

by tapping into this literature.

III.3 Reverse Brownian filtration and diffusion driven

Loewner chains

In this section we apply Theorem III.1.4 to the class of diffusion processes. To appreciate

better, we restrict to the processes of form Ut=F(t, Bt) for nice enough (say smooth)

functions F. The general diffusions can be handled similarly but is technically more

involved. For a fixed time t > 0, the process βs=βt

s=Ut−Ut−sis the time reversal of

U. Note that Ws=Bt−Bt−sis another Brownian motion and a martingale w.r.t. to

its natural completed filtration Fssatisfying usual hypothesis. We recall the following

classical result on expansion of filtration. See [[69], Chapter 6] for details.

Theorem III.3.1 ([69]).Brownian motion Wremains a semimartingale w.r.t. expanded

filtration ˜

Fs:= Fs∨σ(Wt) = Fs∨σ(Bt). Moreover,

Ws=˜

Ws+∫s

Wt−Wr

t−rdr

where ˜

Wis a Brownian motion adapted to the filtration ˜

We prove here that βsis a semimartingale w.r.t. to filtration ˜

Fand provide its explicit

decomposition into martingale and bounded variation part.

Lemma III.3.2. The process βis a semimartingale w.r.t. ˜

Fwith the decompostion

βs=∫s

F′(t−r, Bt−r)dWr+∫s

0(˙

F(t−r, Bt−r)−1

2F′′ (t−r, Bt−r))dr

=∫s

F′(t−r, Bt−r)d˜

+∫s

0(˙

F(t−r, Bt−r)−1

2F′′ (t−r, Bt−r) + F′(t−r, Bt−r)Bt−r

t−r)dr

Proof. By Ito’s formula,

βs=∫t

t−s

F′(r, Br)dBr+∫s

0(˙

F(t−r, Bt−r) + 1

2F′′ (t−r, Bt−r))dr

Note that by computing the difference between forward (Ito) and backward stochastic

integral,

∫t

t−s

F′(r, Br)dBr=∫s

F′(t−r, Bt−r)dWr−∫s

F′′ (t−r, Bt−r)dr

which completes the proof.

Corollary III.3.3. If |F′(t, x)| ≤ √κ < √2and for α > α0(κ)

E[exp(α∫T

0{˙

F(r, Br)−1

2F′′ (r, Br) + F′(r, Br)Br

r}2

dr)]<∞(46)

Then Loewner chain driven by Ut=F(t, Bt)is generated by a curve on [0, T].

To note down an interesting example of Corollary III.3.3, we first recall a classical

result on integrability of Wiener chaos of Gaussian measures.

Theorem III.3.4 ([44]).Let (E, ||.||)be a real separable Banach space equipped with

Borel σ-algebra Band µbe a centered Gaussian measure on E. Then for an element Ψ

in the d-th (homogenous) Wiener chaos, there exist η0>0such that,

E[exp(η||Ψ||2

d)]<∞ ⇐⇒ η < η0

In fact, the constant η0can be given an explicit formula in terms of Ψ. See [[44],

Chapter 5] for a proof and further details.

Corollary III.3.5. For p > 0,Ut=tpBtgenerates a trace almost surely on [0, T]for T

(deterministic) small enough.

Proof. With F(t, x) = tpx,|F′(t, x)| ≤ 1 for Tsmall enough. For checking condition 46,

note that ∫T

(B2

r−r)r2p−2dr

lies in the second Wiener chaos of Brownian motion and use of Theorem III.3.4 completes

the proof.

III.4. REGULARITY AND STABILITY UNDER APPROXIMATION OF THE TRACE71

Remark III.3.6. If function F′(t, x)is not space depedent, e.g. F(t, x) = tpxor

F(t, x) = √κx, we can apply the formula

∫t

t−s

F′(r)dBr=∫s

F′(t−r)dWr

Note that RHS is indeed a martingale w.r.t. the filtration Fand we do not have to work

with expanded filtration ˜

F. In this case the canonical decomposition of βis given by

βs=∫s

F′(t−r)dWr+∫s

F(t−r, Bt−r)dr

and Theorem III.1.4 again can be applied considering βas a semimartingale w.r.t. the

filtration F.

Remark III.3.7. It is possible to give an intuitive explanation to the existence of trace

for Ut=tpBton [0, T], T small enough. If Tis small enough, fluctuations of Uwill be

dominated by that of √κBt,κlarge enough, for which the trace exists. However, to the

authors best knowledge, there is no such comparison principle known. In fact, there is a

counter example, which appeared in [46], to a similar question posed by Omer Angel:

If Ugenerates a trace, it is true that for r < 1,rU generates a trace ?

III.4 Regularity and stability under approximation

of the trace

In this section, we study the regularity and stability under approximation of the trace,

specially in the case when Uis Cameron-Martin path. Bound obtained in Proposition

III.2.7 is the key to following results.

Theorem III.4.1. If Uis a Cameron-Martin path, then,

1. The trace γis 1

2-Holder with

||γ||1

2≤g(||U||H)

for some continuous function g: [0,∞)→(0,∞).

2. For each ϵ > 0,γis Lipchitz on [ϵ, T]. Moreover the map t→γ(t2)is Lipchitz on

[0, T]. In particular, γis bounded variation curve on [0, T].

Remark III.4.2. The form of function gcan be explicitly seen in the proof below and in

fact it suffices to take

g(x) = Cecx2

Proof. •Proof of Part 1 Define

v(t, y) := ∫y

0|f′

t(ir +Ut)|dr

Note that,

|γ(t)−ft(iy +Ut)| ≤ v(t, y)

and by an application of Koebe’s one-quater Theorem,

v(t, y)≥y

4|f′

t(iy +Ut)|(47)

In the proof below, we will choose y=√t−s. Now,

|γ(t)−γ(s)| ≤|γ(t)−ft(Ut+iy)|

+|γ(s)−fs(Us+iy)|

+|ft(Us+iy)−fs(Us+iy)|

+|ft(Ut+iy)−ft(Us+iy)|

The first two terms are bounded by v(t, y) and v(s, y) respectively. For the third

term, Lemma 3.5 in [37] and 47 implies,

|ft(Us+iy)−fs(Us+iy)| ≤ Cv(s, y)

For the fourth term,

|ft(Ut+iy)−ft(Us+iy)|≤|Ut−Us|sup

r∈[0,1] |f′

t(rUt+ (1 −r)Us+iy)|

Note that

|Ut−Us| ≤ y||U||1

and By Lemma 3.6 (Koebe’s distortion Theorem) in [37], there exist constant C

and αsuch that

|f′

t(rUt+ (1 −r)Us+iy)| ≤ Cmax{1,(|Ut−Us|

y)α}|f′

t(iy +Ut)|

≤Cmax{1,||U||α

2}|f′

t(iy +Ut)|

and using 47 again,

|ft(Ut+iy)−ft(Us+iy)| ≤ C||U||1

2max{1,||U||α

2}v(t, y)

Finally, from Proposition III.2.7

v(t, y)≤yexp{1

4||U||2

||U||1

2≤ ||U||H

giving us

|γt−γs| ≤ yg(||U||H)

completing the proof.

III.4. REGULARITY AND STABILITY UNDER APPROXIMATION OF THE TRACE73

•Proof of part 2. We will use the results from [33] for the proof of this part. In

particular, we recall from [33] that if ||U||1

2<4 (which we can assume without loss

of generality since U is a Cameron-Martin path), then there exist a σ, c > 0 such

that for all y > 0,

σ√t≤Im(ft(iy +Ut)) ≤√y2+ 4t

and

|Re(ft(iy +Ut))| ≤ c√t

so that trace γlies inside a cone at 0 and |ft(i√t+Ut)| ≤ c√t. From the proof of

part 1, we have

|γt−γs|.v(t, √t−s) + v(s, √t−s)

If s, t ≥ϵ, using Proposition III.2.7,

v(t, √t−s) + v(s, √t−s).(1

)(t−s).1

√ϵ(t−s)

which implies γis Lipchitz on [ϵ, T]. For proving |γt2−γs2|.|t−s|, note that we

can assume s= 0, for otherwise we can consider the image of γunder conformal

map gs2−Us2whose derivative of the inverse f′(.+Us2) remains bounded in a cone.

Finally again using Proposition III.2.7,

|γt2| ≤ |γt2−ft2(it +Ut2)|+|ft2(it +Ut2)|

.v(t2, t) + t

.t2

Yt2

which completes the proof.

Theorem III.4.3. If Unis a sequence of Cameron-Martin paths with ||Un−U||∞→0

and

sup

n||Un||H+||U||H<∞

then

||γn−γ||∞→0

In fact, for any α < 1

||γn−γ||α→0

Proof. We have,

|γn(t)−γ(t)| ≤|γn(t)−fn

t(iy +Un

t)|

+|ft(iy +Ut)−γ(t)|

+|fn

t(iy +Un

t)−ft(iy +Ut)|

Note that for fixed y > 0,

lim

n→∞|fn

t(iy +Un

t)−ft(iy +Ut)|= 0

uniformly in ton compacts. From Proposition III.2.7,

|γn(t)−fn

t(iy +Un

t)|+|ft(iy +Ut)−γ(t)| ≤ vn(t, y) + v(t, y)

≤y(exp{1

4||Un||2

H}+ exp{1

4||U||2

H})

Thus,

lim

y→0+ |γn(t)−fn

t(iy +Un

t)|+|ft(iy +Ut)−γ(t)|= 0

uniformly in nand t. Since ycan be chosen arbitrarily small,

lim

n→∞||γn−γ||∞= 0

Finally note that from Theorem III.4.1

sup

n||γn||1

2<∞

and standard interpolation argument concludes the proof.

Remark III.4.4. Variant of Theorem III.4.1 and III.4.3 also holds in the case when U

is semimartingale. As a by product of the proof of Theorem III.1.4, we have the bound

|f′

t(iy +Ut)| ≤ y−θ(48)

for some θ < 1. We can also optimise on the value of θby optimising the choice of

p, ϵ so that α=qb

4ϵis as small as possible for given κ( this amounts to solve some

quadratic inequalities). As shown in [37], Holder regularity of the trace can be derived

from 48 and smaller the value of θwe can choose, better is the Holder regularity. Thus

applying the same method as in [37], we can find an Holder exponent for trace driven by

semimartingales.

As for stability under approximation, bound 48 again allows us to apply the deterministic

result obtained in [79] which guarantees the convergence in the uniform topology of square

root or straight line interpolation schemes. See [79] for details.

Part IV

Slow points and the trace of

Loewner chains

IV.1. INTRODUCTION 77

IV.1 Introduction

Phenomena of existence of trace for Loewner chains is very subtle and complicated.

There is very little known about it from a deterministic point of view. It was proven in

[62, 47] that the trace exist if the driver Uis 1

2-H¨older with ||U||1

2<4. Even though

this deterministic result fails to apply to SLE(κ), the trace still exist for SLE(κ). In

the random situations, the proofs are based on probabilistic techiniques and it gives no

understanding of pathwise properties of the driver responsible for the trace. In this article

we provide deterministic conditions on driver responsible for the trace with an aim to

understand the random situations better. We will using the notations from Part III. The

main result of this article is the following Theorem relating trace to the slow points of

the driver. We first recall the definition of slow points.

Definition IV.1.1. A point t > 0is called α-slow point (from below) for Uif

lim sup

s→0+

|Ut−Ut−s|

√s≤α

Theorem IV.1.2. Let Ube bounded variation path. For t > 0and s∈[0, t], define

βs=βt

s:= Ut−Ut−sand ||β||s:= ||β||1−var,[0,s]. Assume for all t > 0,

lim sup

s→0+

||β||s

√s<2 (49)

and ∫t

√rd||β||r<∞(50)

Then the Loewner chain driven by Uis generated by a simple curve.

If U=U1−U2is the difference of monotonic increasing path U1and U2, then the

condition 49 is precisely saying that all t > 0 is a slow point for U1+U2.

The slow points are known to exist for Brownian motion for any α > 1, see [65] for details.

Thus Theorem IV.1.2 relates to the existence of trace for SLE(κ) from a deterministic

point of view, at least for small κ. In fact we believe that the restriction to the small κis

only due to some technicalities. We propose an another direct approach to the trace, again

involving slow points of the driver, which possibly explains the trace of SLE(κ), κ < 16.

IV.2 Proof of Theorem IV.1.2

We will be using Theorem III.1.3 for the proof of Theorem IV.1.2. Classically, as one

can see in [71], [19], [33], the approach is to verify the conditions of Theorem III.1.3 by

obtaining a constant θ < 1 such that for all t∈[0, T] and y > 0 small enough,

|f′

t(iy +Ut)| ≤ y−θ(51)

In the case of Uwith ||U||1

2<4, inequality 51 was proved in [33] by obtaining a crucial

lower bound

Im(ft(iy +Ut)) ≥σ√t(52)

for some constant σuniformly for y > 0.

In the case of random drivers, inequality like 52 is hard to obtain (and unlikely to

be true), but 51 can still be proved by obtaining first appropriate moment bounds on

|f′

t(iy +Ut)|and then applying Borel-Cantelli lemma. This approach works elegantly for

all SLE(κ), κ = 8 and fails for κ= 8 due to some technical reasons . Please see [71] for

details.

For the proof of Theorem IV.1.2, we will verify the conditions of Theorem III.1.3

directly as follows without relying on the inequality 51. We will assume the conditions

put forward in Theorem IV.1.2 throughout the proof.

We split the proof into two parts. The first part will be dedicated to prove the existence

of the limit

γt= lim

y→0+ ft(iy +Ut) (53)

and second part will prove the continuity of curve t→γt.

IV.2.1 Existence of the limit 53

In this subsection, we will keep the time index t > 0 fixed. Define βt

s=Ut−Ut−s,s∈[0, t].

As an abuse of notation, we will write βsto mean βt

s. The dynamics of ft(iy +Ut) can

be easily obtained by following the reverse flow of the Loewner differential equation as

what following lemma gives.

Lemma IV.2.1. For each fixed t∈(0, T]and z∈H,

ft(z+Ut) = ht(z)

where hs(z)s∈[0,t]is given by the solution of the differential equation

dhs(z) = dβs+−2

hs(z)ds, h0(z) = z(54)

Proof. Note that gt−s(ft(z)) for s∈[0, t] is curve from zto ft(z). Note that g.(ft(z)) is

solution to differential equation 38. Then plugging z+Utis the place of zfollowed by

easy manipulations completes the proof.

In view of Lemma IV.2.1, we need to analyse solution h(iy) as y→0+. It becomes

beneficial to look at the curves φs(−y2) = hs(iy)2. Note that for y > 0, hs(iy)∈Hand

thus φs(−y2)∈C\[0,∞). It can be easily seen that

dφs(iy) = 2hs(iy)dβs−4ds = 2√φs(iy)dβs−4ds, φ0(−y2) = −y2

where √z:C\[0,∞)→His a bijective holomorphic map. In fact, it follows easily from

the existence and uniqueness of equation 54 that

dφs(z0) = 2√φs(z0)dβs−4ds, φs(z0) = z0(55)

admits a well defined (i.e. φs(z0)∈C\[0,∞)) unique solution φ(z0) whenever z0∈

C\[0,∞). We want to consider equation 55 for z0= 0. We will need some definitions

for that purpose.

Definition IV.2.2. For a continuous curve X: [0, T]→C, a branch square roof of X

is a measurable map A: [0, T]→¯

Hsuch that for all t,A2

t=Xt.

IV.2. PROOF OF THEOREM ?? 79

Lemma IV.2.3. For any continuous curve X, there exist a branch square root.

Proof. If Xt/∈[0,∞), define At=√Xt. If Xt∈[0,∞), define At=√|Xt|. It can be

easily checked that this constructs a branch square root of Xand the verification is left

to the reader.

In fact, Xcan have more than one branch square roots in general. With an abuse

of notation, we will denote all branch square roots (or a particular one) by symbol At=

√Xt

b. Note that for any such branch |√Xt

b|=√|Xt|, which is continuous.

Lemma IV.2.4. If a continuous curve v: [0, t]→Cwith a branch square root √vb

satisfies |Re(√vsb)| ≤ ||β||sand

vs= 2 ∫s

√vrbdβr−4s

for all s∈[0, t], then for all s > 0,vs∈C\[0,∞)and thus √vsb=√vs.

Proof. Note that

lim sup

s→0+

s∫s

0|Re(√vrb)|d||β||r≤lim sup

s→0+

s||β||2

s<4

so that

lim sup

s→0+

Re(vs)

s<0

At this point, there exist a s0>0 such that for all s∈(0, s0],

Re(vs)<0

which implies vs∈C\[0,∞) for s∈(0, s0]. Since solution of equation 55 remains in

C\[0,∞) once the starting point z0∈C\[0,∞), we conclude that vs∈C\[0,∞) for

all time s∈(0, t].

Lemma IV.2.4 tells us that considering equation

dφs(0) = 2√φs(0)bdβs−4ds, φ0(0) = 0

with some branch √φs(0)bis same as considering

dφs(0) = 2√φs(0)dβs−4ds, φ0(0) = 0

with the condition that φs(0) ∈C\[0,∞) for all s > 0. In the next lemma, we establish

the uniqueness of solution to such equations.

Lemma IV.2.5. If φifor i= 1,2satisfy |Re(√φi

s)| ≤ ||β||s,φi

s∈C\[0,∞)for all s > 0

and

φi

s= 2 ∫s

0√φi

rdβr−4s

then φ1=φ2.

Proof. Define ψi

s=φi

s+ 4s. From the proof of lemma IV.2.4,

lim sup

s→0+

Re(ψi

s<4

for i= 1,2. Thus there exist s0>0 and δ > 0 such that for i= 1,2,

sup

0<s≤s0

Re(ψi

s−4)≤ −δ

so that for 0 < s ≤s0

⏐⏐⏐⏐√ψ1

s−4 + √ψ2

s−4⏐⏐⏐⏐≥√2δ

Now,

|ψ1

s−ψ2

s|=⏐⏐⏐⏐

2∫s

(√ψ1

r−4r−√ψ2

r−4r)dβr⏐⏐⏐⏐

=⏐⏐⏐⏐

2∫s

√r(√ψ1

r−4−√ψ2

r−4)dβr⏐⏐⏐⏐

≤2

√2δ∫s

√r(|ψ1

r−ψ2

r)d||β||r

√2δ∫s

|ψ1

r−ψ2

√rd||β||r

√2δ∫s

0|ψ1

r−ψ2

r|µ(dr)

where the measure µ(a, b] = ∫b

√rd||β||r<∞. Applying Gronwall’s lemma in measure

form, we see that φ1

s=φ2

sfor all s≤s0. Finally note that φ1

s0=φ2

s0∈C\[0,∞) and

uniqueness of solution to equation 55 for starting point z0∈C\[0,∞) implies φ1

s=φ2

for all s∈[0, t].

Having established the uniqueness of solutions for equation 55 with z0= 0, we now

show the existence of a solution. First we note down a lemma used for proving the

existence.

Lemma IV.2.6. Let Xn, X : [0, T]→Care continuous curves with X0= 0,Xn

0∈

C\(0,∞)and Xn

t∈C\[0,∞)for all nand t > 0. If Xnconverges uniformly to X,

then there exist a branch square root √Xbof Xand a subsequence Xnksuch that √Xnk

converges uniformly to √Xb. In particular, √Xbis continuous.

Proof. Note that family of curves {√Xn}is well defined and uniformly bounded. We

will prove that this family is equicontinuous, which implies, using Arzela-Ascoli Theorem,

there exist a subsequence √Xnkconverging uniformly to a continuous curve A. It can

be easily checked that Ais a branch square root of X. For proving the equicontinuity of

{√Xn}, let ϵ > 0. We need to exhibit a δsuch that if |t−s| ≤ δ, then |√Xn

t−√Xn

s| ≤

ϵfor all n. W.l.o.g. we can assume |Xn

s| ≥ ϵ2

4. Since family Xnis equicontinuous,

choose δsuch that for all n,|Xn

t−Xn

s| ≤ ϵ2

16 and thus |Xn

t| ≥ ϵ2

8. We claim that

IV.2. PROOF OF THEOREM ?? 81

|√Xn

s+√Xn

t| ≥ cϵ for some c > 0. If either Re(Xn

s)≤ϵ2

16 or Re(Xn

t)≤ϵ2

16, then

Im(√Xn

s+√Xn

t)≥cϵ. On the other hand, if Re(Xn

s)≥ϵ2

16, Re(Xn

t)≥ϵ2

16, sign of

Im(Xn

s) and sign of Im(Xn

t) are the same (since curve Xndoesn’t intersect the positive

real axis) and thus |Re(√Xn

s+√Xn

t)| ≥ cϵ . Finally

|√Xn

t−√Xn

s|=|Xn

t−Xn

|√Xn

t+√Xn

s|≤cϵ

for some c > 0, concluding the proof.

We also recall a standard result from analysis which we will not prove here.

Lemma IV.2.7. Let xnis a sequence in a metric space such that for any subsequence

xnk, there is further subsequence xnklwhich converges to a fixed element x, then sequence

xnconverges to x.

Theorem IV.2.8. There is a unique continuous curve φs=φs(0) with |Re(√φs)| ≤

||β||s,φs∈C\[0,∞)for s > 0and

φs= 2 ∫s

0√φrdβr−4s

Proof. Uniqueness is already settled in lemma IV.2.5. For the existence of a solution, note

that curves φ(−y2), y > 0 is a uniformly bounded equicontinuous family. Thus by Arzela-

Ascoli Theorem and lemma IV.2.6, there is a sequence φ(−y2

n) converging uniformly to a

continuous curve φand √φ(−y2

n) converging uniformly to some branch square root √φb

as yn→0+. Then it follows that

φs= 2 ∫s

0√φr

bdβr−4s

Also it follows easily from ODE 54 that if Xs+iYs=hs(iy) = √φs(−y2), then

Xs=βs−1

Ys∫s

βrdYr

In particular, |Re(√φs(−y2))| ≤ ||β||sand thus |Re(√φs

b)| ≤ ||β||s. Finally using lemma

IV.2.4, φs∈C\[0,∞) for all s > 0 and √φb=√φ.

As an immediate corollary, we have the existence of the limit 53.

Corollary IV.2.9. Solution φ(z0)of equation 55 with z0∈C\[0,∞)converges uniformly

to φ(0) as z0→0. In particular ft(iy +Ut) = ht(iy) = √φt(−y2)converges to √φt(0)

as y→0+.

Proof. As in the proof of Theorem IV.2.8, φ(zn

0) converges uniformly to φ(0) along some

subsequence zn

0→0. By using the uniqueness of solution φ(0) and lemma IV.2.7, we

conclude φ(z0) conveges to φ(0) as z0→0. Finally note that φt(0) ∈C\[0,∞) and thus

√φt(z0)→√φt(0) as z0→0.

IV.2.2 Continuity of map t→γt

In this subsection, we prove the continuity of γdefined by equation 53. At this point we

denote the solution constructed in Theorem IV.2.8 as φt

s=φt

s(0) for s∈[0, t]. As seen

in corollary IV.2.9,

γt=√φt

Proposition IV.2.10. The map t→φt

tis continuous. In particular, γis a continuous

curve. Also γis a simple curve.

Proof. Note that for s∈[0, t],

φt

s= 2 ∫s

0√φt

rdβt

r−4s

Since family of curves φtare uniformly bounded, we see that

|φt

t|.||βt||1−var,[0,t]+ 4t

implying continuity at t= 0

lim

t→0+ φt

t= 0

For continuity on (0, T], fix a time t0>0. Then for t∈(t0,2t0), define αt

s=φtt

t0sfor

s∈[0, t0]. Note that |Re(√αt

s)| ≤ ||βt|| t

t0sand

αt

s= 2 ∫s

0√αt

rdβtt

t0r−4t

Again it is easy to check that family of curves αtis uniformly bounded and equicontinuous.

Thus again using Arzela Ascoli Theorem and lemma IV.2.6, along some subsequence

tn↓t0,αtnconverges uniformly to some continuous curve φ′and √αtnconverges uniformly

to some branch square root √φ′bwith |Re(√φ′

b)| ≤ ||βt0||son [0, t0]. As an application

of Portmanteau Theorem, we easily see that

φ′

s= 2 ∫s

0√φ′

bdβt0

r−4s

and again by lemma IV.2.4 and IV.2.5, we conclude φ′

s=φt0

s. Finally lemma IV.2.7

implies that αtconverges uniformly to φt0as t↓t0. In particular φt

t=αt

t0→φt0

t0, implying

the right continuity. Similarly as above, φt

tis also left continuous. Finally note that

φt

t∈C\[0,∞) for all t > 0 and thus γt=√φt

tis also a continuous curve. For simpleness

for γ, suppose on the contrary γs=γs′for s<s′. Note that chain ˜

Kt:= gs(Kt+s\Ks)−Us

is driven by ˜

Ut=Ut+s−Us, which again by above argument is generated by a curve ˜γ

with ˜γt∈H. But since γs=γs′, ˜γs′∈Rwhich is a contradiction.

IV.3 Further discussions

Though Theorem IV.1.2 applies only to bounded variation path, it gives us good under-

standing of relation between slow points and the trace of Loewner chains. For example,

IV.3. FURTHER DISCUSSIONS 83

if one could make an absurd assumption that Brownian motion paths are monotonic,

Theorem IV.1.2 suggests the existence of a simple trace for SLE(κ) given that

lim sup

s→0+

|Bt−Bt−s|

√s<2

√κ(56)

Note that equation 56 holds true whenever k < 4 and tis a 2

√κ-slow point for Brownian

motion. Such slow points exists and form a dense subset of [0, T], suggesting a simple

trace if κ < 4. This is in accordance with the fact that SLE(κ) is indeed a simple curve

iff κ≤4.

We give below an another supporting result for slow points and trace if one is only

interested in the existence of the limit 53 for a fixed time t.

Proposition IV.3.1. For a continuous curve Uand time t > 0, if either

lim sup

s→0+

|βt

√s<1 (57)

lim

s→0+ sup

u,v∈[0,s]

|βt

u−βt

√|u−v|<2,(58)

then limit 53 exists at time tand γt∈H.

Proof. We use a representation formula for |f′

t(iy +Ut)|from [71],

|f′

t(iy +Ut)|= exp[∫t

r−Y2

r+X2

dlog(Yr)],

where Xs+iYs=hs(iy) is the solution of ODE 54. We use an estimate obtained in [[33],

lemma 2.1] that |Xs| ≤ sup{|βr−βs|, r ∈[0, s]}. Thus if either 57 or 58 holds, there exist

aσ < 2 and s0>0 such that |Xs| ≤ σ√sfor s∈[0, s0]. If follows from [[33], lemma 3.2]

that there exist a constant L > 0 such that

Ys≥L√s

for all s∈[0, s0] uniformly in y > 0. This implies that there exist θ < 1 such that for all

y > 0 and s∈[0, t],

s−Y2

s+X2

s≤θ,

so that

|f′

t(iy +Ut)|.y−θ,

which implies the existence of limit 53. Also since Ys0≥L√s0for all y > 0, we conclude

that γt∈H.

We can also use Proposition IV.3.1 to reflect in the reverse direction. Since we know

that SLE(κ) is a simple curve iff κ≤4, it suggests the following sample path property

of Brownian motion.

Conjecture 2. For Brownian motion B,

P[inf

tlim

ϵ→0+ sup

u,v∈[t,t+ϵ]

|Bu−Bv|

√|u−v|= 1] = 1

Actually the relation between the slow points and the trace is not restricted to κ < 4

regime. In view of analysing the solution h(iy) of equation 54 as y→0+, we propose

following approximating scheme to the solution h(iy).

Recall that the √z:C\[0,∞)→His a bijective holomorphic map.

Consider a partition P={0 = s0< s1< s2.. < sk=t}of the interval [0, t]. For y > 0,

starting from zs0=iy, define iteratively

zsj+1 =βsj+1 −βsj

2+√(zsj+βsj+1 −βsj

2)2−4(sj+1 −sj) (59)

Note that Im(zsj) is increasing with j. For s∈[sj, sj+1], define

zs=√z2

sj+s−sj

sj+1 −sj

(z2

sj+1 −z2

sj) (60)

It is easy to check that the curves zP

s(iy) is well defined (at least for small |P| de-

pending on y) with the above chosen branch of complex square root function. For each

partition P,

dzP

s(iy) = dβP

s+−2

s(iy)ds, zP

0(iy) = iy (61)

for some continuous curves βPwith ||βP−β||∞→0 as |P| → 0. Actually even though β

is real valued, the curves βPwill be complex valued. Allowing βPto take complex values

makes it for feasible to solve equation 61 explicitly and which helps us to come up with

the recursion 59 in the first place. It then follows for each y > 0,

lim

|P|→0||zP(iy)−h(iy)||∞= 0

Thus, in the limit |P| → 0, the curves zP(iy) and h(iy) are the same. The key remark

here is that even though it is impossible to make sense of ODE 54 with y= 0, it is

possible to make sense zP(0).

Proposition IV.3.2. If the first point s1of the partition Pis such that

|βt

s1|<4√s1

then recursion zP

sj(0) is well defined. In particular if

lim sup

s→0+

|βt

√s<4,(62)

then recursion zP

sj(0) is well defined for all partition Pwith |P| small enough.

Proof. Note that starting from z0= 0,

zs1=βs1

2+√β2

4−4s1

Since β2

4−4s1<0, square root in well defined and zs1∈H. Since Im(zP

sj(0)) is increasing

in j, the whole recursion remains well defined.

IV.3. FURTHER DISCUSSIONS 85

The equation 62 is satisfied in particular when ||U||1

2<4. It also holds if Ut=√κBt

if κ < 16 and tis a 4

√κ-slow point of B.

Conjecture 3. If equation 62 holds, there exist a continuous curve h(0) such that

lim

|P|→0||zP(0) −h(0)||∞= 0 (63)

Equation 63 gives us a natural limit h(0) of h(iy) as y→0+. One can also deal with

point twhich are not slow points if we are willing to pass to subsequential limit |Pn| → 0

in 63. Note that for y > 0, all the subsequential limit are the same and converge to h(iy).

A related result in the literature on slow points is the following result due to B. Davis

[11],

P[sup

lim inf

h→0+

Bt+h−Bt

√h= 1] = 1

We conjecture the following stronger statement:

Conjecture 4.

P[sup

lim inf

h→0+ |Bt+h−Bt|

√h= 1] = 1

Conjecture 4 allows one to define zPn

s(0) along some subsequence Pnfor all time tif

κ < 16. Again limn→∞ zPn(0) will be the natural limit of limy→0+ h(iy) and in particular

of limy→0+ ft(iy +Ut).

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