Séminaire de probabilités XLVIII

Lecture Notes in Mathematics  2168
Séminaire de Probabilités

Catherine Donati-Martin
Antoine Lejay
Alain Rouault Editors

Séminaire de
Probabilités XLVIII


Lecture Notes in Mathematics
Editors-in-Chief:
J.-M. Morel, Cachan
B. Teissier, Paris
Advisory Board:
Camillo De Lellis, Zurich
Mario di Bernardo, Bristol
Michel Brion, Grenoble
Alessio Figalli, Zurich
Davar Khoshnevisan, Salt Lake City
Ioannis Kontoyiannis, Athens
Gabor Lugosi, Barcelona
Mark Podolskij, Aarhus
Sylvia Serfaty, New York
Anna Wienhard, Heidelberg

2168


More information about this series at />

Catherine Donati-Martin • Antoine Lejay •
Alain Rouault
Editors

Séminaire de Probabilités
XLVIII

123


Editors
Catherine Donati-Martin
Laboratoire de Mathématiques
Université de Versailles-St Quentin
Versailles, France

Antoine Lejay
Campus scientifique
IECL
Vandoeuvre-les-Nancy, France

Alain Rouault
Laboratoire de Mathématiques
Université de Versailles-St Quentin
Versailles, France

ISSN 0075-8434
Lecture Notes in Mathematics
ISBN 978-3-319-44464-2

DOI 10.1007/978-3-319-44465-9

ISSN 1617-9692 (electronic)
ISBN 978-3-319-44465-9 (eBook)

Mathematics Subject Classification (2010): 60G, 60J, 60K
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Preface

After the exceptional 47th volume of the Séminaire de Probabilités dedicated to
Marc Yor, we continue in this 48th volume with the usual formula: some of the
contributions are related to talks given during the Journées de Probabilités held

in Luminy (CIRM) in 2014 and in Toulouse in 2015, and the other ones come
from spontaneous submissions. Apart from the traditional topics such as stochastic
calculus, filtrations and random matrices, this volume continues to explore the
subject of peacocks, recently introduced in previous volumes. Other particularly
interesting papers involve harmonic measures, random fields and loop soups.
We hope that these contributions offer a good sample of the mainstreams of
current research on probability and stochastic processes, in particular those active
in France.
We would like to remind the reader that the website of the Séminaire is
/>and that all the articles of the Séminaire from Volume I in 1967 to Volume XXXVI
in 2002 are freely accessible from the web site
/>We thank the Cellule MathDoc for hosting all these articles within the NUMDAM project.
Versailles, France
Vandoeuvre-lès-Nancy, France
Versailles, France

Catherine Donati-Martin
Antoine Lejay
Alain Rouault

v


Contents

Root to Kellerer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Mathias Beiglböck, Martin Huesmann, and Florian Stebegg

1

Peacocks Parametrised by a Partially Ordered Set . . . . . .. . . . . . . . . . . . . . . . . . . .
Nicolas Juillet

13

Convex Order for Path-Dependent Derivatives: A Dynamic
Programming Approach.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Gilles Pagès

33

Stability Problem for One-Dimensional Stochastic Differential
Equations with Discontinuous Drift.. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Dai Taguchi

97

The Maximum of the Local Time of a Diffusion Process
in a Drifted Brownian Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 123
Alexis Devulder
A Link Between Bougerol’s Identity and a Formula
Due to Donati-Martin, Matsumoto and Yor. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 179
Mátyás Barczy and Peter Kern
Large Deviation Principle for Bridges of Sub-Riemannian
Diffusion Processes .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 189
Ismaël Bailleul
Dévissage of a Poisson Boundary Under Equivariance
and Regularity Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 199
Jürgen Angst and Camille Tardif
Weitzenböck and Clark-Ocone Decompositions for Differential

Forms on the Space of Normal Martingales . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 231
Nicolas Privault

vii


viii

Contents

On the Range of Exponential Functionals of Lévy Processes . . . . . . . . . . . . . . . 267
Anita Behme, Alexander Lindner, and Makoto Maejima
t-Martin Boundary of Killed Random Walks in the Quadrant . . . . . . . . . . . . . 305
Cédric Lecouvey and Kilian Raschel
On the Harmonic Measure of Stable Processes .. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 325
Christophe Profeta and Thomas Simon
On High Moments of Strongly Diluted Large Wigner Random
Matrices . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 347
Oleskiy Khorunzhiy
Dyson Processes on the Octonion Algebra . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 401
Songzi Li
Necessary and Sufficient Conditions for the Existence
of ˛-Determinantal Processes .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 423
Franck Maunoury
Filtrations of the Erased-Word Processes . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 445
Stéphane Laurent
Projections, Pseudo-Stopping Times and the Immersion Property . . . . . . . . 459
Anna Aksamit and Libo Li
Stationary Random Fields on the Unitary Dual of a Compact Group . . . . . 469
David Applebaum

On the Spatial Markov Property of Soups of Unoriented
and Oriented Loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 481
Wendelin Werner


Root to Kellerer
Mathias Beiglböck, Martin Huesmann, and Florian Stebegg

Abstract We revisit Kellerer’s Theorem, that is, we show that for a family of real
probability distributions . t /t2Œ0;1 which increases in convex order there exists a
Markov martingale .St /t2Œ0;1 s.t. St
t.
To establish the result, we observe that the set of martingale measures with
given marginals carries a natural compact Polish topology. Based on a particular
property of the martingale coupling associated to Root’s embedding this allows for
a relatively concise proof of Kellerer’s theorem.
We emphasize that many of our arguments are borrowed from Kellerer (Math
Ann 198:99–122, 1972), Lowther (Limits of one dimensional diffusions. ArXiv eprints, 2007), Hirsch-Roynette-Profeta-Yor (Peacocks and Associated Martingales,
with Explicit Constructions. Bocconi & Springer Series, vol. 3, Springer, Milan;
Bocconi University Press, Milan, 2011), and Hirsch et al. (Kellerer’s Theorem
Revisited, vol. 361, Prépublication Université dÉvry, Columbus, OH, 2012).

Mathematics Subject Classification (2010): Primary 60G42, 60G44; Secondary
91G20

M. Beiglböck ( )
Institut für Stochastik und Wirtschaftsmathematik, Technische Universität Wien,
Wiedner Hauptstraße 8, 1040 Wien, Austria
e-mail:
M. Huesmann

Institut für Angewandte Mathematik, Rheinische Friedrich-Wilhelms-Universität Bonn,
Endenicher Allee 60, 53115 Bonn, Germany
e-mail:
F. Stebegg
Department of Statistics, Columbia University, 1255 Amsterdam Avenue, New York, 10025 NY,
USA
e-mail:
© Springer International Publishing Switzerland 2016
C. Donati-Martin et al. (eds.), Séminaire de Probabilités XLVIII, Lecture Notes
in Mathematics 2168, DOI 10.1007/978-3-319-44465-9_1

1


2

M. Beiglböck et al.

1 Introduction
1.1 Problem and Basic Concepts
We consider couplings between probabilities . t /t2T on the real line, where t ranges
over different choices of time sets T. Throughout we assume that all t have a first
moment. We represent these couplings as probabilities (usually denoted by or P)
on the canonical space ˝ corresponding to the set of times under consideration.
More precisely ˝ may be RT or the space D of càdlàg functions if T D Œ0; 1.
In each case we will write .St / for the canonical process and F D .Ft / for the
natural filtration. ˘.. t // denotes the set of probabilities P for which St P t .
M.. t // will denote the subset of probabilities (“martingale measures”) for which
S is a martingale wrt F resp. the right-continuous filtration F C D .FtC /t2Œ0;1 in
the case ˝ D D. To have M.. t // ¤ ; it is necessary that . t / increases in convex

order, i.e. s .'/ Ä t .'/ for all convex functions ' and s Ä t. This is an immediate
consequence of Jensen’s inequality. We denote the convex order by :
Our interest lies in the fact that this condition is also sufficient, and we shall from
now on assume that . t /t2T increases in convex order, i.e. that . t /t2T is a peacock in
the terminology of [5, 6]. The proof that M.. t /t2T / ¤ ; gets increasingly difficult
as we increase the cardinality of the set of times under consideration.
If T D f1; 2g, this follows from Strassen’s Theorem [18] and we take this result
for granted. The case T D f1; : : : ; ng immediately follows by composition of oneperiod martingale measures k 2 M. k ; kC1 /.
If T is not finite, the fact that M.. t /t2T / ¤ ; is less immediate and to establish
that M.. t /t2T / contains a Markov martingale is harder still; these results were first
proved by Kellerer in [11, 12] and now go under the name of Kellerer’s theorem.
We recover these classical results in a framework akin to that of martingale optimal
transport.

1.2 Comparison with Kellerer’s Approach
Kellerer [11, 12] works with peacocks indexed by a general totally ordered index
set T and the corresponding natural filtration F . He establishes compactness
of martingale measures on RT which correspond to the peacock . t /t2T . Then
Strassen’s theorem allows him to show the existence of a martingale with given
marginals . t /t2T for general T.
To show that M.. t /t2T / also contains a Markov martingale is more involved. On
a technical level, an obstacle is that the property of being a Markovian martingale
measure is not suitably closed. Kellerer circumvents this difficulty based on a
stronger notion of Markov kernel, the concept of Lipschitz or Lipschitz-Markov
kernels on which all known proofs of Kellerer’s Theorem rely. The key step to
showing that M.. t /t2T / contains a Markov martingale is to establish the existence


Root to Kellerer

3

of a two marginal Lipschitz kernel. Kellerer achieves this by showing that there
are Lipschitz-Markov martingale kernels transporting a given distribution to the
extremal points of the set
and subsequently obtaining an appealing Choquettype representation for this set.
Our aim is to give a compact, self contained presentation of Kellerer’s result in a
framework that can be useful for questions arising in martingale optimal transport1
for a continuum of marginals. While Kellerer is not interested in continuity
properties of the paths of the corresponding martingales, it is favourable to work
in the more traditional setup of martingales with càdlàg paths to make sense of
typical path-functionals (based on e.g. running maximum, quadratic variation, etc.).
In Theorem 1 we make it a point to show that the space of càdlàg martingales
corresponding to . t /t2Œ0;1 carries a compact Polish topology. We then note that the
Root solution of the Skorokhod problem yields an explicit Lipschitz-Markov kernel,
establishing the existence of a Markovian martingale with prescribed marginals.

1.3 Further Literature
Lowther [14, 15] is particularly interested in martingales which have a property
even stronger than being Lipschitz Markov: He shows that there exists a unique
almost continuous diffusion martingale whose marginals fit the given peacock.
Under additional conditions on the peacock he is able to show that this martingale
has (a.s.) continuous paths.
Hirsch-Roynette-Profeta-Yor [5, 6] avoid constructing Lipschitz-Markov-kernels
explicitly. Rather they establish the link to the works of Gyöngy [3] and Dupire [2]
on mimicking process/local volatility models, showing that Lipschitz-Markov martingales exist for sufficiently regular peacocks. This is extended to general peacocks
through approximation arguments. On a technical level, their arguments differ from
Kellerer’s approach in that ultrafilters rather than compactness arguments are used
to pass to accumulation points. We also recommend [6] for a more detailed review
of existing results.

2 The Compact Set of Martingales Associated to a Peacock
It is well known and in fact a simple consequence of Prohorov’s Theorem that
˘. 1 ; 2 / is compact wrt the weak topology induced by the bounded continuous
functions (see e.g. [19, Sect. 4] for details). It is also straightforward that the
continuous functions f W R2 ! R which are bounded in the sense that j f .x; y/j Ä

1

An early article to study this continuum time version of the martingale optimal transport problem
is the recent article [10] of Kallblad et al.


4

M. Beiglböck et al.

'.x/ C .y/ for some ' 2 L1 . 1 /;
2 L1 . 2 / induce the same topology on
˘. 1 ; 2 /.
A transport plan 2 ˘. R1 ; 2 / is a martingale measure iff for all continuous,
compact support functions h, h.x/.y x/ d D 0. Hence, M. 1 ; 2 / is a closed
subset of ˘. 1 ; 2 / and thus compact. Likewise, M. 1 ; : : : ; n / is compact.

2.1 The Countable Case
We fix a countable set Q 3 1 which is dense in Œ0; 1 and write MQ for the set of all
martingale measures on RQ . For D Â Q we set:
MQ .. t /t2D / WD fP 2 MQ W St

P

t

for t 2 Dg:

We equip RQ with the product topology and consider MQ with the topology of weak
convergence with respect to continuous bounded functions. Note that this topology
is in fact induced by the functions ! 7! f .St1 .!/; : : : ; Stn .!//, where ti 2 Q and f is
continuous and bounded.
Lemma 1 For every finite D Â Q; D 3 1 the set MQ .. t /t2D / is non-empty and
compact. As a consequence, M.. t /t2Q / D MQ .. t /t2Q / is non-empty and compact.
Proof We first show that MQR. 1 / is compact. To this end, we note that for every
" > 0 there exists n such that .jxj n/C d 1 < ". We then also have
R
.R n Œ .n C 1/; .n C 1// Ä .jxj

R
n/C d Ä .jxj

n/C d

1

<"

for every
1.
For every r W Q ! RC the set Kr WD fg W Q ! R; jgj Ä rg is compact by
Tychonoff’s theorem. Also, for given " > 0 there exists r such that for all P on
RQ with LawP .St /

". Hence Prohoroff’s
1 for all t 2 Q we have P.Kr / > 1
Theorem implies that MQ . 1 / is compact.
Next observe that for any finite set D Â Q; 1 2 D the set MQ .. t /t2D / is
non empty by Strassen’s theorem. Clearly MQ .. t /t2D / is also closed and hence
compact. The family of all such sets MQ .. t /t2D / has the finite intersection property,
hence by compactness
MQ .. t /t2Q / D

T
DÂQ;12D;jDj<1

MQ .. t /t2D / ¤ ;:

2.2 The Right-Continuous Case
We will now extend this construction to right-continuous families of marginals on
the whole interval Œ0; 1.


Root to Kellerer

5

We first note that it is not necessary to distinguish between the terms rightcontinuous and càdlàg in this context: fix a (not necessarily countable) set Q Â
Œ0; 1; Q 3 1, a peacock . t /p
t2Q and a strictly convex function ' which grows at
most linearly, e.g. '.x/ D 1 C x2 . Then the following is straightforward: the
mapping
W Q ! P.R/; q R7! q is càdlàg wrt the weak topology on P.R/ iff
the increasing function q 7! ' d q is right-continuous. In this case we say that

. t /t2Q is a right-continuous peacock.
As we have to deal with right limits we will recall the following:
Lemma 2 Let .Xn /n2
Law.Xn /. If limn! 1

N[f 1g
n

D

be a martingale wrt .Gn /n2 N[f 1g and write
1 , then X 1 D lim Xn a.s. and in L1 .

n

D

Proof Set Y WD limn! 1 Xn which exists (see for instance [16, Theorem II.2. 3]),
has the same law as X 1 and satisfies EŒYjX 1  D X 1 . This clearly implies that
X 1 D Y.
As above, we fix a countable and dense set Q  Œ0; 1 with 1 2 Q and consider
D D fg W Œ0; 1 ! R W g is càdlàg g;
DQ D f f W Q ! R W 9g 2 D s.t. gjQ D f g:
Note that DQ is a Borel subset of RQ . Indeed a useful explicit description of DQ
can be given in terms of upcrossings. For f W Q ! R we write UP. f ; Œa; b/ for the
number of upcrossings of f through the interval Œa; b. Then f 2 DQ iff f is càdlàg
and bounded on Q and satisfies UP. f ; Œa; b/ < 1 for arbitrary a < b (clearly it is
enough to take a; b 2 Q). We also set
FNs WD

T
t2Q;t>s

Ft

(1)

for s 2 Œ0; 1/ and let FN1 D F1 .
Proposition 1 Assume that . t /t2Q is a right-continuous peacock and let P 2
M.. t /t2Q /. Then P.DQ / D 1. For q 2 Q, SN q WD Sq D limt#q;t2Q;t>q St holds P-a.s.
For s 2 Œ0; 1 n Q, limt#s;t2Q;t>s St exists and we define it to be SN s . The thus defined
process .SN t /t2Œ0;1 is a càdlàg martingale wrt .FNt /t2Œ0;1 .
Proof By Lemma 2, Sq D limt#q;t>q;t2Q St for all q 2 Q. Using standard martingale
folklore (cf. [16, Theorem 2.8]), this implies that .St /t2Q is a martingale under wrt
.FNt /t2Q as well and the paths of .St /t2Q are almost surely càdlàg. Moreover these are
almost surely bounded by Doob’s maximal inequality and have only finitely many
upcrossings by Doob’s upcrossing inequality. This proves P.DQ / D 1. As the paths
of .St /t2Q are càdlàg the definition SN s WD limt#s;t2Q;t>s St is well for s 2 Œ0; 1 n Q
and .SN t /t2Œ0;1 is a càdlàg martingale under P wrt .FNt /t2Œ0;1 .
Identifying elements of D and DQ , the right-continuous filtration F C on D
equals the restriction of FN [cf. (1)] to DQ . Since any martingale measure P


6

M. Beiglböck et al.

concentrated on DQ corresponds to a martingale measure e
P on D Proposition 1
yields:

Proposition 2 Let . t /t2Œ0;1 be a right-continuous peacock and Q 3 1; Q  Œ0; 1
a countable dense set. Then the above correspondence
P 7! e
P

(2)

constitutes a bijection between M.. t /t2Q / and M.. t /t2Œ0;1 /.
Through the identification P 7! e
P, the set M.. t /t2Œ0;1 / carries a compact
topology TQ . Superficially, this topology seems to depend on the particular choice
of the set Q but this is not the case. To see this, consider another countable dense
set Q0  Œ0; 1. The set Q [ Q0 gives rise to a topology TQ[Q0 which is a priori finer
than TQ and TQ0 resp. Recall that whenever two compact Hausdorff topologies on
a fixed space are comparable, they are equal. Since TQ ; TQ0 ; TQ[Q0 are compact
Hausdorff topologies, we conclude that TQ D TQ[Q0 D TQ0 . Hence we obtain:
Theorem 1 Let . t /t2Œ0;1 be a right-continuous peacock and consider the canonical process .St /t2Œ0;1 on the Skorokhod space D. The set M.. t /t2Œ0;1 / of martingale
measures with marginals . t / is non empty and compact wrt the topology induced
by the functions
! 7! f .St1 .!/; : : : ; Stn .!//;
where t1 ; : : : ; tn 2 Œ0; 1 and f is continuous and bounded.

2.3 General Peacocks
Kellerer [11] considers the more general case of a peacock . t /t2T where .T; is an abstract total order and s < t implies s
t , moreover no continuity
assumptions on t 7! t are imposed. Notably the existence of a martingale
associated to such a general peacock already follows from the case treated in the
previous section since every peacock can be embedded in a (right-) continuous
peacock indexed by real numbers:

Lemma 3 Let .T; a peacock . s /s2RC which is continuous (in the sense that s 7! s is weakly
continuous) and an increasing function f W T ! RC such that
t

D

f .t/ :

If T has a maximal element we may assume that f W T ! Œ0; 1.


Root to Kellerer

7

Proof
Assume first that T Rcontains a maximal element t . Consider again '.x/ D
p
1 C x2 and set f .t/ WD ' d t for t 2 T. On the image S of f we define . s /
through f .t/ WD t . Then s 7! s is continuous on I and s WD f .t / is a maximal
element of S.
Using tightness of . s /s2S we obtain that s WD limr2S;r!s exists for s 2 S. It
remains to extend . s /s2S to Œ0; s. The set Œ0; s n S is the union of countably many
intervals and on each of these we can define s by linear interpolation. Finally it is
of course possible to replace Œ0; s by Œ0; 1 through rescaling.
If T does not have a maximal
element, we
R
R first pick an increasing sequence

.tn /n 1 in T such that supn ' d tn D supt2T ' d t , then we apply the previous
argument to the initial segments fs 2 T W s Ä tn g.
Above we have seen that M.. t /t2Œ0;1 / ¤ ; for . t /t2Œ0;1 right-continuous and
pasting countably many martingales together this extends to the case of a rightcontinuous peacock . s /s2RC . By Lemma 3 this already implies M.. t /t2T / ¤ ; for
a peacock wrt to a general total order T.

3 Root to Markov
So far we have constructed martingales which are not necessarily Markov. To obtain
the existence of a Markov-martingale with desired marginals, one might try to adapt
the previous argument by restricting the sets MQ .. t /t2D / to the set of Markovmartingales. As noted above, this strategy does not work in a completely straight
forward way as being Markovian is not a closed property wrt weak convergence.
Example 1 The sequence

n

D

1
.ı
2 .1; 1n ;1/

C ı.

weakly converge to the non-Markovian measure

1;

D

/ of Markov-measures

1
n ; 1/
1
.ı
C ı. 1;0; 1/ /.
2 .1;0;1/

3.1 Lipschitz-Markov Kernels
A solution to the two marginal Skorokhod problem B0
;B
gives rise
to the particular martingale transport plan .B0 ; B /. Sometimes these martingale
couplings induced by solutions to the Skorokhod embedding problem exhibit certain
desirable properties. In particular we shall be interested in the Root solution to the
Skorokhod problem.
Theorem 2 (Root [17]) Let
be two probability measures on R. There exists
a closed set (“barrier”) R Â RC R (i.e. .s; x/ 2 R; s < t implies that .t; x/ 2 R)
such that for Brownian motion .Bt /t 0 started in B0
the hitting time R of R
embeds in the sense that B R
and .Bt^ R /t is uniformly integrable.


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M. Beiglböck et al.

Before we formally introduce the Lipschitz-Markov property we recall that the
L1 - Wasserstein distance between two probabilities ˛; ˇ on R is given by

W.˛; ˇ/ D inf

nR

jx

o
nR
fd
2 ˘.˛; ˇ/ D sup

yj d W

R

fd

o
W f 2 Lip1 ;

where ˘.˛; ˇ/ denotes the set of all couplings between ˛ and ˇ and Lip1 denotes
the set of all 1-Lipschitz functions R ! R: The equality of the two terms is a
consequence of the Monge-Kantorovich duality in optimal transport, see e.g. [19,
Sect. 5].
A martingale coupling 2 M. ; / is Lipschitz-Markov iff for some (and then
any) disintegration . x /x of wrt and some set X Â R, .X/ D 1 we have for
x; x0 2 X
W. x ;

x0 /

D jx

x0 j:

(3)

We note that the inequality W. x ; x0 /
jx x0 j is satisfied for arbitrary
2
0
0
M. ; /: for typical x; x ; x < x , the mean of x equals x and the mean of x0 equals
x0 . We thus find for arbitrary 2 ˘. x ; x0 /
R

jy

y0 j d .y; y0 /

ˇR
ˇ
R 0
ˇ y d .y; y0 /
y d .y; y0 /ˇ
ˇ
ˇR
R 0
y d x0 .y0 /ˇ D jx
D ˇ y d x .y/

(4)
x0 j;

hence W. x ; x0 / jx x0 j.
Note also that W. x ; x0 / D jx x0 j holds iff the inequality in (4) is an equality for
the minimizing coupling . This holds true iff there is a transport plan which is
isotone in the sense that it transports x -almost all points y to some y0 y. This is of
course equivalent to saying that x precedes x0 in first order stochastic dominance.
Lemma 4 The Root coupling

R

D Law.B0 ; B R / is Lipschitz-Markov.

Proof Write .Bt /t for the canonical process on ˝ D CŒ0; 1/, W for Wiener
measure started in and R for the Root stopping time s.t. .B0 ; B R / W R 2
M. ; /.
It follows from the geometric properties of the barrier R that for all x < x0 and
! 2 ˝ such that !.0/ D 0
B

R .xC!/

.x C !/ Ä B

R .x

0 C!/

.x0 C !/:

Write x for the distribution of B R given B0 D x and W0 for Wiener measure with
start in 0. Then . x /x defines a disintegration (wrt the first coordinate) of R and for
x < x0 an isotone coupling 2 ˘. x ; x0 / can be explicitly defined by
.A

B/ WD

R

1A B .B

R .xC!/

.x C !/; B

R .x

0 C!/

.x0 C !// W0 .d!/:


Root to Kellerer

9

Remark 1 We thank David Hobson for pointing out that Lemma 4 remains true if
we replace R by Hobson’s solution to the Skorokhod problem [7].2

We also note that this property is not common among martingale couplings. It
is not present e.g. in the coupling corresponding to the Rost-embedding nor the
various extremal martingale couplings recently introduced by Hobson–Neuberger
[9], Hobson–Klimmek [8], Juillet (and one of the present authors) [1], and HenryLabordere–Touzi [4].

3.2 Compactness of Lipschitz-Markov Martingales
To generalize the Lipschitz-Markov property to multiple time steps we first provide
an equivalent formulation in the two step case. Using the Lipschitz-function
characterization of the Wasserstein distance we find that (3) is tantamount to the
following: for every f 2 Lip1 .R/ the mapping
x 7!

R

fd

x

D EŒ f .S2 /jS1 D x

(5)

is 1-Lipschitz (on a set of full -measure).
Let Q  Œ0; 1 be a set which is at most countable. In accordance with (5) we
call a measure/coupling P on RQ Lipschitz-Markov if for any s; t 2 Q; s < t and
f 2 Lip1 .R/ there exists g 2 Lip1 .R/ such that
EP Œ f .St /jFs  D g.Ss/:

(6)

2
Hobson’s solution [7] can be seen as an extension of the Azema-Yor embedding to the case of a
general starting distribution.


10

M. Beiglböck et al.

The Lipschitz-Markov property is closed in the desired sense:
Lemma 5 A martingale measure P on RQ is Lipschitz-Markov iff
EP ŒXf .St / EP ŒY

EP ŒX EP ŒYf .St / Ä

R

X.!/Y.!/j!
N s

!N s j d.P ˝ P/

(7)

for all f 2 Lip1 .R/, s < t 2 Q and X; Y non-negative, bounded, and Fs -measurable.
Proof If P is Lipschitz-Markov, then for a given 1-Lipschitz function f we can
find by definition of a Lipschitz-Markov measure/coupling a 1-Lipschitz function
g satisfying (6). Moreover, as g 2 Lip1 we have for non-negative, bounded X; Y
.g.!s /

g.!N s //X.!/Y.!/
N Ä j!s

!N s jX.!/Y.!/:
N

Integration with respect to P ˝ P and an application of (6) yields (7).
For the reverse implication, by basic properties of conditional expectation there
is a ..Sq /q2Q\Œ0;s /-measurable function such that P-a.s.
.!/ D EP Œ f .St /jFs .!/:
Now from (7) we almost surely have .!/
.!/
N Ä j!s !N s j which shows that
only depends on the s coordinate and is in fact 1-Lipschitz.
For D Â Q we set
LQ .. t /t2D / WD fP 2 MQ W P is Lipschitz-Markov, St

P

t

for t 2 Dg:

Theorem 3 Let Q  Œ0; 1; Q 3 1 be countable. For every finite 1 2 D  Q the set
LQ .. t /t2D / is non-empty and compact. In particular, L.. t /t2Q / WD LQ .. t /t2Q / is
non-empty and compact.
Proof For finite D Â Q it is plain that LQ .. t /t2D / is non-empty: this
follows
by composing of Lipschitz-Markov-kernels. Hence, LQ .. t /t2Q / D
T

DÂQ;jDj<1 LQ .. t /t2D / ¤ ; by compactness.
A martingale on D is Lipschitz-Markov if (6) holds for s < t 2 Œ0; 1 wrt F C .
Theorem 4 Assume that . t /t2Œ0;1 is a right-continuous peacock and let Q 3 1 be
countable and dense in Œ0; 1. If P 2 L.. t /t2Q /, then the corresponding [cf. (2)]
martingale measure e
P 2 M.. t /t2Œ0;1 / is Lipschitz-Markov.
In particular, the set of all Lipschitz-Markov martingales with marginals
. t /t2Œ0;1 is compact and non-empty.
Proof The arguments in the proof of Lemma 5 work in exactly the same way
to show that PQ being Lipschitz-Markov is equivalent to conditions similar to (7)
where X; Y are chosen to be measurable wrt FsC (or FNs , see the remark before
Proposition 2).


Root to Kellerer

11

For arbitrary s; t 2 Œ0; 1; s < t choose sequences sn # s; tn # t in Q. Note that
X; Y are in fact measurable wrt Fsn and we thus have
EP ŒXf .Stn / EP ŒY

EP ŒX EP ŒYf .Stn / Ä

R

X.!/Y.!/j!
N sn

!N sn j d.P ˝ P/.!; !/

N

by Lemma 5. Letting n ! 1 concludes the proof.

3.3 Further Comments
It is plain that a Lipschitz-Markov kernel also has the Feller-property and in
particular a Lipschitz-Markov martingales are strong Markov processes wrt F C
(see [13, Remark 1.70]). As in the previous section, the right-continuity of . t /t2Œ0;1
is not necessary to establish the existence of a Lipschitz-Markov martingale, this
follows from Lemma 3. We also remark that the arguments of Sect. 2 directly extend
to the case of multidimensional peacocks, where the marginal distributions t are
probabilities on Rd . However it remains open whether Theorem 4 extends to this
multidimensional setup.
Acknowledgements Mathias Beiglböck and Florian Stebegg acknowledge support through FWFprojects P26736 and Y782-N25. Martin Huesmann acknowledges support through CRC 1060. We
also thank the Hausdorff Research Institute for Mathematics (HIM) for its hospitality in spring
2015 and Nicolas Juillet and Christophe Profeta for many insightful comments.

References
1. M. Beiglböck, N. Juillet, On a problem of optimal transport under marginal martingale
constraints. Ann. Probab. 44(1), 42–106 (2016)
2. B. Dupire, Pricing with a smile. Risk 7(1), 18–20 (1994)
3. I. Gyöngy, Mimicking the one-dimensional marginal distributions of processes having an Itô
differential. Probab. Theory Relat. Fields 71(4), 501–516 (1986)
4. P. Henry-Labordere, N. Touzi, An explicit Martingale version of Brenier’s theorem. Finance
Stochast. 20(3), 635–668 (2016)
5. F. Hirsch, C. Profeta, B. Roynette, M. Yor, Peacocks and Associated Martingales, with Explicit
Constructions. Bocconi & Springer Series, vol. 3 (Springer, Milan; Bocconi University Press,
Milan, 2011)
6. F. Hirsch, B. Roynette, M. Yor, Kellerer’s Theorem Revisited, vol. 361 (Prépublication
Université dÉvry, Columbus, OH, 2012)

7. D. Hobson, The maximum maximum of a martingale, in Séminaire de Probabilités, XXXII.
Lecture Notes in Mathematics, vol. 1686 (Springer, Berlin, 1998), pp. 250–263
8. D. Hobson, M. Klimmek, Model independent hedging strategies for variance swaps. Finance
Stochast. 16(4), 611–649 (2012)
9. D. Hobson, A. Neuberger, Robust bounds for forward start options. Math. Financ. 22(1),
31–56 (2012)
10. S. Källblad, X. Tan, N. Touzi, Optimal Skorokhod embedding given full marginals and AzemaYor peacocks. Ann. Appl. Probab. (2015, to appear)


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11. H.G. Kellerer, Markov-Komposition und eine Anwendung auf Martingale. Math. Ann. 198,
99–122 (1972)
12. H.G. Kellerer, Integraldarstellung von Dilationen, in Transactions of the Sixth Prague
Conference on Information Theory, Statistical Decision Functions, Random Processes (Tech.
Univ., Prague, 1971; Dedicated to the Memory of Antonín Špaˇcek) (Academia, Prague, 1973),
pp. 341–374
13. T. Liggett, Continuous Time Markov Processes. Graduate Studies in Mathematics, vol. 113
(American Mathematical Society, Providence, RI, 2010). An introduction.
14. G. Lowther, Limits of one dimensional diffusions. Ann. Probab. 37(1), 78–106 (2009)
15. G. Lowther, Fitting martingales to given marginals (2008). arXiv:0808.2319
16. D. Revuz, M. Yor, Continuous Martingales and Brownian Motion. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293, 3rd edn.
(Springer, Berlin, 1999)
17. D.H. Root, The existence of certain stopping times on Brownian motion. Ann. Math. Stat. 40,
715–718 (1969)
18. V. Strassen, The existence of probability measures with given marginals. Ann. Math. Stat. 36,
423–439 (1965)
19. C. Villani, Optimal Transport. Old and New. Grundlehren der mathematischen Wissenschaften,

vol. 338 (Springer, Berlin, 2009)


Peacocks Parametrised by a Partially Ordered
Set
Nicolas Juillet

Abstract We indicate some counterexamples to the peacock problem for families
of (a) real measures indexed by a partially ordered set or (b) vectorial measures
indexed by a totally ordered set. This is a contribution to an open problem of the
book (Peacocks and Associated Martingales, with Explicit Constructions, Bocconi
& Springer Series, Springer, Milan, 2011) by Hirsch et al. and Yor (Problem 7a–7b:
“Find other versions of Kellerer’s Theorem”).
Case (b) has been answered positively by Hirsch and Roynette (ESAIM Probab
Stat 17:444–454, 2013) but the question whether a “Markovian” Kellerer Theorem
hold remains open. We provide a negative answer for a stronger version: A
“Lipschitz–Markovian” Kellerer Theorem will not exist.
In case (a) a partial conclusion is that no Kellerer Theorem in the sense of the
original paper (Kellerer, Math Ann 198:99–122, 1972) can be obtained with the
mere assumption on the convex order. Nevertheless we provide a sufficient condition
for having a Markovian associate martingale. The resulting process is inspired by
the quantile process obtained by using the inverse cumulative distribution function
of measures . t /t2T non-decreasing in the stochastic order.
We conclude the paper with open problems.

1 Introduction
The rich topic investigated by Strassen [16] in his fundamental paper of 1965 was
to determine whether two probability measures and can be the marginals of a
joint law satisfying some constraints. The most popular constraint on Law.X; Y/
is probably P.X Ä Y/ D 1. In this case if Ä is the usual order on R, a

necessary and sufficient condition on
and to be the marginals of .X; Y/ is
F
F , where FÁ denotes the cumulative distribution function of Á. Actually
if we note GÁ the quantile function of Á, the random variable .G ; G / answers the
question. Recall that the quantile function GÁ is the generalised inverse of FÁ , that

N. Juillet ( )
Institut de Recherche Mathématique Avancée, UMR 7501, Université de Strasbourg et CNRS,
7 rue René Descartes, 67000 Strasbourg, France
e-mail:
© Springer International Publishing Switzerland 2016
C. Donati-Martin et al. (eds.), Séminaire de Probabilités XLVIII, Lecture Notes
in Mathematics 2168, DOI 10.1007/978-3-319-44465-9_2

13


14

N. Juillet

is the unique nondecreasing functions on 0; 1Œ that is left-continuous and satisfies
.GÁ /# jŒ0;1 D Á. In the case of a general family . t /t2T , the family consisting
of the quantile functions G t on .0; 1Œ; j0;1Œ / is also a process. It proves that
measures are in stochastic order if and only if there exists a process .Xt /t2T with
P.t 7! Xt is non-decreasing/ D 1 and Law.Xt / D t for every t 2 T. This result
is part of the mathematical folklore on couplings. We name it quantile process or
Kamae–Krengel process after the authors of Kamae and Krengel [12] because in
this paper a generalisation for random variables valued in a partially ordered set E

is proven. See also [15] where it appears.
Another type of constraint on Law.X; Y/ that is considered in Strassen article are
the martingale and submartingale constraints, E.Yj .X// D X and E.Yj .X// X
respectively. Strassen proved that measures . t /t2N are the marginals of a martingale
.Xt /t2N if and only if the measures t are in the so-called convex order (see
Definition 2). Kellerer extended this result to processes indexed by R and proved
that the (sub)martingales can be assumed to be Markovian. Strangely enough, but
for good reasons this famous result only concerns R-valued processes indexed by
R or another totally ordered set, which is essentially the same in this problem.
Nevertheless, Strassen-type results have from the start been investigated with
partially ordered set, both for the values of the processes or for the set of indices
(see [5, 12, 13]). Hence the attempt of generalising Kellerer’s theorem by replacing
R by R2 for one of the two sets is a natural open problem that has been recorded as
Problem 7 by Hirsch et al. in their book devoted to peacocks [9].
In Sects. 2 and 3 we define the different necessary concepts, state Kellerer
Theorem and exam the possible generalised statement suggested in [9, Problem 7].
About Problem 7b we explain in Sect. 3.2 why Kellerer could not directly apply his
techniques to the case of R2 -valued martingales. Problem 7a is the topic of the last
two parts. In Sect. 4 we exhibit counterexamples showing with several degrees of
precision that one can not obtain a Kellerer theorem on the marginals of martingales
indexed by R2 , even if the martingales are not assumed to be Markovian. However,
in Sect. 5 we provide a sufficient condition on . t /t2T that is inspired by the quantile
process. We conclude the paper with open problems.

2 Definitions
Let .T; Ä/ be a partially ordered set. In this note, the most important example may
be R2 with the partial order: .s; t/ Ä .s0 ; t0 / if and only if s Ä s0 and t Ä t0 . We
consider probability measures with finite first moment and we simply denote this
set by P.Rd /.
We introduce the concepts that are necessary for our paper. Martingales indexed

by a partially ordered set were introduced in the 1970. Two major contributions were
[3, 17]. The theory was known under the name “two indices”.


Peacocks Parametrised by a Partially Ordered Set

15

Definition 1 (Martingale Indexed by a Partially Ordered Set) Let .Xt /t2T be the
canonical process associated to some P on .Rd /T . For every s 2 T we introduce
Fs D .Xr j r Ä s/.
A probability measure P on .Rd /T is a martingale if and only if for every .s; t/ 2
2
T satisfying s Ä t it holds E.Xt j Fs / D Xs . In other words it is a martingale
if and only if for every s Ä t, n 2 N and sk Ä s for k 2 f0; 1; : : : ; ng we have
EP .Xt j Xs ; Xs1 ; : : : ; Xsn / D Xs .
The convex order that we introduce now is also known under the names second
stochastic order or Choquet order.
Definition 2 (Convex Order) The measures ;R 2 P.RRd / are said to be in convex
order if for every convex function ' W Rd ! R, ' d Ä ' d : This partial order
is obviously transitive and we denote it by
C .
Note that in Definition 2, ' may not be integrable but the negative part is integrable
because ' is convex.
The next concept of peacock is more recent. To our best knowledge it appeared
the first time in [8] as the acronym PCOC, that is Processus Croissant pour l’Ordre
Convexe. Both the writing peacock and the problem have been popularised in
the book by Hirsch et al.: Peacocks and Associated Martingales, with Explicit
Constructions [9].
Definition 3 (Peacock) The family . t /t2T is said to be a peacock if for every s Ä t

we have s C t .
Because of the conditional Jensen inequality, if .Xt /t2T is a martingale, the family
t D Law.Xt / of marginals is a peacock. More generally if for some peacock
. t /t2T a martingale .Yt /t2T satisfies for every t, Law.Yt / D t , the martingale
is said to be associated to the peacock . t /t2T .
Definition 4 (Kantorovich Distance) The Kantorovich distance between  and
 0 2 P.Rd / is
W.Â; Â 0 / D sup
f

Z

Z
f dÂ

f d 0
Rd

where f describes the set of 1-Lipschitz functions from Rd to R.
Definition 5 (Lipschitz Kernel) A kernel k W x 7! Âx transporting to D k
is called Lipschitz if there exist a set A Â Rd satisfying .A/ D 1 such that kjA is
Lipschitz of constant 1 from .A; k:kRd / to .P.Rd /; W/.
As .P.Rd /; W/ is a complete geodesic metric space a simple extension procedure
that we describe now permits us to extend k to a 1-Lipschitz function on R. First
the kernel k seen as a map is uniformly continuous so that one can extend it in a
N The connected components of R n AN are open intervals a; bŒ and
unique way on A.
the linear interpolation t 7! .b a/ 1 ..t a/k.b/ C .b t/k.a// is also a geodesic

16

N. Juillet

curve for the Kantorovich distance. Therefore it gives a solution for extending k and
making it a 1-Lipschitzian curve on R.
To our best knowledge, the next concept is the key of all known proofs of Kellerer
Theorem. Unlike Markov martingales, converging sequences of Lipschitz–Markov
martingale have Markovian limits (in fact Lipschitz–Markov). In his original proof
Kellerer uses a similar concept where the Kantorovich distance is replaced by the
Kantorovich distance build on d.x; y/ D min.1; jy xj/.
Definition 6 (Lipschitz–Markov Martingale) A process .Xt /t2T is a Lipschitz–
Markov martingale if it is a Markovian martingale and the Markovian transitions
are Lipschitz kernels.
For surveys with examples of Lipschitz kernels and Lipschitz–Markov martingales,
one can refer to [10] or [2].

3 The Kellerer Theorem and Trying to Generalise It
3.1 Problem 7a
Theorem 1 is a reformulation of Theorem 3 by Kellerer [14] in terms of the peacock
terminology.
Theorem 1 ([14]) Let . t /t2T be a family of integrable probability measure on
P.R/ indexed by the totally ordered set T (for simplicity thing of T D Œ0; C1Œ).
The following statements are equivalent
1.
2.
3.
4.

t

t
t
t

is a peacock,
is associated to a martingale process .Xt /t2T ,
is associated to a Markovian martingale process .Xt /t2T ,
is associated to a Lipschitz–Markovian martingale process .Xt /t2T .

Note that the implications 4 ) 3 ) 2 ) 1 are obvious. Theorem 2
that we prove in Sect. 4 contradicts the converse implications if T is merely a
partially ordered set. This is a negative answer to Problem 7a that we quote: “Let
0/ be a two-parameter peacock. Does there exist an associated two.Xt; I t;
parameter martingale .Mt; I t;
0/?”. Note that with our definition of peacock,
one should read Law.Xt; / in place of Xt; .
Theorem 2 Let .T; Ä/ be f0; 1g2, R2C or R2 with the partial order. For every choice
of T, we have the following:
• There exists a peacock indexed by T that is not associated to a martingale,
• there exists a peacock indexed by T that is associated to a martingale process but
not to a Markovian martingale process,
• there exists a peacock indexed by T that is associated to a Markovian martingale
process but not to a Lipschitz-Markovian martingale process.


Peacocks Parametrised by a Partially Ordered Set

17

3.2 Problem 7b

For completeness we explain what is known on Problem 7b: “Is a Rn -valued peacock
a Rn -valued 1-martingale?”, which with our notations means nothing but: Can any
peacock on Rd be associated to an Rd -valued martingale? Hirsch and Roynette
provided a positive answer in [7].
Theorem 3 Let . t /t2T be a family of integrable probability measures on P.Rd /
indexed by the totally ordered set T. The following statements are equivalent
1.
2.

t
t

is a peacock,
is associated to a martingale process .Xt /t2E .

Nevertheless it is to our knowledge still an open problem whether the full Kellerer
theorem may hold in the vectorial case: Can every peacock be associated to a
Markovian martingale? (equivalence of (1) and (3) in Theorem 1). We prove in
Proposition 1 that (1) and (4) are not equivalent. Actually, the existence of a
Lipschitz kernel for
is an essential step of each known proof of Kellerer
C
Theorem, but for dimension d > 1 it does not exist for any pairs. This fact was
very likely known by Kellerer (see the last paragraph of the introduction of Kellerer
[14]1 ). We provide a short proof of it.
Proposition 1 There exists a peacock . t /t2T indexed by T D f0; 1g and with
P.R2 / that is not associated to any Lipschitz-Markov martingale.
As a trivial corollary, the same also holds for T D Œ0; C1Œ defining t D
Œ0; 1Œ and t D 1 for t 2 Œ1; C1Œ.

t
0

2

on

Proof Let 0 D jŒ0;1 ı0 2 P.R2 / and k the dilation .x; 0/ 7! 12 .ı.x;f .x// C
ı.x; f .x// /. Let 1 be 0 k. If 1 D 0 k0 for another dilation k0 , the projection of k0 on
the Ox-axis must be identity so that k0 D k. We choose a non continuous function f
as for instance f D Œ1=2;1 , and the proof is complete because k is not a Lipschitz
kernel.

4 Proof of Theorem 2
In the three examples, we define a peacock on T D f0; 1; 10 ; 2g Á f0; 1g2 where
the indices 1, 10 stand for the intermediate elements, 0 Á .0; 0/ is the minimal
and 2 Á .1; 1/ the maximal element. One will easily check that . i /i2T is really
a peacock from the fact that we indicate during the proof martingale transitions
between 0 and 1 , 10 as well as between 1 , 10 and 2 .
To complete the statement of Theorem 2 we need to explain what are the
peacocks for T D R2C or T D R2 . In fact for .s; t/ 2 f0; 1g2 , the measures s;t are
1

Kellerer: “[. . . ], während die Übertragung der im zweiten Teil enthaltenen Ergebnisse etwa auf
den mehrdimensionalen Fall ein offenes Problem darstellt”.