Dirichlet Forms Methods for Poisson Point Measures and Lévy Processes: With Emphasis on the Creation-Annihilation Techniques

333 Pages·2015·3.55 MB·English

by  Nicolas Bouleau

Probability Theory and Stochastic Modelling 76
Nicolas Bouleau
Laurent Denis
Dirichlet Forms
Methods for Poisson
Point Measures and
Lévy Processes
With Emphasis on the
Creation-Annihilation Techniques
Probability Theory and Stochastic Modelling
Volume 76
Editors-in-Chief
Søren Asmussen, Aarhus, Denmark
Peter W. Glynn, Stanford, USA
Thomas G. Kurtz, Madison, WI, USA
Yves Le Jan, Orsay, France
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Martin Hairer, Coventry, UK
Peter Jagers, Gothenburg, Sweden
Ioannis Karatzas, New York, NY, USA
Frank P. Kelly, Cambridge, UK
Andreas E. Kyprianou, Bath, UK
Bernt Øksendal, Oslo, Norway
George Papanicolaou, Stanford, CA, USA
Etienne Pardoux, Marseille, France
Edwin Perkins, Vancouver, BC, Canada
Halil Mete Soner, Zürich, Switzerland
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Nicolas Bouleau Laurent Denis
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Dirichlet Forms Methods
for Poisson Point Measures
é
and L vy Processes
With Emphasis on the Creation-
Annihilation Techniques
123
NicolasBouleau Laurent Denis
L’École des Ponts ParisTech Laboratoire ManceaudeMathématique
UniversitéParis-Est UniversitéduMaine
Marnela Vallée LeMans
France France
ISSN 2199-3130 ISSN 2199-3149 (electronic)
Probability Theoryand Stochastic Modelling
ISBN978-3-319-25818-8 ISBN978-3-319-25820-1 (eBook)
DOI 10.1007/978-3-319-25820-1
LibraryofCongressControlNumber:2015954981
MathematicsSubjectClassification(2010): 60H07,60G57,60G51,60J45
SpringerChamHeidelbergNewYorkDordrechtLondon
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L’intérêt est assez difficile à prévoir,
les chaos de Wiener, par exemple, ont été
découverts par Wiener dans les années 30,
il en a donné la définition et personne
ne s’y est intéressé…
On prend conscience du fait qu’une chose
qui n’a l’air de rien du tout est la clef,
l’idée autour de laquelle toute sorte
de choses s’organisent…
Paul-André Meyer 1997
L. Denis dedicates this book to Carole,
Coline and Corentin
Preface
Inthelandscapeofstochasticanalysis,Dirichletformsareparticularlyendearingto
those who have met their amazing properties. After having opened the finite
dimensionalpotentialtheorytoHilbertiantechniques,theyhaveprovedveryuseful
ininfinitedimensionsforthestudyofWienerspaceanddiffusionprocessesaswell
as for an error calculus adapted to stochastic analysis, while retaining some
mystery––it must be said––due to the fact that they have kept several astonishing
conjectures such as the energy image density property which remains a quite
fascinating enigma to this day.
The purpose of this book is to implement their properties to build new tech-
niques concerning Poisson measures and Lévy processes that are central to the
study of processes with jumps. More precisely, we develop a new approach to
MalliavincalculusforPoissonmeasuresandjumpprocesses.Themethodgivesrise
to a new explicit calculus: if one considers a Poisson measure as a random system
of particles, it consists in adding a particle and taking it back after computing the
gradienti.e.“derivating”withrespecttothepositionoftheparticle(thesizeofthe
jumps in the case of a Lévy process). This method permits to establish absolute
continuityofPoissonfunctionalssuchasLévyareas,solutionsofSDE’sdrivenby
Poisson measure and many other examples. By iterating the gradient we also
construct associated Sobolev spaces of any order and get some criteria ensuring
smoothness of the law of Poisson functionals.
The work of many mathematicians have put us on the path of such a program,
especiallythatofPaulMalliavin,MasatoshiFukushima,SergioAlbeverio,Michael
Roeckner and particularly Francis Hirsch to whom we are particularly acknowl-
edging. Our ideas gradually took their final form during various lectures and
conferencesinseveraluniversitieswherefruitfuldiscussionshavebeenpossible:in
Japan thanks to Iroshi Sugita, Yasushi Ishikawa and Arturo Kohatsu-Higa, at
Swansea Wales thanks to Niels Jacob, Eugene Lytvynov, in the Mittag-Leffler
Institute Stockholm in September–October 2007 thanks to Marta Sanz, in Hong
Kong in June 2009 thanks to Nicolas Privault and at Roscoff in 2012 thanks to
Rainer Buckdahn. We acknowledge these colleagues gratefully. Frequent
ix
x Preface
discussions at the university of Marne-la-Vallée with Vlad Bally, Damien
Lamberton,BenjaminJourdain,attheuniversityofEvrywithDenisFeyel,Arnaud
de la Pradelle, Thomas Simon, as well as with Suleyman Üstünel, Christophe
Chorro and Simone Scotti, and also with Jin Ma at the University of Southern
California, helped us to better define the specificity of our approach and the scope
ofthelentparticlemethodwhichwillbeexposedinthebook.Thiscollectivework,
where we should quote many other researchers, allowed us to give a course at the
Institut Henri Poincaré in Paris in 2011, from which the present book is directly
derived. This series of lectures was given on the pretext of its applications in
mathematicalfinance,whicharereal,butwhichwedonotespeciallyemphasizein
the present book, preferring to focus on mathematical ideas.
Nicolas Bouleau
Laurent Denis
Contents
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Introduction to the Theory of Dirichlet Forms. . . . . . . . . . . . . . . 9
2.1 Unbounded Operators, Semigroups and Closed Forms. . . . . . . 10
2.1.1 Self-adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.2 Semigroup and Resolvent Associated
to a Non-negative Self-adjoint Operator. . . . . . . . . . . 11
2.1.3 Closed Forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 Dirichlet Forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.1 Definition and Fundamental Relationships . . . . . . . . . 14
2.2.2 Carré du Champ, Gradient and the (EID) Property . . . 15
2.3 The Ornstein–Uhlenbeck Structure on the Wiener Space . . . . . 19
2.4 Sufficient Conditions for (EID) Property . . . . . . . . . . . . . . . . 22
2.4.1 A Sufficient Condition on ðRr;BðRrÞÞ. . . . . . . . . . . . 23
2.4.2 The Case of a Product Structure . . . . . . . . . . . . . . . . 27
2.4.3 The Case of Structures Obtained
by Injective Images. . . . . . . . . . . . . . . . . . . . . . . . . 28
3 Reminders on Poisson Random Measures . . . . . . . . . . . . . . . . . . 31
3.1 Poisson Random Measures. . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 Lévy Processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3 Framework Adopted in the Sequel . . . . . . . . . . . . . . . . . . . . 34
3.3.1 Density Lemmas. . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.3.2 Some Basic Formulas . . . . . . . . . . . . . . . . . . . . . . . 36
3.4 Multiple Integrals and Chaos Decomposition
(Without Dirichlet Forms) . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4 Construction of the Dirichlet Structure on the Upper Space . . . . . 41
4.1 Operators εþ, ε(cid:2) and Related Formulas. . . . . . . . . . . . . . . . . 41
4.2 The Ornstein–Uhlenbeck Structure on the Poisson Space. . . . . 43
4.2.1 Definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2.2 The Kabanov Integral . . . . . . . . . . . . . . . . . . . . . . . 46
xi