Concentrated Advanced Course on
Lévy Processes
and
Branching Processes
Lectures by
Jean Bertoin (Université Pierre et Marie Curie)
and
JeanFrançois Le Gall (ENS, Paris)
Monday August 28  Friday September 1, 2000
Auditorium G1, Department of Mathematical Sciences
University of Aarhus
In the abovementioned week, MaPhySto organized a
Concentrated Advanced Course on Lévy Processes
and
Branching processes.
The course took place at the Department of Mathematical Sciences,
University of Aarhus. Each day there were 24 hours of lectures plus
exercise sessions.
The course was part of a longer thematic period on Lévy Processes and their
applications  the first event in this respect was the
Conference on
Lévy Processes: Theory and Applications held by MaPhySto in January 1999;
see the Conference miniproceedings.
Furthermore, Keniti Sato gave
in the week January 2428, 2000 a Concentrated Advanced
Course on
Lévy Processes.
Content of main course
The main goal of this series of lectures was to present some connections between Lévy
processes with no negative jumps and
branching processes or random trees. The
lectures by J. Bertoin described some
of the basic theory of Lévy processes,
including subordinators, connections
with Markov processes and fluctuation theory
in the case of processes with no negative jumps.
The lectures by J.F. Le Gall dealt more
specifically with the coding of the genealogy
of continuous branching processes, including
applications to limit theorems for discrete GaltonWatson
trees and to the construction of superprocesses.
Part A (J. Bertoin)

Subordinators. (3 hours)

Preliminaries on Poisson measure, compensation and exponential formulas.

Construction of subordinators (LévyIt\^o), Laplace exponent and
LévyKhintchine formula.

Examples, connections with excursions of Markov processes,
regenerative sets and local times.

Passage times, renewal theory and DynkinLamperti theorem.

Lévy processes with no negative jumps. (3 hours)

Elementary study of subordinators with a negative drift.

General case, LévyKhintchine formula and the subordinator of passage
times.

Fluctuation theory: Markov property of the reflected process, duality
lemma and calculation of the exponents.

The scale function and its applications (twosided exit problem,
points of increase)

Continuous state branching processes. (2 hours)

Branching property and connection with subordination.

Lamperti's construction and applications.
Part B (J.F. Le Gall)

Discrete and continuous branching processes. (2 hours)

GaltonWatson processes and their scaling limits.

The construction of superprocesses and their Laplace functionals.

The quadratic branching case. (2 hours)

Coding the genealogy of Feller's diffusion by Brownian excursions.

Aldous' continuum random tree, ISE and connections with
statistical mechanics.

The Brownian snake approach to quadratic superprocesses.

Lévy processes and the general branching case. (4 hours)

The height process of a Lévy process with no negative jump.

The connection with branching processes: A generalized
RayKnight theorem.

Limit theorems for GaltonWatson trees and applications to
reduced trees.

The genealogical structure of general continuous trees.

Applications to superprocesses.
The following books were references:
 Jean Bertoin:

Lévy processes. Cambridge Tracts in Mathematics, 121.
Cambridge University Press, 1996. ISBN: 0521562430
 JeanFrançois Le Gall:

Spatial branching processes, random
snakes and partial differential equations.
Lectures in Mathematics. Birkhäuser, 1999. ISBN: 3764361263
Additional lectures
In addition to the main lectures there was a minicourse
(4 hours) with
lectures by Ole E. BarndorffNielsen (MaPhySto) on
Lévy Processes from a modelling perspective. This minicourse
will, in particular, treat applications to financial economics.
Furthermore there were the following two survey lectures:
 Sergei Levendorskii (RostovonDon):

Option pricing under Levy processes and boundary
problems for pseudodifferential operators.
 Jan Rosinski (University of Tennessee):

Continuity and extrema of stochastic integrals
with respect to Lévy
processes.
Schedule
Note: 1011 really means 10.1511.00, and so forth.
 Monday Tuesday Wednesday Thursday Friday


910  registration OEBN OEBN OEBN OEBN

1011  Bertoin Le Gall Le Gall Le Gall Le Gall

1112  Bertoin Le Gall Le Gall Le Gall Le Gall

1415  Bertoin Bertoin Bertoin Bertoin

1516  Levendorskii Bertoin Rosinski Bertoin

1617  Exercises Exercises Exercises Exercises
A very informal workshop took place
on Monday and Tuesday in the week after the course on Lévy Processes
and Branching Processes.
The main idea
of the workshop was to have some of our many guests during that time (see
'People at
MaPhySto') to give a talk on
their current research. So some of the talks may be related to
the subject of the course, whereas others may focus on other
areas of stochastics.
Schedule
Monday (4 September); in Auditorium G2
 11.1512.00: Fred Espen Benth (University of Oslo):
 Portfolio optimization in Levy markets.
ABSTRACT:
We consider optimal portfolio selection and consumption in a market
where the logreturns of the risky assets are not normally distributed.
Recent empirical studies show that the normal inverse
Gaussian distribution is a very good and flexible model for
logreturns capturing nonGaussian effects like, e.g., heavy tails and
skewness. This distribution leads to a geometric (purejump) Levy process
dynamics of the stock prices.
We treat a portfolio optimization problem where the investor derives
her utility from present and past consumption through HindyHuangKreps
preferences. The value function of the (singular) stochastic control
problem is shown to be the unique (constrained) viscosity solution of
the HamiltonJacobiBellman equation, which will be an integrodifferential
equation subject to gradient constraints in our case. We study markets
both with and without transaction costs, and derive explicit solutions
in some special cases.
 13.0013.45: Elisa Nicolato (MaPhySto & CAF, University of Aarhus)
and Emmanuel Venardos (Nuffield College):
 Option pricing in stochastic volatility models
of the OrnsteinUhlenbeck type with a leverage effect.
ABSTRACT:
In order to capture key features of stock returns
BarndorffNielsen and Shephard (1999) have introduced a new class of stochastic
volatility models characterized by the use of processes of the OrnsteinUhlenbeck
type and allowing for a leverage effect. In this work we discuss these
models from the viewpoint of derivative asset analysis.
Instead of selecting a priori one particular equivalent martingale
measure (EMM), we study the subset of EMMs M'
under which the Lévy property of the process driving the volatility
is preserved. For each of these possible pricing measures Q, a
closed form formula for the price p(Q) of a
European call option is determined. Moreover M' is rich enough
for allowing the pricing function p(Q) to span the interval of
values that the option might take and is
sufficiently tractable for implementing calibration procedures.
Finally, for several concrete examples we discuss different numerical
approaches to the actual computation of option values.
 14.0014.45: Alexander Cherny (Moscow State University):
 Qualitative behaviour of solutions
of Stochastic Differential Equations with
singular coefficients.
ABSTRACT:
We consider a onedimensional homogeneous stochastic differential equation of
the form
dX_t = b(X_t)dt + \sigma(X_t) dB_t, X_0 = x,
where b and \sigma are supposed to be measurable functions and \sigma is
nonzero. No assumptions of boundedness (or boundedness away from zero)
are imposed
We introduce a class of points which will be called isolated singular
points, and investigate the weak existence as well as the uniqueness
of the solution in the neighborhood of such a point. A complete qualitative
classification if these points is presented: there are 63 different types.
It has been found that, for 59 types, there exists a unique solution
in the neighborhood of an isolated singular point. (This solution is
defined up to the moment it leaves some interval.) Moreover, the solution
is a strong Markov process.
The remaining 4 types of isolated singular points (we call them
branch types) disturb the uniqueness. One can construct various 'bad'
solutions in the neighborhood of a branch point. In particular, there
exist nonMarkov solutions.
As an application of the obtained results, we consider the equations of
the form
dX_t = \mu X_t^\alpha dt + \nu X_t^\beta dB_t
and present the classification for this case.
 15.1516.00: Steen Thorbjørnsen (University of Southern Denmark, Odense):
 Selfdecomposability in Free Probability.
ABSTRACT:
Free Probability was founded in the 1980's by D.V. Voiculescu
(U.C. Berkeley). The theory is a noncommutative analog to
``classical'' probability, in which random variables are replaced by
linear operators on an infinite dimensional Hilbert space, and the notion of
independence is replaced by a new concept: Freeness. This concept
may be encountered as the asymptotic relation between independent
(Gaussian) random matrices, as the size of the matrices increase to
infinity, and thus free probability provides a concrete model for the
joint asymptotic behavior of independent random matrices.
The first part of the talk will be a short introduction to free
probability, and subsequently I shall focus on the free version of the
theory of selfdecomposability.
The talk is on joint work with O.E. BarndorffNielsen.
 16.1517.00: Albert N. Shiryaev (Steklov Mathematical Institute):
 Cumulant's algebra for semimartingales
and Esscher's change of measures.
Tuesday (5 September); in Auditorium G2 before lunch; in D2 after lunch
 9.159.45: Jan Pedersen (University of Aarhus):
 Selfdecomposability and stability in
multivariate subordination I.
 10.1511.00: Keniti Sato (Nagoya University):
 Selfdecomposability and stability in
multivariate subordination II.
 15.1515.45: Francesco Mainardi (University of Bologna):
 Fractional Diffusion Processes I: analytical properties and special
functions.
ABSTRACT: The general onedimensional diffusion equation, fractional
both in space (of order \alpha) and in time (of order \beta), is
discussed. Its fundamental solution is the probability density
(evolving in time) governing the modelled stochastic process.
For a wide range of parameter this fundamental solution is shown to
be representable by aid of a Fox Hfunction with the similarity
variable x/t^{\alpha/\beta} in the argument. Convergent and
asymptotic series for its approximation are given.
 15.4516.15: Rudolf Gorenflo (Free University of Berlin):
 Fractional Diffusion Processes II: types of random walk
models and transition to the limit of vanishing stepsizes.
ABSTRACT: Four types of random walk models of Markov type (in one
spacedimension) are considered and their interrelations via
passages to the limit of vanishing space or time step (separately
or in a correctly scaled manner simultaneously in space and time)
are considered. The transition probabilities are chosen in the
domain of attraction of Levy stable probability distributions so
that these random walks approximate LevyFeller diffusion processes
that are governed by a pseudodifferential evolution equation
generalizing the classical diffusion equation. The four types of
random walk are distinguished via being discrete OR continuous IN
space OR time. Finally, a sketch is presented how to generalize the
theory to random walks with memory, thus approximating diffusion
processes that are fractional also in time.
 16.3017.15: Goran Peskir (University of Aarhus):
 Newtonian Finance.
Notes
The following notes were used for the course:
 Jean Bertoin:

Subordinators, Lévy processes with no negative jumps,
and branching processes.
Download in
[
gzipped postscriptformat 
pdfformat
]
 JeanFrançois Le Gall:

Random Trees and Spatial Branching Processes.
Download in
[
gzipped postscriptformat 
pdfformat
]
The Notes will appear in the MaPhySto Lecture Notes Series.
Participants

Peter Andrew
University of Manchester
9 Whitehall Road
Blackburn BB2 6DU
England
pandrew@ma.man.ac.uk

Yuri Bakhtin
Department of Probability
Faculty of Mechanics and Mathematics
Moscow State University
Moscow 119899, Russia
bakhtin@mech.math.msu.su

Ole E. BarndorffNielsen
MaPhySto
Department of Mathematical Sciences
University of Aarhus
DK8000 Aarhus C, Denmark
oebn@imf.au.dk

Fred Espen Benth
Department of Mathematics
University of Oslo
P.O. Box 1053 Blindern
N0316 Oslo, Norway
fredb@math.uio.no

Jean Bertoin
Laboratoire de Probabilités
Université Pierre et Marie Curie
4 Place Jussieu
F75252 Paris Cedex 05, France
jbe@ccr.jussieu.fr

Matthias Birkner
Fachbereich Mathematik
Universität Frankfurt
Postfach 111932
D60054 Frankfurt am Main, Germany
birkner@math.unifrankfurt.de

Jochen Blath
Fachbereich Mathematik
Universität Kaiserslautern
GerhartHauptmann Str. 24/251
D67663 Kaiserslautern, Germany
blath@mathematik.unikl.de

Angharad BrynJones
Manchester University
35 Lebanon Park
TW1 3DH Twickenham
England
angharad@maths.man.ac.uk

Alexander Cherny
35 Leningradsky Prospekt
Apt. 18
125284 Moscow
Russia
cherny@mech.math.msu.su

Jose Manuel Corcuera
Faculty of Mathematics
University of Barcelona
E08007 Barcelona
Spain
corcuera@mat.ub.es

Irene Crimaldi
Scuola Normale Superiore (Studio 99)
Piazza dei Cavalieri 77
I56126 Pisa
Italy
crimaldi@paley.dm.unipi.it

Thomas Duquesne
DMIENS
45 rue d'Ulm
F75230 Paris Cedex 05
France
duquesne@cmla.enscachan.fr

Susanne Emmer
Center for Mathematical Sciences
Munich University of Technology
D80290 Munich
Germany
emmer@ma.tum.de

Søren Fournais
Department of Mathematical Sciences
University of Aarhus
Ny Munkegade, Building 530
DK8000 Aarhus C, Denmark
fournais@imf.au.dk

JeanFrancois Le Gall
DMIENS
45 rue d'Ulm
F75230 Paris Cedex 05
France
legall@dmi.ens.fr

Svend Erik Graversen
Department of Mathematical Sciences
University of Aarhus
Ny Munkegade, Building 530
DK8000 Aarhus C, Denmark
matseg@imf.au.dk

Rudolf Gorenflo
Mathematik I
FU Berlin
Arnimallee 3
D14195 Berlin, Germany
gorenflo@math.fuberlin.de

Niels Hansen
LangeMullers Gade 9, 1.th.
DK2100 Copenhagen
Denmark
richard@math.ku.dk

Walter Hoh
Fakultät für Mathematik
Universität Bielefeld
Postfach 100131
D33501 Bielefeld, Germany
hoh@mathematik.unibielefeld.de

Martin Jacobsen
Department of Statistics
University of Copenhagen
Universitetsparken 5
DK2100 Copenhage Ø, Denmark
martin@math.ku.dk

Andreas Kyprianou
Mathematics Institute
Utrecht University
Budapestlaan 6
NL3584 Utrecht, Netherlands
kyprianou@math.uu.nl

Kasper Larsen
Grønlandsgade 4, I
DK5000 Odense C
Denmark
Email: vicky\_adler@vip.cybercity.dk

Sergei Levendorskii
Rostov State Academy of Economics
69 B. Sadovaya
RostovonDon 344007
Russia
leven@ns.rnd.runnet.ru

Francesco Mainardi
Department of Physics
University of Bologna
Via Irnerio 46
I40126 Bologna, Italy
mainardi@bo.infn.it

Anders MartinLöf
Department of Mathematical Statistics
Stockholm University
S10691 Stockholm
Sweden
andersml@matematik.su.se

Francesco Morandin
Scuola Normale Superiore de Pisa
c/o Collegio Carducci
Via Turati 35
I56125 Pisa, Italy
morandin@cibs.sns.it

Morten Mosegaard
Bjørnevangen 12
5260 Odense S
Denmark
mosegaard@odense.kollegienet.dk

Elisa Nicolato
Department of Mathematical Sciences
University of Aarhus
DK8000 Aarhus C
Denmark
elisa@imf.au.dk

Eulalia Nualart
Ecole Polytechnique Federale de Lausanne
E.P.F.L.D.M.A.
CH1015 Lausanne
Switzerland
eulalia.nualart@epfl.ch

Anthony Pakes
Department of Mathematics and Statistics
University of Western Australia
6907 Nedlands WA
Australia
pakes@maths.uwa.edu.au

Jacob Krabbe Pedersen
Department of Mathematical Sciences
University of Aarhus
DK8000 Aarhus C
Denmark
krabbe@imf.au.dk

Jan Pedersen
Department of Mathematical Sciences
University of Aarhus
DK8000 Aarhus C
Denmark
jan@imf.au.dk

Jesper Lund Pedersen
Department of Mathematical Sciences
University of Aarhus
DK8000 Aarhus C
Denmark
jesperl@imf.au.dk

Mads Kvist Pedersen
Solsikkemarken 34, 1.th
DK5260 Odense S
Denmark
kvist@odense.kollegienet.dk

Marta Perez
Dept. Matematica Aplicada II
Universitat Politecnica de Catalunya
Pau Gargallo 5
E08028 Barcelona, Spain
perez@ma2.upc.es

Goran Peskir
Department of Mathematical Sciences
University of Aarhus
DK8000 Aarhus C
Denmark
goran@imf.au.dk

Martijn Pistorius
Utrecht University
Budapestlaan 6
NL3584 CD Utrecht
Netherlands
pistorius@math.uu.nl

Jan Rosinski
Department of Mathematics
University of Tennessee
379961300 Tennessee
U.S.A.
rosinski@math.utk.edu

Keniti Sato
Hachimayama 11015103
Tenpakuku
Nagoya 4680074
Japan
keniti.sato@nifty.ne.jp

Yumiko Sato
Dept. of Management and Inf. Systems
Aichi Institute of Technology
Yachigusa 1247, Yakusacho
4700392 Toyota, Japan
ysato@ge.aitech.ac.jp

Rene Schilling
Mathematics Department
Nottingham Trent University
Burton Street
Nottingham NG1 4BU, England
rls@maths.ntu.ac.uk

Tomasz Schreiber
Faculty of Mathematics and Computer Science
Nicholas Copernicus University
Ul. Chopina 12/18
87100 Torun, Poland
tomeks@mat.uni.torun.pl

Josel Lluis Sole i Clivilles
Dep. de Matematiques
Universitat Autonoma de Barcelona
E08193 Bellaterra (Barcelona)
Spain
jllsole@mat.uab.es

Anja Sturm
University of Oxford
New College
Oxford OX1 3BN
England
sturm@maths.ox.ac.uk

Ana Camelia Tiplea
Scuola Normale Superiore
Piazza dei Cavalieri 7
I56100 Pisa
Italy
tiplea@cibs.sns.it

Guillermo VazquezCoutino
Univ. Autonoma MetropolitanaIztapalapa
Av. Michocan y la Purisima S/N
A.P. 55534, Col. Vicentina Del Iztapalapa
09340 Mexico D.F., Mexico
gavc@xanum.uam.mx

Emmanuel Venardos
Nuffield College
University of Oxford
New Road
Oxford OX1 1NF, England
emmanuel.venardos@nuf.ox.ac.uk

Vincent Vigon
INSA de Rouen
F76130 Mt. St. Aignan
France
vigon@lmi.insarouen.fr

Pascal Vogt
Arzheimer Hauptstrasse 92
D76829 Landau
Germany
pvogt@rhrk.unikl.de

Shiva Zamani
CMAPX, NRS UMRI 7641
Ecole Polytechnique
F91128 Palaiseau
France
shiva@cmapx.polytechnique.fr
More Information
Do not hesitate to contact the MaPhySto secretariat
(at maphysto@maphysto.dk
)
or the local organizer
Goran Peskir
for more information.
This document,
http://www.maphysto.dk/oldpages/events/LevyBranch2000/index.html,
was last modified
January 19, 2004.
Questions or comments to the contents of this document should
be directed to
maphysto@maphysto.dk.