[MaPhySto logo]
MaPhySto
Centre for Mathematical Physics and Stochastics
Department of Mathematical Sciences, University of Aarhus

Funded by The Danish National Research Foundation

Concentrated Advanced Course on

Lévy Processes

Lectures by Ken-iti Sato (Nagoya University)

and mini-course on Fractional Calculus

Lectures by Francesco Mainardi (University of Bologna)
and Rudolf Gorenflo (Free University of Berlin)

January 24-28, 2000

University of Aarhus

In the above-mentioned week, MaPhySto organized a Concentrated Advanced Course on LÚvy Processes. The course took place in Auditorium G1 at the Department of Mathematical Sciences, University of Aarhus. Each day there were 2-4 hours of lectures plus exercise sessions and additional lectures (see schedule below).

The course is part of a longer thematic period on LÚvy Processes and their applications - the first event in this respect was the Conference on LÚvy Processes: Theory and Applications held by MaPhySto in January 1999; see the Conference mini-proceedings. The present course will be followed by an ordinary graduate course at the Department of Mathematical Sciences, University of Aarhus. Furthermore, a follow-up concentrated advanced course on further aspects of the theory of LÚvy Processes and some of its applications will take place in the early fall, 2000.

Content of Main Course

LÚvy processes are stochastic processes on the Euclidean space, stochastically continuous and with stationary independent increments. Examples are Brownian motion, Poisson processes, stable processes (such as Cauchy processes), and subordinators (such as Gamma-processes). They form a basic class in stochastic analysis. This course aims at giving an introduction to elementary properties of LÚvy process and to transformations between LÚvy processes. Familiarity with the method of characteristic functions and some knowledge of Brownian motion, Poisson processes, and infinitely divisible distributions are expected. The following are the main contents of the lectures.

  1. Characterization of LÚvy processes by the LÚvy-Khintchine representation of infinitely divisible distributions. Probabilistic meaning of the characterization.
  2. Transformations of LÚvy processes to LÚvy processes. Especially, the subordination invented by Bochner and the density transformation (mutual absolute continuity in the path space in finite time) of Skorohod, Newman, and Kunita-S.Watanabe will be discussed in detail.
  3. Large time behaviors of LÚvy processes. Especially recurrence, transience, and oscillation are characterized. (Chung-Fuchs, Spitzer, Port-Stone, Shepp, Kesten, Erickson)
  4. Time evolution of unimodality and multimodality of the distributions of LÚvy processes on the line. (Wolfe, Yamazato, Sato, Toshiro Watanabe)

The following book will be a reference: K. Sato, LÚvy Processes and Infinitely Divisible Distributions (Cambridge Studies in Advanced Mathematics Vol. 68). Cambridge University Press, 1999.

Notes

The notes by Ken-iti Sato handed out during the course have been revised and expanded, and have appeared in the MaPhySto Lectur Notes Series, from where you may down-load them.

Content of mini-course

Francesco Mainardi (University of Bologna) and Rudolf Gorenflo (Free University of Berlin)
Fractional calculus in the theory of probability and stochastic processes

As an introduction we briefly review the essentials of the fractional calculus according to different approaches that can be useful for our applications in the theory of probability and stochastic processes.

We discuss the linear operators of fractional integration and fractional differentiation, which were introduced in pioneering works by Abel, Liouville, Riemann, Weyl, Marchaud, M. Riesz, Feller and Caputo. Particular attention is devoted to the techniques of Fourier and Laplace transforms for treating these operators in a way accessible to applied scientists, avoiding unproductive generalities and excessive mathematical rigor.

Furthermore, we discuss the approach based on limit of difference quotients, formerly introduced by GrŘnwald and Letnikov, which provides a discrete view-point to the fractional calculus. Such approach is very useful for actual numerical computation and is complementary to the previous integral approaches, which provide the continuous view-point.

We also give some information on the transcendental functions of the Mittag-Leffler and Wright type which, together with the most common Eulerian functions, turn out to play a fundamental role in the theory and applications of the fractional calculus.

Fractional calculus allows one to generalize the linear, one-dimensional, diffusion equation by replacing either the second space derivative or the first time derivative by a space or time derivative of fractional order, $\alpha $ or $\beta$, respectively. Correspondingly, the generalized equation is referred to as the space-fractional diffusion equation of order $\alpha $ or the time-fractional diffusion equation of order $\beta$, provided that its fundamental solution for the Cauchy problem can be interpreted as a (no longer Gaussian) probability density evolving in time.

For the space-fractional diffusion equation of order $\alpha$ ($0<\alpha \le 2$) we generate the class of LÚvy stable densities of index $\alpha$ according to the Feller parameterization. We thus obtain a special class of Markovian processes, called stable LÚvy motions, which for $ \alpha \ne 2$ exhibit infinite variance associated to the possibility of arbitrarily large jumps (LÚvy flights).

For the time-fractional diffusion equation of order $\beta $ ($0<\beta < 2$) we generate a class of symmetric densities whose moments of order $2n$ are proportional to the $n\, \beta$ power of time. We thus obtain a class of stochastic processes which for $\beta \ne 1$ are non-Markovian and exhibit a variance consistent with anomalous diffusion.

We also briefly consider the space-time-fractional diffusion equation, namely the cases with $\alpha \ne 2$ and $\beta \ne 1\,. $

In the space-fractional case we approximate these processes by random walk models among which we roughly distinguish four types:

(a) discrete in space, discrete in time,
(b) discrete in space, continuous in time,
(c) continuous in space, discrete in time,
(d) continuous in space, continuous in time.

In type (a) jumps of an integer multiple of a basic step-length occur at equidistant instants of time. In type (b) such jumps can occur at any instant of time, the waiting time from one jump to the next being characterized by an exponential waiting time distribution. In type (c) jumps (obeying a continuous jump-width distribution) to any point in space occur at equidistant time instants. In type (d) jumps to any point on space can occur at any instant of time and again there is exponentially distributed waiting time between two successive jumps.

We give examples for these four types and show how, via properly scaled transition to vanishing space or time step or by proper scaling of the independent variables, these four types are related to space-fractional diffusion processes and among each other. Then, as an example of a more exotic model we discuss the famous Weierstrass random walk.

Finally, we present a model of type (a) for the time-fractional diffusion process.

Notes

The notes will be expanded and revised after the course, and will appear in the MaPhySto lecture notes series.

Guest lectures

In addition to the lectures there were be three guest lectures during the course:

Sergei Levendorskii (Rostov State Academy of Economics)
Generalization of the Black-Scholes Equation For Truncated LÚvy Processes, with Applications (joint work with S.I.Boyarchenko)

ABSTRACT: We define a class of generalized truncated LÚvy processes (GTLP), which contains variance gamma processes, hyperbolic processes, processes of normal inverse gaussian type and LÚvy processes of Koponen's family.
With a market of a riskless bond and a stock, whose returns follow a GTLP, and an equivalent martingale measure (EMM), we associate GTLP-analogs of the Black-Scholes formula and equation; the GTLP-analog of the Black-Scholes equation is a pseudo-differential equation.
We discuss possible ways of fitting parameters of GTLP and EMM to data, and study main properties of GTLP--analogs of the Black-Scholes equation. We show that these properties essentially depend on a process but not on a choice of EMM.
We construct a locally risk-minimizing portfolio and produce numerical examples to show how option prices and hedging ratios depend on characteristics of a process and on a choice of EMM.
We apply the generalized Black-Scholes Equations to Pricing of the Perpetual American Put and Barrier Options.

Roderick McCrorie (University of Essex)
Applications of the fractional calculus to econometrics

ABSTRACT: In this talk we explain how the fractional calculus has been used to solve the problem of finding the exact distribution of the Wald statistic and how it underlies a method of constructing efficient tests of non-stationary hypotheses. In statistical distribution theory, the Weyl definition of a fractional integral (or derivative) has been more useful than the Riemann--Liouville definition.

Barbara Rüdiger (Institut für Angewandte Mathematik, Universität Bonn)
Construction by subordination of processes with jumps on infinite dimensional state spaces and corresponding non local Dirichlet forms

ABSTRACT: We show how symmetric non local , quasi -regular Dirichlet forms on infinite dimensional state spaces can be constructed by subordination. We start by considering any symmetric Markov semigroup $(T_t)_{t\geq 0}$ on $L^2(X,m)$ , with $(X,m)$ any measure space, and consider the subordinate Markov semigroup $(T_t^f)_{t \geq 0}$ with subordinator corresponding to any Bernstein function $f$. We construct the generator on its whole domain. We analyze properties of essentially self -adjointness of the subordinate generators, and properties of closability and quasi -regularity of the subordinate Dirichlet forms. We then show how these results can be applied to construct processes with jumps on infinite dimensional state spaces and apply them as a particular example to the subordinate of Ornstein -Uhlenbeck processes.

List of Participants

  • Silvia Valeria Annibaldi
    Dublin Institute for Advanced Studies
    School of Cosmic Physics
    5 Merrion Square
    Dublin 2, Ireland
    sa@cp.dias.ie

  • Fredrik Armerin
    Matematiska Institutionen
    KTH
    S-100 44 Stockholm
    Sweden
    armerin@math.kth.se

  • S°ren Asmussen
    Department of Mathematical Statistics
    University of Lund
    Box 118
    S-221 00 Lund, Sweden
    asmus@maths.lth.se

  • Fernando Avila
    Centro de Investigaciˇn en Matematicßs
    Xalisco s/n
    Valenciana
    36240 Guanajuato, Mexico
    avila@cimat.mx

  • Ole E. Barndorff-Nielsen
    MaPhySto
    Department of Mathematical Sciences
    University of Aarhus
    DK-8000 Aarhus C, Denmark
    oebn@imf.au.dk

  • Fred Espen Benth
    Matematisk Institutt
    Universitetet i Oslo
    PO Box 1053 Blindern
    N-0316 Oslo, Norway
    fredb@math.uio.no

  • Sara Biagini
    Scuola Normale Superiore
    Via Dei Gelsi 6
    I-19123 La Spezia
    Italy
    biagin@paley.dm.unipi.it

  • Toni Blomster
    Dep of Mathematics
    University of Helsinki
    PL 4 Yliopistonkatu 5
    SF-00014 Helsinki, Finland
    toni.blomster@helsinki.fi

  • Irene Crimaldi
    Scuola Normale Superiore
    Via Delle Tagliate 180
    I-55100 Lucca
    Italy
    crimaldi@paley.dm.unipi.it

  • JosÚ Luis Bat˙n Cutz
    CIMAT
    Apartado Postal 402
    36000 Guanajuato
    Mexico
    batun@fractal.cimat.mx

  • Giulia Di Nunno
    Department of Mathematical Statistics
    Pavia University
    Corso Lodi 47
    I-20139 Milano, Italy
    giulia@dimat.unipv.it

  • Boualem Djehiche
    Matematiska Institutionen
    KTH
    S-100 44 Stockholm
    Sweden
    boualem@math.kth.se

  • Omar El Dakkak
    Bocconi University
    Alzaia Naviglio Pavese 8
    I-20100 Milano
    Italy
    omareldakkak@yahoo.it

  • Marco Fantozzi
    Scuola Normale Superiore
    Via Turati n.35
    I-56100 Pisa
    Italy
    fantozzi@cibs.sns.it

  • Matthias Fischer
    Lange Zeile 35
    D-90419 NŘrnberg
    Germany
    matthias.fischer@wiso.uni-erlangen.de

  • Nils Chr. Framstad
    Matematisk Institutt
    Universitetet i Oslo
    PO Box 1053 Blindern
    N-0316 Oslo, Norway
    ncf@math.uio.no

  • Rudolf Gorenflo
    First Institute of Mathematics
    Free University of Berlin
    Arnimallee 2-6
    D-14195 Berlin, Germany
    gorenflo@math.fu-berlin.de

  • Svend Erik Graversen
    Department of Mathematical Sciences
    Theoretical Statistics
    University of Aarhus
    DK-8000 Aarhus C, Denmark
    matseg@imf.au.dk

  • Niels VŠver Hartvig
    Dept. of Theoretical Statistics
    University of Aarhus
    Ny Munkegade
    8000 Aarhus C, Denmark
    vaever@imf.au.dk

  • John Hawkes
    Department of Mathematics
    University College of Swansea
    Swansea
    Wales
    j.hawkes@bbk.ac.uk

  • Daniel Hlubinka
    Charles University of Prague
    Sokolovska 83
    CZ-186 75 Praha 8
    Czech Republic
    hlubinka@karlin.mff.cuni.cz

  • Wenjiang Jiang
    Department of Mathematical Sciences
    University of Aarhus
    DK-8000 Aarhus C
    Denmark
    wenjiang@imf.au.dk

  • Vladimir Kalashnikov
    Laboratory of Actuarial Mathematics
    University of Copenhagen
    Universitetsparken 5
    DK-2100 Copenhagen ě, Denmark
    vkalash@math.ku.dk

  • Joanna Karlowska-Pik
    Faculty of Mathematics and Computer Science
    Nicholas Copernicus University
    Ul. Chopina 12/18
    PL-87100 Torun, Poland
    joanka@mat.uni.torun.pl

  • Rodrigo Labouriau
    Forskningscenter Foulum
    DK-8830 Tjele
    Denmark
    rodrigo@foulum.min.dk

  • Sergei Levendorskii
    Rostov State Academy of Economics
    69 B. Sadovaya
    Rostov-on-Don 344007
    Russia
    leven@ns.rnd.runnet.ru

  • Francesco Mainardi
    Department of Physics
    University of Bologna
    Via Irnerio 46
    I-40126 Bologna, Italy
    mainardi@bo.infn.it

  • Roderick McCrorie
    Department of Economics
    University of Essex
    Wivenhoe Park
    CO4 3SQ Colchester, England
    mccro@essex.ac.uk

  • Francesco Morandin
    Scuola Normale Superiore
    Via Turati n.35
    I-56100 Pisa
    Italy
    morandin@cibs.sns.it

  • Eva Naumann
    Institut fŘr Mathematik
    Humboldt Universitńt Berlin
    Unter den Linden 6
    D-10999 Berlin, Germany
    naumann@mathematik.hu-berlin.de

  • Elisa Nicolato
    Department of Mathematical Sciences
    University of Aarhus
    DK-8000 Aarhus C
    Denmark
    elisa@imf.au.dk

  • Bent Nielsen
    Nuffield College
    New Road
    OX1 5PG Oxford
    England
    bent.nielsen@nuf.ox.ac.uk

  • Morten Nielsen
    Department of Economics
    University of Aarhus
    DK-8000 Aarhus C
    Denmark
    monielsen@econ.au.dk

  • Rolf Nielsen
    Department of Statistics and Demography
    SDU Odense University
    Campusvej 55
    DK-5230 Odense M, Denmark
    roleif@imada.sdu.dk

  • Fehmi Oezkan
    University of Freiburg
    Eckerstrasse 1
    D-79104 Freiburg
    Germany
    oezkan@fdm.uni-freiburg.de

  • Jacob Krabbe Pedersen
    Department of Mathematical Sciences
    University of Aarhus
    8000 Aarhus C
    Denmark
    krabbe@imf.au.dk

  • Jan Pedersen
    Department of Mathematical Sciences
    University of Aarhus
    8000 Aarhus C
    Denmark
    jan@imf.au.dk

  • Jesper Lund Pedersen
    Department of Mathematical Sciences
    University of Aarhus
    DK-8000 Aarhus C
    Denmark
    jesperl@imf.au.dk

  • Goran Peskir
    Department of Mathematical Sciences
    Theoretical Statistics
    University of Aarhus
    DK-8000 Aarhus C, Denmark
    goran@imf.au.dk

  • Silvio Pinamonti
    Politecnico di Torino (i)
    Corso Salvemini 41
    I-10137 Torino
    Italy
    s.pinamonti@studenti.to.it

  • Igor Pruenster
    Dip. di Matematica
    Univ. degli Studi di Pavia
    Via Ferrata 1
    I-27100 Pavia, Italy
    igor@dimat.unipv.it

  • Alfonso Rocha
    CIMAT
    Apartado Postal 402
    36000 Guanajuato
    Mexico
    arteaga@uas.uasnet.mx

  • Barbara RŘdiger
    Institut fŘr Angewandte Mathematik
    Abteilung Stochastik
    Wegelerstr. 6
    D-53115 Bonn, Germany
    ruediger@wiener.iam.uni-bonn.de

  • Deimante Rusaityte
    Laboratory of Actuarial Mathematics
    University of Copenhagen
    Universitetsparken 5
    DK-2100 Copenhagen \O, Denmark
    deimante@math.ku.dk

  • Marina Santacroce
    University of Rome
    Via Livio Andronico 56
    I-00136 Roma
    Italy
    sanma@rmi.acnet.ge

  • Matthew N. Sathekge
    The Judge Institute of Management Studies
    University of Cambridge
    Trumpington Street
    Cambridge CB2 1AG, UK
    mns22@eng.cam.ac.uk

  • Ken-iti Sato
    Prof. Emer., Nagoya University
    Mailing address:
    Hachiman-yama 1101-5-103, Tenpaku-ku
    Nagoya, 468-0074, Japan
    ken-iti.sato@nifty.ne.jp

  • Wim Schoutens
    K.U. Leuven
    EURANDOM
    Celestijnenlaan 200 B
    B-3001 Leuven, Belgium
    wim.schoutens@wis.kuleuven.ac.be

  • Takaaki Shimura
    The Institute of Statistical Mathematics
    4-6-7 Minami-Azabu Minato-ku
    106-8569 Tokyo
    Japan
    shimura@ism.ac.jp

  • Jose Lluis Sole
    Dept. de Matematiques
    Universitat Autonoma de Barcelona
    E-08193 Bellaterra (Barcelona)
    Spain
    jllsole@mat.uab.es

  • Dario Spano
    Dip. di Matematica
    Univ. degli Studi di Pavia
    Via Ferrata 1
    I-27100 Pavia, Italy
    dario@dimat.unipv.it

  • Lars Stentoft
    Dept. of Economics
    University of Aarhus
    DK-8000 Aarhus C
    Denmark
    lstentoft@econ.au.dk

  • Steen Thjorbj°rnsen
    Dept. of Mathematics and Computer Science
    SDU Odense University
    DK-5230 Odense M
    Denmark
    steenth@imada.sdu.dk

  • Frederik Utzet
    Dep. de Matematiques
    Universitat Autonoma de Barcelona
    E-08193 Bellaterra (Barcelona)
    Spain
    utzet@mat.uab.es

  • Mikhail Vronski
    Steklov Mathematical Institute
    Gubkina str. 8
    117966 Moscow
    Russia
    mvronski@mail.ru

  • Matthias Winkel
    Lab. de ProbabilitÚs et ModŔles AlÚatoires
    UniversitÚ Pierre et Marie Curie
    4 Place Jussieu
    F-75252 Paris Cedex 05, France
    winkel@proba.jussieu.fr

  • Jeannette Woerner
    Institut fŘr Mathematische Stochastik
    Universitńt Freiburg
    Eckerstr. 1
    D-79104 Freiburg, Germany
    woerner@stochastik.uni-freiburg.de

  • Jacopo Zani
    Dept. of Applied Mathematics
    Politecnico of Turin
    Via Pediano 6/B
    I-40026 Imola (BO), Italy
    jacopozani@libero.it

  • Pio Andrea Zanzotto
    Dip. di Matematica
    Universita di Pisa
    Via F. Buonarroti
    2-56127 Pisa, Italy
    zanzotto@dm.unipi.it

  • Cristina Zucca
    Dept. of Mathematics
    University of Turin
    Via Carlo Alberto 100
    I-10123 Torino, Italy
    zucca@dm.unito.it

Schedule

Time Monday Tuesday Wednesday Thursday Friday
9.00-9.50 registration Mainardi Mainardi Gorenflo Gorenflo
BREAK
10.10-11.00 Sato Sato Sato Sato Sato
11.10-12.00 Sato Sato Sato Sato Sato
LUNCH (from 12.30)
14.00-14.50 B. Rüdiger S. Levendorskii McCrorie/Exercises Exercises
BREAK
15.20-16.20 Exercises Exercises Exercises Exercises

More Information

Do not hesitate to contact the MaPhySto secretariat (at maphysto@maphysto.dk) or the local organizers Ole E. Barndorff-Nielsen and Goran Peskir for more information.


This document, http://www.maphysto.dk/oldpages/events/LevyCAC2000/index.html, was last modified January 19, 2004. Questions or comments to the contents of this document should be directed to maphysto@maphysto.dk.