5th International Conference on Lévy Processes: - Institut for ...

<strong>5th</strong> <strong>International</strong> <strong>Conference</strong> on
Lévy Processes:
Theory and Applications
Copenhagen
August 13-17, 2007
Updated with corrections 21-08-2007

The Fifth <strong>International</strong> <strong>Conference</strong> on Lévy Processes: Theory and Applications is hosted by the Department
of Applied Mathematics and Statistics of the University of Copenhagen. It is jointly organized with and financially
supported by the Stochastic Centre of Chalmers University Gothenburg, the Center of Mathematics and its Applications
at the University of Oslo, the Danish Natural Science Research Council, the European Mathematical Society,
the Thiele Center at the University of Aarhus and the Graduate School for Applied Mathematics at the Universities
of Copenhagen and Aarhus.
The First and Second <strong>Conference</strong>s were held in Aarhus (1999, 2002), the Third in Paris (2003), and the Fourth in
Manchester (2005). As in the previous meetings, the 2007 meeting will schedule review papers and original research
on all aspects of Lévy process theory and its applications.
It is the aim of the conference to bring together a wide range of researchers, practitioners, and graduate students
whose work is related to Levy processes and infinitely divisible distributions in a wide sense. Topics of interest
include:
• Structural results for Lévy processes: distribution and path properties
• Lévy trees, superprocesses and branching theory
• Fractal processes and fractal phenomena
• Stable and infinitely divisible processes and distributions
• Applications in finance, physics, biosciences and telecommunications
• Lévy processes on abstract structures
• Statistical, numerical and simulation aspects of Levy processes
• Lévy and stable random fields.
Scientific Organizing Committee
Gennady Samorodnitsky (Chair) (Cornell)
Søren Asmussen (Aarhus)
Jean Bertoin (Paris VI)
Serge Cohen (Toulouse)
Ron Doney (Manchester)
Niels Jacob (Swansea)
Claudia Klüppelberg (TU Munich)
Makoto Maejima (Keio)
Thomas Mikosch (Copenhagen)
Bernt Øksendal (Oslo)
Holger Rootzén (Chalmers)
Local Organizing Committee
Thomas Mikosch (Copenhagen)
Michael Sørensen (Copenhagen)
Niels Richard Hansen (Copenhagen)
Anders Tolver Jensen (Copenhagen)
Jeffrey Collamore (Copenhagen)

8.00-9.00 Registration
8.50 Opening
Scientific Program
Monday 13 Tuesday 14 Wednesday 15 Thursday 16 Friday 17
9.00-9.30 Ole Barndorff-Nielsen Sid Resnick Holger Rootzén Ron Doney Jean-François Le Gall
9.30-10.00 Michael B. Markus Ingemar Kaj Vicky Fasen François Roueff Zenghu Li
10.00-10.30 Coffee Coffee Coffee Coffee Coffee
10.30-11.00 Niels Richard Hansen Jeanette Wörner Bernt Øksendal Claudia Klüppelberg Narn-Rueih Shieh
11.00-11.30 Jean Bertoin Loïc Chaumont Thomas Simon Donatas Surgailis Michaela Prokeˇsová
11.30-12.00 Andreas Kyprianou Giulia di Nunno Zoran Vondracek Friedrich Hubalek Josep Lluís Solé
12.00-14.00 Lunch Lunch Lunch Lunch
14.00-14.30 Filip Lindskog Serge Cohen Lunch Makoto Maejima Jan Rosiński
14.30-15.00 Victor Perez-Abreu Henrik Hult and/or Alexander Lindner Jean Jacod
15.00-15.30 Coffee Coffee Excursion Coffee Closing
15.30-16.00 Jan Kallsen Jay Rosen Yimin Xiao
16.00-16.30 Davar Khoshnevisan Niels Jacob Mark Meerschaert
16.30-17.00 Rama Cont René Schilling
17.00-19.00 Poster Session with
food and drinks
19.00- <strong>Conference</strong> Dinner

Contents
Invited Speakers 5
BARNDORFF-NIELSEN, OLE E.
UPSILON TRANSFORMATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
BERTOIN, JEAN
REFLECTING A LANGEVIN PROCESS AT AN INELASTIC BOUNDARY . . . . . . . . . . . . . . . . . . . . . . . 8
CHAUMONT, LO ÏC
SOME EXPLICIT IDENTITIES ASSOCIATED WITH POSITIVE SELF-SIMILAR MARKOV PROCESSES . . . . . . . . 9
COHEN, SERGE
TAIL BEHAVIOR OF RANDOM PRODUCTS AND STOCHASTIC EXPONENTIALS . . . . . . . . . . . . . . . . . . 10
CONT, RAMA
HEDGING OPTIONS IN MODELS WITH JUMPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
DI NUNNO, GIULIA
LÉVY RANDOM FIELDS: STOCHASTIC DIFFERENTIATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
DONEY, RON
THE REFLECTED PROCESS OF A LÉVY PROCESS OR RANDOM WALK . . . . . . . . . . . . . . . . . . . . . . 13
FASEN, VICKY
ASYMPTOTIC RESULTS FOR SAMPLE AUTOCOVARIANCE FUNCTIONS AND EXTREMES OF INTEGRATED GE-
NERALIZED ORNSTEIN-UHLENBECK PROCESSES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
HANSEN, NIELS RICHARD
LEVY PROCESSES REFLECTED AT A GENERAL BARRIER . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
HULT, HENRIK
LARGE DEVIATIONS FOR POINT PROCESSES BASED ON HEAVY-TAILED SEQUENCES . . . . . . . . . . . . . . 16
JACOB, NIELS
SUBORDINATION WITH RESPECT TO STATE SPACE DEPENDENT BERNSTEIN FUNCTIONS. . . . . . . . . . . . 17
JACOD, JEAN
DISCRETIZATION OF SEMIMARTINGALES AND NOISE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
JENSEN, ANDERS TOLVER
AN EM ALGORITHM FOR REGIME SWITCHING LÉVY PROCESSES . . . . . . . . . . . . . . . . . . . . . . . . 19
KAJ, INGEMAR
SELF-SIMILAR RANDOM FIELDS AND RESCALED RANDOM BALLS MODELS . . . . . . . . . . . . . . . . . . . 20
KALLSEN, JAN
ON QUADRATIC HEDGING IN AFFINE STOCHASTIC VOLATILITY MODELS . . . . . . . . . . . . . . . . . . . . 21
KHOSHNEVISAN, DAVAR
LÉVY PROCESSES AND STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS . . . . . . . . . . . . . . . . . . . 22
KLÜPPELBERG, CLAUDIA
THE COGARCH MODEL: SOME RECENT RESULTS AND EXTENSIONS . . . . . . . . . . . . . . . . . . . . . . 23
KYPRIANOU, ANDREAS
OLD AND NEW EXAMPLES OF SCALE FUNCTIONS FOR SPECTRALLY NEGATIVE LÉVY PROCESSES . . . . . . .
LE GALL, JEAN-FRANÇOIS
24
RANDOM TREES AND PLANAR MAPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1

LINDNER, ALEXANDER
CONTINUITY PROPERTIES AND INFINITE DIVISIBILITY OF STATIONARY DISTRIBUTIONS OF SOME GENER-
ALISED ORNSTEIN-UHLENBECK PROCESSES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
LINDSKOG, FILIP
INFINITE DIVISIBILITY AND PROJECTIONS OF MEASURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
MAEJIMA, MAKOTO
TO WHICH CLASS DO KNOWN DISTRIBUTIONS OF REAL VALUED INFINITELY DIVISIBLE RANDOM VARIABLES
BELONG? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
MARCUS, MICHAEL B.
L p MODULI OF CONTINUITY OF LOCAL TIMES OF SYMMETRIC LÉVY PROCESSES
MEERSCHAERT, MARK M.
. . . . . . . . . . . . . . . 29
TRIANGULAR ARRAY LIMITS FOR CONTINUOUS TIME RANDOM WALKS
ØKSENDAL, BERNT
. . . . . . . . . . . . . . . . . . . . 30
A MALLIAVIN CALCULUS APPROACH TO STOCHASTIC CONTROL OF JUMP DIFFUSIONS
PEREZ-ABREU, VICTOR
. . . . . . . . . . . . 31
REPRESENTATION OF INFINITELY DIVISIBLE DISTRIBUTIONS ON CONES . . . . . . . . . . . . . . . . . . . . 32
PROKEˇ SOVÁ, MICHAELA
LÉVY DRIVEN COX POINT PROCESSES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
RESNICK, SIDNEY
LÉVY PROCESSES AND NETWORK MODELING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
ROOTZÉN, HOLGER
EMPIRICAL PROCESS THEORY FOR EXTREME CLUSTERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
ROSEN, JAY
LARGE DEVIATIONS FOR RIESZ POTENTIALS OF ADDITIVE PROCESSESS . . . . . . . . . . . . . . . . . . . . 36
ROSI ŃSKI, JAN
SPECTRAL REPRESENTATIONS OF INFINITELY DIVISIBLE PROCESSES AND INJECTIVITY OF THE
Υ-TRANSFORMATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
ROUEFF, FRANÇOIS
SOME SAMPLE PATH PROPERTIES OF LINEAR FRACTIONAL STABLE SHEET . . . . . . . . . . . . . . . . . . 38
SCHILLING, RENÉ L.
ON THE FELLER PROPERTY FOR A CLASS OF DIRICHLET PROCESSES . . . . . . . . . . . . . . . . . . . . . 39
SIMON, THOMAS
LOWER TAILS OF HOMOGENEOUS FUNCTIONALS OF STABLE PROCESSES . . . . . . . . . . . . . . . . . . . . 40
SOLÉ CLIVILLÉS, JOSEP LLUÍS
ON THE POLYNOMIALS ASSOCIATED WITH A LÉVY PROCESS . . . . . . . . . . . . . . . . . . . . . . . . . . 41
SURGAILIS, DONATAS
A QUADRATIC ARCH MODEL WITH LONG MEMORY AND LÉVY-STABLE BEHAVIOR OF SQUARES . . . . . . . .
VONDRACEK, ZORAN
42
ON INFIMA OF LEVY PROCESSES AND APPLICATION IN RISK THEORY . . . . . . . . . . . . . . . . . . . . . 43
WÖRNER, JEANNETTE H.C.
POWER VARIATION FOR REFINEMENT RIEMANN-STIELTJES INTEGRALS WITH RESPECT TO STABLE PROCESSES 44
XIAO, YIMIN
MODULI OF CONTINUITY FOR INFINITELY DIVISIBLE PROCESSES . . . . . . . . . . . . . . . . . . . . . . . . 45
ZENGHU, LI
BRANCHING PROCESSES AND STOCHASTIC EQUATIONS DRIVEN BY STABLE PROCESSES . . . . . . . . . . . . 46
Posters 47
AOYAMA, TAKAHIRO
NESTED SEQUENCE OF SOME SUBCLASSES OF THE CLASS OF TYPE G SELFDECOMPOSABLE DISTRIBUTIONS
ON R d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
BARAN, S ÁNDOR
MEAN ESTIMATION OF A SHIFTED WIENER SHEET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2

BAYER, CHRISTIAN
CUBATURE ON WIENER SPACE FOR INFINITE DIMENSIONAL PROBLEMS . . . . . . . . . . . . . . . . . . . . 51
BENE ˇ S, VIKTOR
APPLICATION OF FILTERING IN LÉVY BASED SPATIO-TEMPORAL POINT PROCESSES . . . . . . . . . . . . . . 52
EDER, IRMINGARD
THE QUINTUPLE LAW FOR SUMS OF DEPENDENT LÉVY PROCESSES . . . . . . . . . . . . . . . . . . . . . . 53
ESMAEILI, HABIB
PARAMETER ESTIMATION OF LÉVY COPULA
FAZEKAS, ISTV
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
ÁN
ALMOST SURE LIMIT THEOREMS FOR SEMI-SELFSIMILAR PROCESSES
GRZYWNY, TOMASZ
. . . . . . . . . . . . . . . . . . . . . 55
ESTIMATES OF GREEN FUNCTION FOR SOME PERTURBATIONS OF FRACTIONAL LAPLACIAN
HINZ, MICHAEL
. . . . . . . . . 56
APPROXIMATION OF JUMP PROCESSES ON FRACTALS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
HUBALEK, FRIEDRICH
ON TRACTABLE FINITE-ACTIVITY LÉVY LIBOR MARKET MODELS
ILIENKO, ANDRII
. . . . . . . . . . . . . . . . . . . . . . . 58
STOCHASTICALLY LIPSCHITZIAN FUNCTIONS AND LIMIT THEOREMS FOR FUNCTIONALS OF SHOT NOISE PROCESSES 59
ISHIKAWA, YASUSHI
COMPOSITION OF POISSON VARIABLES WITH DISTRIBUTIONS
IVANOVA, NATALIA
. . . . . . . . . . . . . . . . . . . . . . . . . 60
MULTIVARIATE IBNR CLAIMS RESERVING MODEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
JACH, AGNIESZKA
61
ROBUST WAVELET-DOMAIN ESTIMATION OF THE FRACTIONAL DIFFERENCE PARAMETER IN HEAVY-TAILED
TIME SERIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
JAKUBOWSKI, TOMASZ
PERTURBATIONS OF FRACTIONAL LAPLACIAN BY GRADIENT OPERATORS . . . . . . . . . . . . . . . . . . . 63
JÖNSSON, HENRIK
EXOTIC OPTION PRICING ON SINGLE NAME CDS UNDER JUMP MODELS . . . . . . . . . . . . . . . . . . . . 64
KADANKOVA, TETYANA
TWO-SIDED EXIT PROBLEMS FOR A COMPOUND POISSON PROCESS WITH EXPONENTIAL NEGATIVE JUMPS
AND ARBITRARY POSITIVE JUMPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
KASSMANN, MORITZ
APPROXIMATION OF SYMMETRIC JUMP PROCESSES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
KELLER-RESSEL, MARTIN
YIELD CURVE SHAPES IN AFFINE ONE-FACTOR MODELS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
KHOKHLOV, YURY
ASYMPTOTIC PROPERTIES OF RANDOM SUMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
KONLACK, VIRGINIE
PRICING EQUITY SWAPS IN AN ECONOMY DRIVEN BY GEOMETRIC ITÔ-LÉVY PROCESSES . . . . . . . . . . .
KWA
69
´ SNICKI, MATEUSZ
UNIFORM BOUNDARY HARNACK INEQUALITY AND MARTIN REPRESENTATION FOR α-HARMONIC FUNCTIONS
LEMPA, JUKKA
70
ON MINIMAL β-HARMONIC FUNCTIONS OF RANDOM WALKS . . . . . . . . . . . . . . . . . . . . . . . . . . 71
LIEBMANN, THOMAS
MINIMAL Q-ENTROPY MARTINGALE MEASURES FOR EXPONENTIAL LÉVY PROCESSES . . . . . . . . . . . . . 72
MANSTAVIČIUS, MARTYNAS
HAUSDORFF-BESICOVITCH DIMENSION OF GRAPHS AND P-VARIATION OF SOME LÉVY PROCESSES . . . . . . 73
MASOL, VIKTORIYA
LÉVY BASE CORRELATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
MATSUI, MUNEYA
74
GENERALIZED FRACTIONAL ORNSTEIN-UHLENBECK PROCESSES . . . . . . . . . . . . . . . . . . . . . . . . 75
3

MI̷LO ´ S, PIOTR
OCCUPATION TIME FLUCTUATIONS OF BRANCHING PROCESSES . . . . . . . . . . . . . . . . . . . . . . . . 76
MÜLLER, GERNOT
GARCH MODELLING IN CONTINUOUS TIME FOR IRREGULARLY SPACED TIME SERIES DATA . . . . . . . . . . 77
MWANIKI, IVIVI JOSEPH
GENERALIZED HYPERBOLIC MODEL: EUROPEAN OPTION PRICING IN DEVELOPED AND EMERGING MARKETS 78
PIIL, RUNE
STOCHASTIC SIMULATION OF CORRELATION EFFECTS IN CLOUDS OF ULTRA COLD ATOMS EXPOSED TO AN
ELECTROMAGNETIC FIELD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
PIPIRAS, VLADAS
HEAVY TRAFFIC SCALINGS AND LIMIT MODELS IN A WIRELESS SYSTEM WITH LONG RANGE DEPENDENCE
AND HEAVY TAILS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
RAKKOLAINEN, TEPPO
OPTIMAL DIVIDENDS IN PRESENCE OF DOWNSIDE RISK . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
SAKUMA, NORIYOSHI
CHARACTERIZATIONS OF THE CLASS OF FREE SELF DECOMPOSABLE DISTRIBUTIONS AND ITS SUBCLASSES . 82
SCALAS, ENRICO
STATISTICAL PHYSICS APPROACH TO HIGH-FREQUENCY FINANCE . . . . . . . . . . . . . . . . . . . . . . . 83
SCHAEL, MANFRED
NON-DANGEROUS RISKY INVESTMENTS FOR INSURANCE COMPANIES . . . . . . . . . . . . . . . . . . . . . 84
SEMERARO, PATRIZIA
EXTENDING TIME-CHANGED LÉVY ASSET MODELS THROUGH MULTIVARIATE SUBORDINATORS . . . . . . . .
SHIEH, NARN-RUEIH
85
MULTIFRACTALITY OF PRODUCTS OF GEOMETRIC ORNSTEIN-UHLENBECK TYPE PROCESSES . . . . . . . . .
SHIMURA, TAKAAKI
86
QUESTIONABLE RESULTS ON CONVOLUTION EQUIVALENT DISTRIBUTIONS
SIAKALLI, MICHAILINA
. . . . . . . . . . . . . . . . . . 87
STOCHASTIC STABILIZATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
STELZER, ROBERT
MULTIVARIATE CONTINUOUS TIME LÉVY-DRIVEN GARCH PROCESSES . . . . . . . . . . . . . . . . . . . . .
SZTONYK, PAWE̷L
89
REGULARITY OF HARMONIC FUNCTIONS FOR ANISOTROPIC FRACTIONAL LAPLACIAN . . . . . . . . . . . . 90
TIKANMÄKI, HEIKKI
SERIES APPROXIMATION OF THE DISTRIBUTION OF LÉVY PROCESS . . . . . . . . . . . . . . . . . . . . . .
VERAART, ALMUT
91
FEASIBLE INFERENCE FOR REALISED VARIANCE IN THE PRESENCE OF JUMPS . . . . . . . . . . . . . . . . .
VETTER, MATHIAS
92
ESTIMATION OF INTEGRATED VOLATILITY IN THE PRESENCE OF NOISE AND JUMPS
VIGNAT, CHRISTOPHE
. . . . . . . . . . . . . 93
STUDENT RANDOM WALKS AND RELATED PROBLEMS
WULFSOHN, AUBREY
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
ORNSTEIN-UHLENBECK PROCESSES IN PHYSICS AND ENGINEERING . . . . . . . . . . . . . . . . . . . . . . 95
List of Participants 97
4

Invited Speakers
5

UPSILON TRANSFORMATIONS
BARNDORFF-NIELSEN, OLE E. University of Aarhus, Denmark, oebn@imf.au.dk
Infinite divisibility; Lévy measures; Lévy processes; stochastic integrals
Stochastic integrals of deterministic functions are infinitely divisible and thus, in particular, associate the Lévy
measure of the Lévy process to the Lévy measure of the integral. For certain types of integrands the mapping thus
established have interesting special properties and is referred to as an Upsilon transformation. More broadly, this
term is used for injective regularising mappings on the class of Lévy measures into itself. The talk will survey the
properties of such transformations, relating to classical and free infinite divisibility.
References
[1] Barndorff-Nielsen, O.E. and Lindner, A. (2006): Lévy copulas and transforms of Upsilon type. Scand. J. Statist.
34, 298-316.
[2] Barndorff-Nielsen, O.E. and Maejima M. (2007): Dynamic Upsilon Transformations. (Submitted.)
[3] Barndorff-Nielsen, O.E., Maejima, M. and Sato, K. (2004): Some classes of multivariate infinitely divisible
distributions admitting stochastic integral representation. Bernoulli 12, 1-33.
[4] Barndorff-Nielsen, O.E. and Pérez-Abreu, V. (2007): Matrix subordinators and related Upsilon transformations.
Theory of Probability and Its Applications 52 (To appear).
[5] Barndorff-Nielsen, O.E., Pedersen, J. and Sato, K. (2001): Multivariate subordination, selfdecomposability and
stability. Adv. Appl. Prob. 33, 160-187.
[6] Barndorff-Nielsen, O.E., Rosinski, J. and Thorbjørnsen, S. (2007): General Upsilon transformations. (Submitted.)
[7] Barndorff-Nielsen, O.E. and Thorbjørnsen, S. (2004): A connection between free and classical infinite divisibility.
Inf. Dim. Anal. Quantum Prob. 7, 573-590.
[8] Barndorff-Nielsen, O.E. and Thorbjørnsen, S. (2005): Classical and Free Infinite Divisibility and Lévy Processes.
In U. Franz and M. Schürmann (Eds.): Quantum Independent Increment Processes II. Quantum Lévy processes,
classical probabililty and applications to physics. Heidelberg: Springer, 33-160.
[9] Barndorff-Nielsen, O.E. and Thorbjørnsen, S. (2006): Regularising mappings of Lévy measures, Stoch. Proc.
Appl. 116, 423-446
[10] Jurek, Z.J. (1985): Relations between the s-selfdecomposable and selfdecomposable measures, Annals Prob. 13,
592-608.
[11] Sato, K. (1999): Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.
[12] Sato, K. (2005): Unpublished notes.
[13] Sato, K. (2006a): Two families of improper stochastic integrals with respect to Lévy processes. Alea (Latin
American Journal of Probability and Mathematical Statistics), 1, 47-87.
[14] Sato, K. (2006b): Additive processes and stochastic integrals. Illinois J. Math. 50, 825-851.
[15] Sato, K. (2007): Transformations of infinitely divisible distributions via improper stochastic integrals, Preprint.
7

REFLECTING A LANGEVIN PROCESS AT AN INELASTIC
BOUNDARY
BERTOIN, JEAN Université Paris 6, France, jbe@ccr.jussieu.fr
We consider a Langevin process with white noise random forcing. We suppose that the energy of the particle
is instantaneously absorbed when it hits some fixed obstacle. We point out that nonetheless, the particle can be
instantaneously reflected, and present some properties of this reflecting solution. The study relies partly on an
underlying stable Lévy process and more precisely, on Rogozin’s solution to the two-sided exit problem for the latter.
8

SOME EXPLICIT IDENTITIES ASSOCIATED WITH POSITIVE
SELF-SIMILAR MARKOV PROCESSES
CHAUMONT, LOÏC University of Angers, France, loic.chaumont@univ-angers.fr
Kyprianou, A.E. University of Bath, UK
Pardo, J.C. University of Bath, UK
Positive self-similar Markov processes, Lamperti representation, conditioned stable Lévy processes, first exit time,
first hitting time, exponential functional.
We consider some special classes of Lévy processes whose Lévy measure is of the type π(dx) = e γx ν(e x − 1)dx,
where ν is the density of the stable Lévy measure and γ is a positive parameter which depends on its characteristics.
These processes were introduced in [1] as the underlying Lévy processes in the Lamperti representation of conditioned
stable Lévy processes. We compute explicitly the law of these Lévy processes at their first exit time from a finite or
semi-finite interval and the law of their exponential functional.
References
[1] M.E. Caballero and L. Chaumont: Conditioned stable Lévy processes and Lamperti representation. J.
Appl. Prob., 43, 967–983, (2006).
[2] L. Chaumont, A. Kyprianou and J.C. Pardo: Wiener-Hopf factorization and some explicit identities associated
with positive self-similar Markov processes. Work in progess.
[3] J.W. Lamperti (1972): Semi-stable Markov processes. Z. Wahrscheinlichkeitstheorie verw. Gebiete, 22, 205-
225.
9

TAIL BEHAVIOR OF RANDOM PRODUCTS AND STOCHASTIC
EXPONENTIALS
COHEN, SERGE Université Paul Sabatier, France, Serge.Cohen@math.ups.tlse.fr
Mikosch, Thomas University of Copenhagen, Denmark, mikosch@math.ku.dk
Random product, stable process, stochastic differential equation, tail behavior:
In this paper we study the distributional tail behavior of the solution to a linear stochastic differential equation driven
by infinite variance α-stable Levy motion. We show that the solution is regularly varying with index α. An important
step in the proof is the study of a Poisson number of products of independent random variables with regularly varying
tail. The study of these products deserves its own interest because it involves interesting saddle-point approximation
techniques.
10

HEDGING OPTIONS IN MODELS WITH JUMPS
CONT, RAMA Columbia University, New York, Rama.Cont@columbia.edu
Tankov, Peter Université de Paris VII
Voltchkova, Ekaterina Université de Toulouse
Lévy process, Poisson random measure, Kunita–Watanabe decomposition, quadratic hedging, option pricing,
integro-differential equations:
We study the problem of hedging options when the underlying asset is described by a process with jumps. We
compare various hedging strategies using the underlying asset and a set of traded options and examine the properties
of the hedging error, both theoretically and through numerical experiments. We obtain a representation for the
hedging strategies that allows a direct comparison with the diffusion case and illustrates the relation between hedge
ratios and sensitivities. We illustrate in particular that using sensitivities to compute ∆-neutral and Γ-neutral hedge
ratios can lead to a large hedging error, and illustrate how such strategies can be improved by using a risk-minimizing
approach to hedging and by taking positions in options. We give numerical examples illustrating the applicability of
the approach to exponential Lévy models and stochastic volatility models with jumps.
References
[1] Cont, R., Tankov, P., Voltchkova, E. (2006) Hedging with options in models with jumps, Stochastic analysis &
applications , (Abel symposia, Vol. 2) 197–217.
[2] Cont, R., Tankov, P., Voltchkova, E. (2007) Hedging options in presence of jumps, Working Paper.
11

LÉVY RANDOM FIELDS: STOCHASTIC DIFFERENTIATION
DI NUNNO, GIULIA University of Oslo, Norway, giulian@math.uio.no
Non-anticipating integral; non-anticipating derivative; Skorohod integral; Malliavin derivative:
We present some elements of stochastic calculus with respect to stochastic measures with independent values on
a space-time product. In relation with the problem of finding explicit integrands in the Ito integral representation of
square integrable random variables, we will consider stochastic differentiation both in the non-anticipating (Ito) and
in the anticipating (Malliavin/Skorohod) framework. The relationship between the two will be exploited in order to
obtain explicit formulae for the integrand. We will present and discuss in detail the derivative operators involved
and their adjoint operators.
References
[1] Di Nunno, G.(2002) Random Fields Evolution: non-anticipating integration and differentiation, Theory of Probability
and Math. Statistics, 66, 82-94.
[2] Di Nunno, G.(2004) On orthogonal polynomials and the Malliavin derivative for Lévy stochastic measures. To
appear in Seminaires et Congres. Preprint Series in Pure Mathematics, 10.
[3] Di Nunno, G.(2006) Random Fields: non-anticipating derivative and differentiation formulae. To appear in Infin.
Dimens. Anal. Quantum Probab. Relat. Top.(2007 September). Preprint Series in Pure Mathematics, 1.
12

THE REFLECTED PROCESS OF A LÉVY PROCESS OR RANDOM
WALK
DONEY, RON University of Manchester, England, rad@maths.man.ac.uk
Maller, Ross Australian National University, Australia
Savov, Mladen University of Manchester, England
Reflected process, Curve crossing, power-law boundaries, Renewal theorems:
The reflected process is defined by Rn = max0≤r≤n Sr−Sn, n ≥ 0 if S is a random walk, and by Rt = sup 0≤s≤t Xs−
Xt, t ≥ 0 if X is a Lévy process. It has been used in several applications areas, including queuing theory, genetics,
and finance. Here we survey some investigations into it as a stochastic process in its own right, being particularly
interested in its rate of growth. We formulate our results in terms of passage times over horizontal or power- law
boundaries, but implicitly they are results about the maximum of the process and the maximum of a normalised
version of the process. We give NASCs for such passage times to be almost surely finite, and in some cases for the
expectation of the passage time to be finite. These results are proved for the discrete time case in [2], and extended
to the continuous time case in [4], using the stochastic bounds in [1]. In the case of horizontal boundaries there
are more detailed results in [3], including renewal-type theorems, and information about the overshoot. Finally the
small-time version is also treated in [4].
References
[1] Doney, R. A. (2004) Stochastic bounds for Lévy processes, Ann. Probab., 32, 1545–1552.
[2] Doney, R. A. and Maller, R. A. (2007) Curve crossing for random walks reflected at their maximum, Ann. Probab.,
35, 1351-1373.
[3] Doney, R. A., Maller, R. A., and Savov, M. (2007) Renewal theorems and stability for the reflected process.
(Preprint.)
[4] Savov, M. (2007) Curve crossing for the reflected Lévy process at zero and infinity. (Preprint.)
13

ASYMPTOTIC RESULTS FOR SAMPLE AUTOCOVARIANCE
FUNCTIONS AND EXTREMES OF INTEGRATED GENERALIZED
ORNSTEIN-UHLENBECK PROCESSES
FASEN, VICKY, Munich University of Technology, Germany, fasen@ma.tum.de
Continuous time GARCH process; extreme value theory; generalized Ornstein-Uhlenbeck process; integrated
generalized Ornstein-Uhlenbeck process; point process; regular variation; sample autocovariance function; stochastic
recurrence equation :
We consider a positive stationary generalized Ornstein-Uhlenbeck process
Vt = e −ξt
�� t
e ξs− �
dηs + V0 for t ≥ 0,
0
and the increments of the integrated generalized Ornstein-Uhlenbeck process Ik = � k �
Vt− dLt, k ∈ N, where
k−1
(ξt, ηt, Lt)t≥0 is a three-dimensional Lévy process independent of the starting random variable V0, and η is a subordinator.
As a continuous time version of ARCH(1) and GARCH(1, 1) processes, we derive the asymptotic behavior
of extremes and the sample autocovariance function of V and I similar as in the papers of Davis and Mikosch [1]
and Mikosch and Stărică [3] for the discrete-time analogon. Regular variation and point process convergence play
a crucial role in establishing the statistics of V and I. The theory can be applied to the COGARCH(1, 1) and the
Nelson diffusion model.
References
[1] Davis, R., Mikosch, T. (1998) The sample autocorrelations of heavy-tailed processes with applications to ARCH,
Ann. Statist. 26, 2049–2080.
[2] Fasen, V. (2007) Asymptotic results for sample autocovariance functions and extremes of integrated generalized
Ornstein-Uhlenbeck processes, Preprint.
[3] Mikosch, T., Stărică (2000) Limit theory for the sample autocorrelations and extremes of a GARCH(1,1) process,
Ann. Statist. 28, 1427–1451
14

LEVY PROCESSES REFLECTED AT A GENERAL BARRIER
HANSEN, NIELS RICHARD University of Copenhagen, Denmark, richard@math.ku.dk
Levy processes, reflections, non-linear barriers:
The reflection of a Lévy process at a general barrier can be constructed, as in [1], in a manner similar to that
of the reflection at 0. We present results on the tail behaviour of the global maximum for a Lévy process reflected
in a barrier given by a deterministic function. Under the assumption of a finite Laplace exponent ψ(θ) for some
θ > 0 and the existence of a solution θ ∗ > 0 to ψ(θ) = 0 we derive conditions in terms of the barrier for almost
sure finiteness of the maximum. In case it is finite almost surely, we show that the tail of its distribution decays like
K exp(−θ ∗ x). The constant K can be completely characterized, and we present several possible representations. We
also present some special cases where the constant can be computed explicitly, most prominantly is Brownian motion
with a linear or a piecewise linear barrier. The results represent the continuous time generalization of the results
obtained in [2]. In [2] the construction was motivated by algorithms from structural molecular biology. In continuous
time we can consider queue and storage models where the reflection can be interpreted as giving a time-dependent
maximal capacity. In risk theory the reflected process can be interpreted as the risk process where the barrier gives
a time-dependent strategy for (continuous) dividend payout. If time permits, we will touch upon an in-homogeneous
Poisson approximation of the times of exceedance of a high threshold. The result is shown for a special class of nice
barriers where the maximum is infinite almost surely.
References
[1] Kella, Offer; Boxma, Onno and Mandjes, Michel. (2006) A Lévy process reflected at a Poisson age process. J.
Appl. Prob. 43, 221-230.
[2] Hansen, N. R. (2006) The maximum of a random walk reflected at a general barrier The Annals of Applied
Probability, 16, No. 1, 15-29.
15

LARGE DEVIATIONS FOR POINT PROCESSES BASED ON
HEAVY-TAILED SEQUENCES
HULT, HENRIK Brown University, USA, henrik hult@brown.edu
Samorodnitsky, Gennady Cornell University, USA
Point processes; Heavy tails; Regular variation; Large deviations:
A stationary sequence of random variables with regularly varying tails is considered. For this sequence it is
possible that large values arrive in clusters. That is, there may be many large values in a relatively short period of
time. The aim is to give a detailed description of the occurrence of large values, not only the size of clusters but also
the structure within a single cluster. To do this a point process based on appropriately scaled points of the stationary
sequence is constructed. A limiting measure, on the space of point measures, describes the joint limiting behavior
of all the large values of the sequence. From this limiting result one can proceed to obtain the functional large
deviations for the partial sum process, ruin probabilities, etc. Examples include, linear processes, random coefficient
ARMA processes, and solutions to stochastic recurrence equations, and stochastic integrals.
References
[1] Davis, R.A. and Hsing, T. (1995) Point process and partial sum convergence for weakly dependent random
variables with infinite variance Ann. Probab. 23 (1995), 879–917.
[2] Hult, H. and Lindskog, F. (2007) Extremal behavior of stochastic integrals driven by regularly varying Lévy
processes Ann. Probab. 35, 309–339.
[3] Hult, H., Lindskog, F., Mikosch, T. and Samorodnitsky, G. (2005) Functional large devitations for multivariate
regularly varying random walks Ann. Appl. Probab. (15), 2651–2680.
[4] Hult, H., Samorodnitsky, G. (2007) Tail probabilities for infinite series of regularly varying random vectors
Preprint.
16

SUBORDINATION WITH RESPECT TO STATE SPACE
DEPENDENT BERNSTEIN FUNCTIONS.
JACOB, NIELS University of Wales, UK, N.Jacob@swansea.ac.uk
Evans, Kristian P.
We show that in many cases a Feller semigroup is obtained when subordinating a giving Feller semigroup with the
help of state space dependent Bernstein functions. This includes the case of fractional powers of variable order. More
generally it fits to the idea to make parameters in characteristic exponents of Levy processes state space dependent,
compare with Barndorff- Nielsen and Levendorski [1].
References
[1] Barndorff- Nielsen, Levendorski, Quantitative Finance 1 (2001), 318 - 331.
17

DISCRETIZATION OF SEMIMARTINGALES AND NOISE
JACOD, JEAN Université Paris VI, France, jj@ccr.jussieu.fr
The motivation is as follows: we consider a semimartingale X on a fixed time interval [0, T], which is observed at
regularly spaced discrete times iT/n. Moreover each observation is subject to an error - the noise - and conditionally
on the path of X the error at time t is centered around the value Xt, and further the errors at different times
are independent; however their conditional laws may depend on t, and also be random. This accommodates i.i.d.
additive errors, and also some sort of rounding errors.
Then if we want to estimate quantities related to X (volatility, jumps,...) we have first to de-noise the process.
This gives rise to a new type of limit theorems for discretized processes, where one consider increments over all
(overlapping) blocks of kn successive discretization times, with kn → ∞. The aim of this talk is to exhibit some of
these central limit type theorems, with or without jumps for the underlying process X.
18

AN EM ALGORITHM FOR REGIME SWITCHING LÉVY
PROCESSES
JENSEN, ANDERS TOLVER University of Copenhagen, Denmark, tolver@life.ku.dk
Regime switching Lévy processes; hidden Markov process; inhomogeneous Markov chain; EM algorithm:
Markov processes evolving in switching environments given by the state of an unobservable continuous time finite
Markov chain may be regarded as the continuous time analog to discrete time hidden Markov models. In this paper
we discuss the perspectives for likelihood inference in regime switching Markov models. We demonstrate that the
conditional distribution of the latent chain given the observable process is that of an inhomogeneous Markov chain.
It is explained how this structural result allows for implementation of a version of the EM algorithm for evaluating
the maximum likelihood estimator. Simulation studies illustrate the performance of the estimation procedure for
different classes of regime switching Lévy processes.
19

SELF-SIMILAR RANDOM FIELDS AND RESCALED RANDOM
BALLS MODELS
KAJ, INGEMAR Uppsala University, Sweden, ikaj@math.uu.se
Biermé, Hermine Université Paris Descartes, France
Estrade, Anne Université Paris Descartes, France
Self-similarity; Generalized random field; Fractional field; Fractional Brownian motion:
In this work we construct essentially all Gaussian, stationary and isotropic, self-similar random fields on R d in a
unified manner as scaling limits of a Poisson germ-grain type model. This is a random balls model that arises
by aggregation of spherical grains attached to uniformly scattered germs given by a Poisson point process in ddimensional
space. The grains have random radius, independent and identically distributed, with a distribution
which is assumed to have a power law behavior either in zero or at infinity. The resulting configuration of mass,
obtained by counting the number of balls that cover any given point in space, suitably centered and normalized
exhibits limit distributions under scaling. For the case of the random balls radius distribution being heavy-tailed at
infinity, the corresponding scaling operation amounts to zooming out over larger areas of space while re-normalizing
the mass. In the opposite case, when the radius of balls is given by an intensity with prescribed power-law behavior
close to zero, the scaling which is applied entails zooming in successively smaller regions of space. Infinitesimally
small microballs will emerge and eventually shape the resulting limit fields.
The rescaled limit configurations are conveniently described in a random fields setting which allows us to construct
in this manner stationary Gaussian self-similar random fields of index H, for arbitrary non-integer H > −d/2. We
obtain also non-Gaussian random fields with interesting properties, in particular a model of the type ”fractional
Poisson motion”.
The focus of the presentation will be on the microballs case, for which we also discuss the extension to nonsymmetric
grains and corresponding non-isotropic fields.
References
[1] Biermé, H., Estrade, A., Kaj, I. (2007) Self-similar random fields and rescaled random balls models, Preprint,
June 2007.
[2] Kaj, I., Leskelä, L., Norros, I., Schmidt, V. (2007) Scaling limits for random fields with long-range dependence,
Ann. Probab. 35:2, 528–550.
[3] Kaj, I., Taqqu, M.S. (2007). Convergence to fractional Brownian motion and to the Telecom process: the integral
representation approach, Brazilian Probability School, 10th anniversary volume, Eds. M.E. Vares, V. Sidoravicius,
Birkhauser 2007 (to appear).
20

ON QUADRATIC HEDGING IN AFFINE STOCHASTIC
VOLATILITY MODELS
KALLSEN, JAN TU München, Germany, kallsen@ma.tum.de
Mean-variance hedging; affine processes:
A key problem in financial mathematics is how to hedge a contingent claim by dynamic trading in the underlying.
Since models based on jump processes are incomplete, perfect replication is typically impossible. As a natural
alternative one may seek to minimize the expected squared hedging error. In this talk we discuss how to compute
the optimal hedge and the corresponding hedging error semi-explicitly in a variety of affine asset price models with
stochastic volatility and jumps.
21

LÉVY PROCESSES AND STOCHASTIC PARTIAL DIFFERENTIAL
EQUATIONS
KHOSHNEVISAN, DAVAR University of Utah, USA, davar@math.utah.edu
Foondun, Mohammud University of Utah, USA
Nualart, Eulalia University of Paris 13, France
Stochastic partial differential equations; isomorphism theorems; potential theory:
We present two problems in the theory of stochastic partial differential equations.
1. Consider the stochastic heat equation, ∂tu(t , x) = (∆xu)(t , x) + ˙ W(t , x), where ˙ W denotes space-time white
noise on R+ × R d , t ∈ R+, and x ∈ R d . It is well known that this stochastic PDE has a function-valued solution if
and only if d = 1. A generally-accepted explanation of this phenomenon is that, when d ≥ 2, the smoothing effect of
the Laplacian is overwhelmed by the roughening result of white noise [2,3,6].
We present a different explanation of this phenomenon which makes it clear that the stochastic heat equation on
R+ × R d has a function-valued solution if and only if Brownian motion in R d has local times. Our results describe
a new family of isomorphism theorems [1,4] for local times of Lévy, and more general Markov, processes. They also
make rigorous the natural assertion that the stochastic heat equation has a function-valued solution in all dimensions
d ∈ (0 , 2).
2. Consider the following system of stochastic wave equations, �uj(t , x) = ˙ Lj(t , x), subject to uj(0 , x) = ∂tuj(0 , x) =
0 for t ≥ 0 and x ∈ R. Here, j = 1, . . .,d, ˙ L := ( ˙ L1 , . . . , ˙ Ld) denotes a family of d [possibly dependent] space-time
Lévy noises, and � denotes the wave operator. In the case that ˙ L is symmetric and sufficiently stable-like, we derive
a necessary and sufficient condition for the solution to hit zero. We also compute the Hausdorff dimension of the
set of points (t , x) where the solution is zero, when zeros exist. Our method hinges on a comparison theorem which
relates the solution to the mentioned SPDE to a family of multiparameter Lévy processes [5].
References
[1] Brydges, David, Fröhlich, Jürg, and Spencer, Thomas (1982). The random walk representation of classical spin
systems and correlation inequalities. Comm. Math. Phys. 83, no. 1, 123–150.
[2] Dalang, Robert C. (1999). Extending the martingale measure stochastic integral with applications to spatially
homogeneous s.p.d.e.’s. Electron. J. Probab. 4, no. 6, 29 pp. (electronic).
[3] Dalang, Robert C., Frangos, N. E. (1998). The stochastic wave equation in two spatial dimensions. Ann. Probab.
26, no. 1, 187–212.
[4] Dynkin, E. B. (1984). Gaussian and non-Gaussian random fields associated with Markov processes. J. Funct.
Anal. 55, no. 3, 344–376.
[5] Khoshnevisan, Davar, Shieh, Narn–Ruieh, and Xiao, Yimin (2007). Hausdorff dimension of the contours of
symmetric additive Lévy processes. To appear in Probability Theory and Related Fields.
[6] Peszat, Szymon; Zabczyk, Jerzy (2006). Stochastic heat and wave equations driven by an impulsive noise. In:
Stochastic Partial Differential Equations and Applications—VII, pp. 229–242, Lect. Notes Pure Appl. Math. 245,
Chapman & Hall/CRC, Boca Raton, FL.
22

THE COGARCH MODEL: SOME RECENT RESULTS AND
EXTENSIONS
KLÜPPELBERG, CLAUDIA Munich University of Technology, Germany, cklu@ma.tum.de
Continuous time GARCH; stochastic volatility:
The continuous-time GARCH [COGARCH(1,1)] model, driven by a single Lévy noise process, exhibits the same
second order properties as the discrete-time GARCH(1,1) model. Moreover, the COGARCH(1,1) model has heavy
tails and clusters in the extremes. We present such properties of the COGARCH(1,1) model, which also prove to be
useful for statistical fitting of the model. Certain extensions of the model have been suggested and investigated and
we discuss some examples.
References
[1] Fasen, V. (2007) Asymptotic results for sample autocovariance functions and extremes of integrated generalized
Ornstein-Uhlenbeck processes. Preprint. TU München. Submitted.
[2] Klüppelberg, C., Lindner, A., Maller, R. (2004) A continuous time GARCH process driven by a Lvy process:
stationarity and second order behaviour. J. Appl. Prob. 41(3), 601-622.
[3] Maller, R.A., Müller, G. and Szimayer, A. (2007) GARCH modelling in continuous time for irregularly spaced
time series data. Preprint. ANU Canberra and TU München. Under revision.
[4] Stelzer, R. J. (2007) Multivariate Continuous Time Stochastic Volatility Models Driven by a Lévy Process. Dissertation,
TU München, submitted.
23

OLD AND NEW EXAMPLES OF SCALE FUNCTIONS FOR
SPECTRALLY NEGATIVE LÉVY PROCESSES
KYPRIANOU, ANDREAS University of Bath, U.K., a.kyprianou@bath.ac.uk
Hubalek, Freidrich Technical University Vienna, Austria
Spectrally negative Lévy processes; Wiener-Hopf factorization; scale functions:
We give a review of the state of the art with regard to the theory of scale functions for spectrally negative Lévy
processes. From this introduce a new multi-parameter family of scale functions giving attention to special cases as
well as cross-referencing their analytical behaviour against known general considerations.
24

RANDOM TREES AND PLANAR MAPS
LE GALL, JEAN-FRANÇOIS Ecole normale supérieure de Paris, France, legall@dma.ens.fr
Random tree; planar map; random metric space; Gromov-Hausdorff distance:
We discuss the convergence in distribution of rescaled random planar maps viewed as random metric spaces. More
precisely, we consider a random planar map M(n), which is uniformly distributed over the set of all planar maps with
n faces in a certain class. We equip the set of vertices of M(n) with the graph distance rescaled by the factor n −1/4 .
We then discuss the convergence in distribution of the resulting random metric spaces as n tends to infinity in the
sense of the Gromov-Hausdorff distance between compact metric spaces. This problem was stated by Oded Schramm
in his plenary address paper at the 2006 ICM, in the special case of triangulations. In the case of bipartite planar
maps, we first establish a compactness result showing that a limit exists along a suitable subsequence. Furthermore
this limit can be written as a quotient space of the Continuum Random Tree (CRT) for an equivalence relation
which has a simple definition in terms of Brownian labels atttached to the vertices of the CRT. Finally we show that
any possible limiting metric space is almost surely homeomorphic to the 2-sphere. As a key tool, we use bijections
between planar maps and various classes of labeled trees.
References
[1] Le Gall, J.F. (2006) The topological structure of scaling limit of large planar maps, Inventiones Math., in press.
[2] Le Gall, J.F., Paulin, F. (2006) Scaling limits of planar maps are homeorphic to the 2-sphere, submitted.
25

CONTINUITY PROPERTIES AND INFINITE DIVISIBILITY OF
STATIONARY DISTRIBUTIONS OF SOME GENERALISED
ORNSTEIN-UHLENBECK PROCESSES
LINDNER, ALEXANDER University of Marburg, Germany, lindner@mathematik.uni-marburg.de
Sato, Ken-iti Nagoya, Japan
Absolutely continuous; continuous singular; Poisson process; generalised Ornstein-Uhlenbeck process:
In this talk we study properties of the law µ of the integral � ∞
0 c−Nt− dYt, where c > 1 and {(Nt, Yt), t ≥ 0} is a
bivariate Lévy process such that {Nt} and {Yt} are Poisson processes with parameters a and b, respectively. These
integrals arise naturally as stationary distributions of certain generalised Ornstein-Uhlenbeck processes. The law µ
is either continuous-singular or absolutely continuous, and sufficient conditions for each case are given. Under the
condition of independence of {Nt} and {Yt}, it is shown that µ is continuous-singular if b/a is sufficiently small for
fixed c, or if c is sufficiently large for fixed a and b, or if c is an integer bigger than 1. On the other hand, for Lebesgue
almost every c, µ is absolutely continuous if b/a is sufficiently large. The law µ is infinitely divisible if {Nt} and {Yt}
are independent, but not in general. We obtain a complete characterisation of infinite divisibility for µ. The talk is
based on [5]. Related results from [1] – [4] are mentioned.
References
[1] Bertoin, J., Lindner, A., Maller, R. (2006) On continuity properties of the law of integrals of Lévy processes,
Séminaire de Probabilités. Accepted for publication.
[2] Erickson, K.B., Maller, R.A. (2004) Generalised Ornstein-Uhlenbeck processes and the convergence of Lévy integrals,
in: M. Émery, M. Ledoux, M. Yor (Eds.): Séminaire de Probabilités XXXVIII, Lecture Notes in Mathematics
1857, pp. 70–94. Springer.
[3] Kondo, H., Maejima, M., Sato, K. (2006) Some properties of exponential integrals of Lévy processes and examples,
Elect. Comm. in Probab. 11, 291–303.
[4] Lindner, A., Maller, R. (2005) Lévy integrals and the stationarity of generalised Ornstein-Uhlenbeck processes,
Stoch. Proc. Appl. 115, 1701–1722.
[5] Lindner, A., Sato, K. (2007) Continuity properties and infinite divisibility of stationary distributions of some
generalised Ornstein-Uhlenbeck processes. Submitted.
26

INFINITE DIVISIBILITY AND PROJECTIONS OF MEASURES
LINDSKOG, FILIP KTH Stockholm, Sweden, lindskog@kth.se
Infinite divisibility; projections; uniqueness:
A probability measure on R d with infinitely divisible projections is not necessarily infinitely divisible. Moreover, the
Lévy measure of an infinitely divisible probability measure on R d is not necessarily determined by the Lévy measures
of its projections. Given these negative facts it is natural to look for corresponding positive results. The problems
that appear are closely related to the question when a signed measure which may have infinite mass near the origin
is determined by its projections.
The talk will be partly based on material from joint work with Jan Boman and joint work with Henrik Hult.
27

TO WHICH CLASS DO KNOWN DISTRIBUTIONS OF REAL
VALUED INFINITELY DIVISIBLE RANDOM VARIABLES
BELONG?
MAEJIMA, MAKOTO Keio University, Japan, maejima@math.keio.ac.jp
Infinitely divisible distribution; class B; selfdecomposable distribution; class T :
Recently, subdivision of the class I(R d ) of infinitely divisible distributions on R d has been developed. Especially,
many subclasses of I(R d ) can be characterised in terms of the radial component νξ(dr), r > 0, ξ ∈ {x ∈ R d : ||x|| = 1},
of the polar decomposition of the Lévy measure. Distributions in the classes discussed in this talk have densities
lξ(r) of νξ such that νξ(dr) = lξ(r)dr, r > 0. Depending on properties of lξ(r), six classes are proposed.
(1) Jurek class U(R d ), where lξ(r) is nonincreasing.
(2) Goldie-Steutel-Bondesson class B(R d ), where lξ(r) is completely monotone.
(3) The class of selfdecomposable distributions L(R d ), where lξ(r) = r −1 kξ(r), where kξ(r) is nonincreasing.
(4) Thorin class T(R d ), where lξ(r) = r −1 kξ(r), where kξ(r) is completely monotone.
(5) The class of type G distributions G(R d ), where lξ(r) = gξ(r 2 ) with a completely monotone function gξ.
(6) The class M(R d ), where lξ(r) = r −1 gξ(r 2 ) with a completely monotone function gξ.
Also, a mapping from I(R d ) (or Ilog(R d ), the class of infinitely divisible functions with finite log-moments) to each
class can be defined, and iterating this mapping, a sequence of decreasing subclasses of each class is constructed.
In this talk, a lot of known distributions of real valued infinitely divisible random variables are examined for
determining the class to which they belong. Among many other examples, there are gamma distribution, the distribution
of logarithm of gamma random variable, tempered stable distribution, the distribution of limit of generalized
Ornstein-Uhlenbeck process, the distribution of product of standard normal random variables, the distribution of
random excursion of some Bessel processes.
There remain a lot of infinitely divisible distributions to be specified.
References
[1] Bondesson, L. (1992) Generalized Gamma Convolutions and Related Classes of Distributions and Densities,
Lecture Notes in Statistics, No. 76, Springer.
[2] Sato, K. (1999) Lévy Processes and Infinitely Divisible Distributions, Cambridge University Pres.
[3] Steutel, F.W., Van Harn, K. (2004) Infinitely Divisibility of Probability Distributions on the Real Line, Dekker.
28

L p MODULI OF CONTINUITY OF LOCAL TIMES OF
SYMMETRIC LÉVY PROCESSES
MARCUS, MICHAEL B. CUNY, USA, mbmarcus@optonline.net
Rosen, Jay CUNY, USA
Lévy processes; local times; Gaussian processes :
Let X = {X(t), t ∈ R+} be a real valued symmetric Lévy process with continuous local times {L x t , (t, x) ∈ R+ × R}
and characteristic function Ee iλX(t) = e −tψ(λ) . Let
σ 2 4
0 (x − y) =
π
�∞
0
2 λ(x−y)
sin 2
ψ(λ)
If σ2 0 (h) is concave, and satisfies some additional very weak regularity conditions, then for any p ≥ 1, and all t ∈ R+
lim
h↓0
� b
a
�
�
�
�
− Lx �p
�
t �
σ0(h) � dx = 2 p/2 E|η| p
L x+h
t
dλ.
� b
|L
a
x t |p/2 dx
for all a, b in the extended real line almost surely, and also in Lm , m ≥ 1. (Here η is a normal random variable with
mean zero and variance one.)
This result is obtained via the Eisenbaum Isomorphism Theorem, (see [1]), and depends on the related result for
Gaussian processes with stationary increments, {G(x), x ∈ R1 }, for which E(G(x) − G(y)) 2 = σ2 0 (x − y);
for all a, b ∈ R 1 , almost surely.
lim
h→0
� b
�
�p
�
�
G(x + h) − G(x) �
�
� σ0(h) � dx = E|η| p (b − a)
a
References
[1] Marcus, M.B., Rosen, J. (2006) Markov Processes, Gaussian Processes and Local Times, Cambridge Univ. Press.
29

TRIANGULAR ARRAY LIMITS FOR CONTINUOUS TIME
RANDOM WALKS
MEERSCHAERT, MARK M. Michigan State University, USA, mcubed@stt.msu.edu
Scheffler, Hans-Peter University of Siegen, Germany
Random walk; Anomalous diffusion; Random waiting time; Fractional calculus:
A continuous time random walk (CTRW) is a random walk subordinated to a renewal process, used in physics to
model anomalous diffusion. Transition densities of CTRW scaling limits solve fractional diffusion equations. Here
we develop more general limit theorems, based on triangular arrays, for sequences of CTRW processes. The array
elements consist of random vectors that incorporate both the random walk jump variable and the waiting time
preceding that jump. The CTRW limit process consists of a vector-valued Lévy process whose time parameter
is replaced by the hitting time process of a real-valued nondecreasing Lévy process (subordinator). We provide
a formula for the distribution of the CTRW limit process and show that their densities solve abstract space-time
diffusion equations. Applications to finance are discussed, and a density formula for the hitting time of any strictly
increasing subordinator is developed.
References
[1] Becker-Kern, P., Meerschaert, M.M., Scheffler, H.P. (2004) Limit theorem for continuous time random walks with
two time scales. J. Applied Probab. 41, 455–466.
[2] Becker-Kern, P., Meerschaert, M.M., Scheffler, H.P. (2004) Limit theorems for coupled continuous time random
walks. Ann. Probab. 32, 730–756.
[3] Meerschaert, M.M., Scheffler, H.P. (2007) Triangular array limits for continuous time random walks. Preprint
available at http://www.stt.msu.edu/ mcubed/triCTRW.pdf
30

A MALLIAVIN CALCULUS APPROACH TO STOCHASTIC
CONTROL OF JUMP DIFFUSIONS
ØKSENDAL, BERNT Center of Mathematics for Applications (CMA), University of Oslo, Norway,
oksendal@math.uio.no
Xunyu Zhou Chinese University of Hong Kong, China
Keywords: Stochastic control; maximum principle; Malliavin calculus; jump diffusions; partial information
We use Malliavin calculus to prove a general maximum principle for partial information optimal control of jump
diffusions.
References
[1] Øksendal, B., Zhou, X. (2007) A Malliavin calculus approach to a general maximum principle for stochastic
control, Manuscript, June 2007.
31

REPRESENTATION OF INFINITELY DIVISIBLE DISTRIBUTIONS
ON CONES
PEREZ-ABREU, VICTOR, CIMAT, Guanajuato, Mexico, pabreu@cimat.mx
Rosinski, Jan University of Tennessee, USA
Cone valued Lévy process; Regular Lévy-Khintchine representation
Lévy processes taking values in cones of Euclidean or more general vector spaces are determined by infinitely
divisible distributions concentrated on cones. Skorohod (1991) showed that an infinitely divisible distribution µ on
R d is concentrated on a normal closed cone K if and only its Fourier or Laplace transform admit the so called regular
Lévy-Khintchine representation on cone. We ask whether similar fact holds in infinite dimensional spaces. We show
that the answer is negative in general. Furthermore, we characterize normal cones K in Fréchet spaces such that
every infinitely divisible probability measure µ concentrated on K has the regular Lévy-Khintchine representation on
cone. Geometrically, the latter property is equivalent to that K does not contain a copy of the cone c + of convergent
nonnegative real sequences. Our result also answers an open question of Dettweiler (1976). In this talk a general
idea of the proof and some examples will be provided.
References
[1] Dettweiler, E. (1976), Infinitely divisible measures on the cone of an ordered locally convex vector space, Ann.
Sci. Univ. Clermont 14, 61, 11-17.
[2] Skorohod, A. V. (1991), Random Processes with Independent Increments, Kluwer Academic Publisher, Dordrecht,
Netherlands (Russian original 1986).
32

LÉVY DRIVEN COX POINT PROCESSES
PROKEˇSOVÁ, MICHAELA University of Aarhus, Denmark, prokesov@imf.au.dk
Hellmund, g. University of Aarhus, Denmark
Jensen, E.B.V. University of Aarhus, Denmark
Cox point process; Lévy basis; Inhomogeneous spatial point process:
Cox point processes constitute one of the most important and versatile class of point process models for spatial
clustered point patterns. In the presented work we consider Cox processes driven by Lévy bases – e.g. Cox processes
with a random intensity function that can be expressed in terms of an integral of a weight function with respect
to a Lévy basis. Such definition includes several classes of previously studied Cox point processes as well as new
models. We derive descriptive characteristics of the suggested models and investigate further the ability of Lévy
driven Cox processes to model point patterns with different geometric properties and/or exhibiting different types of
inhomogeneities.
References
[1] Hellmund, G., Prokeˇsová, M., Jensen, E.B.V. (2007) Lévy driven Cox point processes, In preparation.
[2] Møller, J. (2003) Shot noise Cox processes, Adv. Appl. Prob. 35, 614–640.
[3] Møller, J., Syversveen, A. R., Waagepetersen, R. P.: Log Gaussian Cox processes, Scand. J. Statist. 25, 451–482.
[4] Wolpert, R.L., Ickstadt, K. (1998) Poisson/gamma random field models for spatial statistics, Biometrika 85,
251–267.
33

LÉVY PROCESSES AND NETWORK MODELING
RESNICK, SIDNEY Cornell University, Ithaca, NY USA, sir1@cornell.com
Heavy tails; data networks; long range dependence; stable processes:
Data networks can be analyzed at large time scales (Mikosch et al., 2002; Resnick, 2006) or small time scales (D’Auria
and Resnick, 2006, 2007) with time scaling either approaching ∞ or 0. One can try to characterize multi-user input
traffic or single user inputs (Mikosch and Resnick, 2006). Tails of payload per session can be heavy with Pareto
parameter α ∈ (1, 2) as in Mikosch et al., 2002 or D’Auria and Resnick, 2006, 2007 or even so heavy that the mean
is infinite as in Mikosch and Resnick, 2006 and Resnick and Rootzen, 2000. One sees the impact of stable and Lévy
processes in various ways in all these circumstances depending on interaction of heavy tails and input rates.
References
[1] D’Auria, B., Resnick, S.I. (2006) Data network models of burstiness. Adv. in Appl. Probab., 38(2):373–404.
[2] D’Auria, B., Resnick, S.I. (2007) The influence of dependence on data network models. Technical report, Cornell
University, 2006b Report #1449, Available at legacy.orie.cornell.edu/ sid.
[3] Mikosch, T., Resnick, S.I. (2006) Activity rates with very heavy tails. Stochastic Process. Appl., 116, 131–155.
[4] Mikosch, T., Resnick, S.I., Rootzén, H. and Stegeman, A.W. (2002) Is network traffic approximated by stable
Lévy motion or fractional Brownian motion? Ann. Appl. Probab., 12 (1), 23–68.
[5] Resnick, S.I. (2006) Heavy Tail Phenomena: Probabilistic and Statistical Modeling. Springer Series in Operations
Research and Financial Engineering. Springer-Verlag, New York, ISBN: 0-387-24272-4.
[6] Resnick, S.I.,Rootzén, H. (2000) Self-similar communication models and very heavy tails. Ann. Appl. Probab.,
10, 753–778.
34

EMPIRICAL PROCESS THEORY FOR EXTREME CLUSTERS
ROOTZÉN, HOLGER Chalmers, Sweden, rootzen@math.chalmers.se
Drees, Holger University of Hamburg, Germany
Clusters, empirical processes, asymptotic statistics, stationary processes, mixing, tightness, extreme value statistics.:
Large values of stationary sequences often occur in clusters – in particular this is typically the case for heavytailed
processes. This talk is about an attempt to construct an empirical limit theory for the size and structure of
these clusters. Such a theory poses intriguing and sometimes hard problems: What are suitable cluster definitions;
which spaces should one work with; which variants of empirical process theory should be used; and which kind of
dependence restriction are appropriate? We present preliminary (but by no means the definite) answer to these
problems. Our results are aimed at a general approach to central limit theory for a range of statistical procedures
based on clusters of extreme values.
35

LARGE DEVIATIONS FOR RIESZ POTENTIALS OF ADDITIVE
PROCESSESS
ROSEN, JAY College of Staten Island, USA, jrosen3@earthlink.net
Bass, Richard
Chen, Xia
We study functionals of the form
ζt =
� t
0
· · ·
� t
0
|X1(s1) + · · · + Xp(sp)| −σ ds1 · · · dsp
where X1(t), · · · , Xp(t) are i.i.d. d-dimensional symmetric stable processes of index 0 < α ≤ 2. We prove results
about the large deviations and laws of the iterated logarithm for ζt.
36

SPECTRAL REPRESENTATIONS OF INFINITELY DIVISIBLE
PROCESSES AND INJECTIVITY OF THE Υ-TRANSFORMATION
ROSIŃSKI, JAN University of Tennessee, USA, rosinski@math.utk.edu
Infinitely divisible process; stochastic integral representations; Υ − transformation:
Given a Lévy measure η on R, consider the class IDη(Rd ) of infinitely divisible distributions on Rd with no Gaussian
parts and having Lévy measures of the form
�
ν(A) = ρ(x −1 A)η(dx) (1)
R
for some measure ρ on R d . The class IDη(R d ) can be characterized in terms the range of an extended Υηtransformation.
Indeed, (1) can be written as Υη(ρ) = ν which is a straightforward extension of the Υ-transform
[1] to the case of Lévy measures η defined on the whole real line. Using [1] one can also characterize the inclusion
relation among classes IDη(R d ) in terms of the defining Lévy measures η.
A stochastic process X = {X(t) : t ∈ T } is said to be in the class IDη if its finite dimensional distributions (f.d.d.)
belong to IDη(R d ), d ≥ 1. For example, stable processes are in the class IDη1 with η1(dx) = x −α−1 1x>0 dx; symmetric
stable processes are in the class IDη2 with η2(dx) = |x| −α−1 dx; tempered stable processes are in the class IDη3 with
η3(dx) = x −α−1 e −x 1x>0 dx. Taking η4(dx) = x −1 100 dx identifies processes with f.d.d. in the Thorin
class of generalized gamma convolutions; etc.
Given a process X = {X(t)}t∈T from a from class IDη we investigate existence and uniqueness of its spectral
representation in the form
�
Xt =
S
f(t, s)M(ds) a.s., (2)
where M is an independently scattered random measure on a Borel space S equipped with a σ-finite measure m
such that L(M(A)) has the generating triplet (0, m(A)η, m(A)a) for some a ∈ R, and f is a deterministic kernel.
We show that when Υη is injective, then it is possible to obtain ’canonical’ spectral representations (to be defined
at the talk) that are unique up to a measure preserving isomorphism between spaces (S, m). As a consequence, we
give spectral representations of stationary processes from classes IDη in terms of measure preserving flows acting
on (S, m). This extends our previous results for stable processes [2], [3] to many new classes. Applications include
classes IDηk specified above, where k ≥ 3.
References
[1] Barndorff-Nielsen, O., Rosiński, J., Thorbjørnsen, S. (2007) General Upsilon-transformations, Preprint.
[2] Rosiński, J. (1995) On the structure of stationary stable processes, Ann. Probab., 23, 1163–1187.
[3] Rosiński, J. (2001) Decomposition of stationary α-stable random fields, Ann. Probab., 28, 1797–1813.
37

SOME SAMPLE PATH PROPERTIES OF LINEAR FRACTIONAL
STABLE SHEET
ROUEFF, FRANÇOIS ENST, CNRS LTCI, France, roueff@tsi.enst.fr
Ayache, Antoine Université Lille 1, France
Xiao, Yimin Michigan State University, USA
Wavelet analysis; stable processes; Linear Fractional Stable Sheet; modulus of continuity; Hausdorff dimension.
We consider Linear Fractional Stable Sheet with values in R d . Let 0 < α < 2 and H = (H1, . . . , HN) ∈ (0, 1) N
be given. We define an α-stable field X0 = {X0(t), t ∈ R N } with values in R by
�
X0(t) =
R N
κ
N� �
(tℓ − sℓ)
ℓ=1
1 Hℓ− α
+
− (−sℓ)
1 Hℓ− α
+
�
Zα(ds),
where Zα is a strictly α-stable random measure on RN with Lebesgue measure as its control measure and β(s)
as its skewness intensity, κ > 0 is a normalizing constant, and t+ = max{t, 0}. The random field X0 is called a
linear fractional α-stable sheet defined on RN (or (N, 1)-LFSS for brevity) in R with index H. We will also consider
(N, d)-LFSS, with d > 1, that is a linear fractional α-stable sheet defined on RN and taking its values in Rd . The
(N, d)-LFSS that we consider is the stable field X = {X(t), t ∈ RN + } defined by
X(t) = � X1(t), . . . , Xd(t) � , ∀t ∈ R N + ,
where X1, . . .,Xd are d independent copies of X0.
We will present some of the results on sample path properties of Linear Fractional Stable Sheet detailed in [1,2],
namely: estimates of the modulus of continuity, and conditions for the existence and joint continuity of the local time.
Some of these results generalize the ones obtained by K. Takashima in [4] in the case N = 1 but using a completely
different method. First we introduce a random wavelet series representation of real-valued Linear Fractional Stable
Sheet. Using this representation, in the case where the paths are continuous, a uniform and quasi-optimal estimate
of the modulus of continuity of the sample path is obtained as well as an upper bound of its behavior at infinity and
around the coordinate axes. Hausdorff dimensions of the range and graph of multi-dimensional Linear Fractional
Stable Sheet are also derived showing that, in contrast with the Gaussian case (see [3]), the modulus of continuity
does not provide an optimal bound for these dimensions.
References
[1] Ayache, A., Roueff, F., Xiao, Y. (2007) Local and asymptotic properties of linear fractional stable sheets, C. R.
Acad. Sci. Paris, Ser. I. 344(6):389–394.
[2] Ayache, A., Roueff, F., Xiao, Y. (2007) Joint continuity of the local times of linear fractional stable sheets, C. R.
Acad. Sci. Paris, Ser. I. 344(10):635–640.
[3] Ayache, A., Xiao, Y. (2005) Asymptotic properties and hausdorff dimensions of fractional brownian sheets, J.
Fourier Anal. Appl. 11:407–439.
[4] Takashima, K. (1989) Sample path properties of ergodic self-similar processes, Osaka J. Math. 26(1):159–189.
38

ON THE FELLER PROPERTY FOR A CLASS OF DIRICHLET
PROCESSES
SCHILLING, RENÉ L. Universität Marburg, Germany, schilling@mathematik.uni-marburg.de
Uemura, Toshihiro Kobe, Japan
Feller property; Dirichlet form; pseudo-differenial operator; Lévy-type operator:
We establish sufficient criteria for a class of processes given via Dirichlet forms to have the Feller property. This
means, in particular, that the usual exceptional set occurring with such Dirichlet processes is empty. As a by-product
of our approach we obtain an ‘integration by parts’ formula for the generator of the Dirichlet form and it turns out
that this operator is a pseudo-differential operator generating a Levy-type process. This is joint work with T. Uemura
(Kobe, Japan).
39

LOWER TAILS OF HOMOGENEOUS FUNCTIONALS OF STABLE
PROCESSES
SIMON, THOMAS Evry University, France, tsimon@univ-evry.fr
Fluctuating additive functional; Lower tail probability; Self-similar process; Stable process:
Let Z be a strictly α-stable real Lévy process (α > 1) and X be a fluctuating β-homogeneous additive functional
of Z. We investigate the (polynomial) tail-asymptotics of the law of the first passage-time of X above 1, and give a
general upper bound. When Z has no negative jumps, we prove that this bound is optimal. When Z has negative
jumps we prove that this bound is, somewhat surprisingly, not optimal, and state a general conjecture on the value
of the exponent.
40

ON THE POLYNOMIALS ASSOCIATED WITH A LÉVY PROCESS
SOLÉ CLIVILLÉS, JOSEP LLUÍS Universitat Autònoma de Barcelona, Catalunya, jllsole@mat.uab.cat
Utzet, Frederic. Universitat Autònoma de Barcelona, Catalunya.
Lévy processes; Teugels martingales; cumulants; time-space harmonic polynomials.
On one hand, given a stochastic process X = {Xt, t ∈ R+} with finite moments of convenient order, a time–
space harmonic polynomial relative to X is a polynomial Q(x, t) such that the process Mt = Q(Xt, t) is a martingale
with respect to the filtration associated to X. We will give a closed form and a recurrence relation for a family of
time–space harmonic polynomials relative to a Lévy process, The polynomials that we propose here are related with
the polynomials that give the moments of a random variable in function of the cumulants. We will present some
examples.
On the other hand, we can construct a sequence of orthogonal polynomials pσ n (x) with respect to the measure
σ2δ0(dx) + x2 ν(dx), where σ2 is the variance of the Gaussian part of X and ν its Lévy measure. Nualart and
Schoutens proved that these polynomials, denoted as the Teugels polynomials, are the building blocks of a kind
of chaotic representation for square functionals of the Lévy process. The objective of this section is to study the
properties of this family of polynomials. Also, using the Gauss-Jacobi mechanical quadrature theorem, we give a
sequence of simple Lévy processes that converge in the Skorohod topology to X, such that all variations and iterated
integrals of the sequence converge to the variations and iterated integrals of X.
References
[1] Goswami, A. and Sengupta, A. (1995) Time–space polynomial martingales generated by a discrete time martingale,
J. Theoret. Probab., 8, 417–432.
[2] Nualart, D. and Schoutens, W. (2000) Chaotic and predictable representation for Lévy processes. Stochastic
Process. Appl., 90, 109–122.
[3] Schoutens, W. and Teugels, J. L. (1998) Lévy processes, polynomials and martingales, Comm. Statist. Stochastic
Models, 14 (1 & 2), 335–349.
[4] Sengupta, A. (2000) Time-space harmonic polynomials for continuous-time processes and an extension. J. Theoret.
Probab., 13, 951–976.
[5] Solé, J.L. and Utzet, F. Time–space harmonic polynomials. Accepted in Bernouilli.
[6] Solé, J.L. and Utzet, F. On the orthogonal polynomials associated with a Lévy process. Accepted in Annals of
Probability.
[7] Szegö, G. (1939). Orthogonal Polynomials. American Mathematical Society, Providence.
41

A QUADRATIC ARCH MODEL WITH LONG MEMORY AND
LÉVY-STABLE BEHAVIOR OF SQUARES
SURGAILIS, DONATAS Vilnius Institute of Mathematics and Informatics, Lithuania, sdonatas@ktl.mii.lt
ARCH process; Long memory; Scaling limit; Lévy stable process; Fractional Brownian motion :
We introduce a new modification of Sentana’s (1995) Quadratic ARCH (QARCH), the Linear ARCH (LARCH)
(Giraitis et al., 2000, 2004) and the bilinear models (Giraitis and Surgailis, 2002), which can combine the following
properties:
(a.1) conditional heteroskedasticity
(a.2) long memory
(a.3) the leverage effect
(a.4) strict positivity of volatility
(a.5) Lévy-stable limit behavior of partial sums of squares
Sentana’s QARCH model is known for properties (a.1), (a.3), (a.4), and the LARCH model for (a.1), (a.2), (a.3).
Property (a.5) is new.
References
[1] Giraitis, L., Robinson, P.M., Surgailis, D. (2000) A model for long memory conditional heteroscedasticity , Ann.
Appl. Probab. 10, 1002–1024.
[2] Giraitis, L., Surgailis, D. (2002) ARCH-type bilinear models with double long memory , Stoch. Process. Appl.
100, 275–300.
[3] Giraitis, L., Leipus, R., Robinson, P.M., Surgailis, D. (2004) LARCH, leverage and long memory , J. Financial
Econometrics 2, 177–210.
[4] Sentana, E. (1995) Quadratic ARCH models , Rev. Econ. Stud. 3, 77–102.
42

ON INFIMA OF LEVY PROCESSES AND APPLICATION IN RISK
THEORY
VONDRACEK, ZORAN University of Zagreb, vondra@math.hr
Let Y be a one-dimensional Levy process, C an independent subordinator and X = Y − C. We discuss the
infimum process of X. To be more specific, we are interested in times when a new infimum is reached by a jump
of the subordinator C. We give a necessary and sufficient condition that such times are discrete. A motivation for
this problem comes from the ruin theory where X can be interpreted as a perturbed risk process. When X drifts to
infinity, decomposition of the infimum at those times leads to a Pollaczek-Khintchine-type formula for the probability
of ruin.
43

POWER VARIATION FOR REFINEMENT RIEMANN-STIELTJES
INTEGRALS WITH RESPECT TO STABLE PROCESSES
WÖRNER, JEANNETTE H.C. University of Göttingen, Germany, woerner@math.uni-goettingen.de
Corcuera, José Manuel University of Barcelona, Spain
Nualart, David University of Kansas, USA
Central limit theorem; power variation; refinement Riemann-Stieltjes integral; stable process:
Over the recent years classical stochastic volatility models based on Brownian motion have been generalized to Lévytype
stochastic volatility models, where the Brownian motion is replaced by a pure jump Lévy process, which leads
to a model of the form
� t
Xt = Yt +
0
σsdLs,
for the log-price process, where L denotes a Lévy process, Y a mean process, possibly possessing jumps, and σ a
fairly general volatility process.
For these types of models the approach of estimating the integrated volatility by the quadratic variation does not
work as in the Brownian setting, since here the quadratic variation is simply the sum of the squares of the jumps
of X. However, the approach of normed power variation can be modified in a suitable way to estimate � t
0 σp sds. In
this case the norming sequence has to be chosen as ∆1−p/β with p < β, where ∆ denotes the distance between the
observations and β the Blumenthal-Getoor index of the driving Lévy process, i.e. a measure for the activity of the
jumps.
We consider the special case that L is a stable process and view the integral as a refinement Riemann-Stieltjes
integral. Compared to the classical Itô integral this has the advantage that an arbitrary correlation structure between
the volatility process and the driving stable process, as needed for modelling leverage effects, may be included without
any complications to the proofs. The key point is to use Young’s inequality to estimate the difference between the
integral and a discretization. Therefore some regularity conditions on the sample paths of the volatility process are
needed, which differ from the classical Itô setting. We analyze these conditions, provide consistency of the power
variation estimators and a functional central limit theorem.
References
[1] J. M. Corcuera, D. Nualart and J. H. C. Woerner (2007) A functional central limit theorem for the realized power
variation of integrated stable processes, Stochastic Analysis and Applications, 25, 169-186.
44

MODULI OF CONTINUITY FOR INFINITELY DIVISIBLE
PROCESSES
XIAO, YIMIN Michigan State University, U.S.A., xiao@stt.msu.edu
Infinitely divisible processes; stable processes; modulus of continuity; harmonizable fractional stable motion; linear
fractional stable motion.
Let X = {X(t), t ∈ RN } be an infinitely divisible process of the form
�
X(t) = f(t, x)M(dx),
R N
where f is a deterministic function and M is an independently scattered infinitely divisible random measure. Continuity
of such processes has been studied by many authors. See Marcus and Pisier (1984), Nolan (1989), Kwapień
and Rosiński (2004), Marcus and Rosiński (2005), Talagrand (1990, 2006).
When X is a stable process such as a harmonizable fractional stable motion or a linear fractional stable motion,
Kôno and Maejima (1991a, 1991b) studied the uniform modulus of continuity of X by using series representations
for X and the conditional Gaussian argument. Ayache, Roueff and Xiao (2007) proved similar results for a linear
fractional stable sheet X by first establishing a wavelet expansion for X.
In this talk, we present a different method for establishing local and uniform moduli of continuity for X. When
applied to stable processes, our results improve the previous theorems of Kôno and Maejima (1991a, 1991b) for
harmonizable fractional stable motion and linear fractional stable motion.
References
[1] Ayache, A., Roueff, F. and Xiao, Y. (2007), Local and asymptotic properties of linear fractional stable sheets. C.
R. Acad. Sci. Paris, Ser. A. 344, 389–394.
[2] Kôno, N. and Maejima, M. (1991a), Self-similar stable processes with stationary increments. In: Stable processes
and related topics (Ithaca, NY, 1990), pp. 275–295, Progr. Probab., 25, Birkhäuser Boston, Boston, MA.
[3] Kôno, N. and Maejima, M. (1991b), Hölder continuity of sample paths of some self-similar stable processes. Tokyo
J. Math. 14, 93–100.
[4] Marcus, M. B. and Pisier, G. (1984), Some results on the continuity of stable processes and the domain of
attraction of continuous stable processes. Ann. H. Poincaré, Section B, 20, 177–199.
[5] Marcus, M. B. and Rosinski, J. (2005), Continuity and boundedness of infinitely divisible processes: a Poisson
point process approach. J. Theoret. Probab. 18, 109–160.
[6] Nolan, J. (1989), Continuity of symmetric stable processes. J. Maultivariate Anal. 29, 84–93.
[7] Talagrand, M. (1990), Sample boundedness of stochastic processes under increments conditions. Ann. Probab.
18, 1-49.
[8] Talagrand, M. (2006), Generic Chaining. Springer-Verlag, New York.
45

BRANCHING PROCESSES AND STOCHASTIC EQUATIONS
DRIVEN BY STABLE PROCESSES
ZENGHU, LI Beijing Normal University, China, lizh@bnu.edu.cn
Continuous state branching process; One-sided stable process; Stochastic equation; Affine process:
A continuous state branching process with immigration (CBI-process) arises as the high density limit of a sequence
of Galton-Watson branching processes with immigration. The simplest CBI-process is the strong solution of the
stochastic differential equation
dx(t) = x(t) 1/2 dB(t) + dt, (1)
which involves critical binary branching. A slight generalization of the above equation is
dx(t) = x(t−) 1/α dz(t) + dt, (2)
where {z(t)} is a one-sided stable process with index α ∈ (1, 2]. While the weak existence and uniqueness of the
solution of (2) can be derived from a result of Kawazu and Watanabe (Theory Probab. Appl. 1971), the pathwise
uniqueness of the solution still remains open.
In this talk, we present some Yamada-Watanabe type criterions for stochastic equations of non-negative processes
with non-negative jumps. From one of those results, the CBI-process defined by (2) is characterized as the unique
strong solution of another stochastic equation which are much easier to handle. Using similar ideas, we define some
catalytic branching models which extends the one of Dawson and Fleischmann (J. Theoret. Probab. 1997). We
also specify the connections of the catalytic branching models with the affine Markov models introduced by Duffie,
Filipović and Schachermayer (Ann. Appl. Probab. 2003), which have been used widely in mathematical finance.
References
[1] Dawson, D.A.; Li, Z.H. (2006): Skew convolution semigroups and affine Markov processes. Ann. Probab. 34, 3:
1103–1142.
[2] Fu, Z.F.; Li, Z.H. (2007): Stochastic equations of non-negative processes with non-negative jumps. Preprint.
[3] Li, Z.H.; Ma, C.H. (2007): Catalytic discrete state branching models and related limit theorems. Preprint.
46

Posters
47

NESTED SEQUENCE OF SOME SUBCLASSES OF THE CLASS OF
TYPE G SELFDECOMPOSABLE DISTRIBUTIONS ON R d
AOYAMA, TAKAHIRO Keio University, Japan, taoyama@math.keio.ac.jp
Infinitely divisible distribution on R d ; type G distribution; selfdecomposable distribution; stochastic integral
representation; Lévy process:
The class G(R d ) of type G distributions and the class L(R d ) of selfdecomposable distributions are known as important
two subclasses of infinitely divisible distributions. In Urbanik [5] and Sato [4], they studied nested subclasses of
selfdecomposable distributions and showed relation with the class of stable distributions. Also in Maejima and
Rosiński [3], nested subclasses of type G distributions and relation with the class of stable distributions are studied.
In Aoyama et. al. [2], a subclass of type G and selfdecomposable distributions on R d , denote by M0(R d ), is
studied. It is a strict subclass of the intersection of G(R d ) and L(R d ). An analytic characterization in terms of Lévy
measures and probablistic characterizations by stochastic integral representations for M0(R d ) are shown.
In this presentation, we define nested subclasses of M0(R d ), denote by Mn(R d ), n = 1, 2, · · ·. Analytic characterizations
for Mn(R d ), n = 1, 2, · · · are given in terms of Lévy measures as well as probabilistic characterizations by
stochastic integral representations for all classes are shown.
References
[1] Aoyama, T. (2007) Nested sequence of some subclasses of the class of type G selfdecomposable distributions on
R d , submitted.
[2] Aoyama, T., Maejima, M., Rosiński, J. (2007) A Subclass of type G selfdecomposable distributions, to appear in
J. Theoretic. Probab..
[3] Maejima, M. and Rosiński, J. (2001) The class of type G distributions on R d and related subclasses of infinitely
divisible distributions, Demonstratio Math. 34, 251–266.
[4] Sato, K. (1980) Class L of multivariate distributions and its subclasses, J. Multivar. Anal., 10, 207–232.
[5] Urbanik, K. (1973) Limit laws for sequences of normed sums statisfying some stability conditions, Multivariate
Analysis–III (ed. Krishnaiah, P.R., Academic Press, New York), 225–237.
49

MEAN ESTIMATION OF A SHIFTED WIENER SHEET 1
BARAN, SÁNDOR University of Debrecen, Hungary, barans@inf.unideb.hu
Pap, Gyula University of Debrecen, Hungary
Van Zuijlen, Martien C. A. Radboud University Nijmegen, The Netherlands
Wiener sheet; L 2 -Riemann integrals; L 2 -processes along a curve; Radon-Nykodim derivative.
Let {W(s, t) : s, t ≥ 0} be a standard Wiener sheet and consider the process Z(s, t) := W(s, t) + mg(s, t) with
some given function g : R 2 + → R and with an unknown parameter m ∈ R. Let [a, c] ⊂ (0, ∞) and b1, b2 ∈ (a, c),
let γ1,2 : [a, b1] → R and γ0 : [b2, c] → R be continuous, strictly decreasing functions and let γ1 : [b1, c] → R and
γ2 : [a, b2] → R be continuous, strictly increasing functions with γ1,2(b1) = γ1(b1) > 0, γ2(b2) = γ0(b2), γ1,2(a) = γ2(a)
and γ1(c) = γ0(c). Using discrete approximation we show that the maximum likelihood estimator of the unknown
parameter m based on the observation of the process Z on the set G which contains the points bounded by the
functions γ0, γ1, γ2 and γ1,2 has the form �m = ζ/A, where
and
A := g(b1, γ1,2(b1)) 2
b1γ1,2(b1) +
+
γ1,2(a) �
γ1,2(b1)
�b1
a
� ∂2g(γ −1
1,2 (t), t)� 2
γ −1
1,2 (t)
ζ := g(b1, γ1,2(b1))Z(b1, γ1,2(b1))
b1γ1,2(b1)
�
+
+
a
b1
�
g(s, γ1,2(s)) − s∂1g(s, γ1,2(s)) �2 s2 �
ds +
γ1,2(s)
dt +
�
+
b1
c
γ2(b2) �
γ2(a)
� ∂2g(γ −1
2 (t), t)� 2
∂1g(s, γ1(s))
γ1(s)
γ −1
2 (t)
b1
c
��
dt +
G
��
Z(ds, γ1(s)) +
�
g(s, γ1,2(s)) − s∂1g(s, γ1,2(s)) �
�
Z(s, γ1,2(s))ds − sZ(ds, γ1,2(s)) �
γ1,2(a) �
γ1,2(b1)
s 2 γ1,2(s)
∂2g(γ −1
1,2 (t), t)
γ −1
1,2 (t)
Z(γ −1
1,2 (t), dt) +
γ2(b2) �
γ2(a)
∂2g(γ −1
2 (t), t)
γ −1
2 (t)
G
�
∂1g(s, γ1(s)) �2 ds
γ1(s)
� ∂1∂2g(s, t) � 2 ds dt,
∂1∂2g(s, t)Z(ds, dt)
Z(γ −1
2 (t), dt).
The obtained result is a generalization of the results of [1] and [2] and the structure of �m is similar to that of the
estimator considered in [3] where Z is observed on a rectangle.
References
[1] Arató, N. M. (1997) Mean estimation of Brownian sheet, Comput. Math. Appl. 33, 13–25.
[2] Baran, S., Pap, G. and Zuijlen, M. v. (2004) Estimation of the mean of a Wiener sheet, Stat. Inference Stoch.
Process. 7, 279–304.
[3] Baran, S., Pap, G. and Zuijlen, M. v. (2003) Estimation of the mean of stationary and nonstationary Ornstein-
Uhlenbeck processes and sheets, Comput. Math. Appl. 45, 563–579.
2005.
1 Research has been supported by the hungarian scientific research fund under grants no. OTKA-F046061/2004 and OTKAT048544/
50

CUBATURE ON WIENER SPACE FOR INFINITE DIMENSIONAL
PROBLEMS
BAYER, CHRISTIAN Vienna University of Technology, Austria, cbayer@fam.tuwien.ac.at
Teichmann, Josef Vienna University of Technology, Austria
Stochastic partial differential equations; numerical approximation; cubature on Wiener space:
Let H be a separabel real Hilbert space and consider a stochastic partial differential equation in the sense of da Prato
and Zabczyk [3], i. e.
d�
drt = (Art + α(rt))dt + βi(rt)dB i t , (3)
where A : D(A) ⊂ H → H is the generator of a C0-semi-group (St)t≥0, α, β1, . . . , βd : H → H are smooth vector
fields and Bt = (B 1 t , . . . , B d t ) is a d-dimensional Brownian motion. By a (mild) solution of the above equation one
understands a stochastic process (rt)t≥0 with values in H such that
� t
rt = Str0 +
0
St−sα(rs)ds +
i=1
d�
i=1
� t
0
St−sβi(rs)dB i s . (4)
Naturally, (3) can only be solved in very special cases, and usually one has to use numerical approximations based
on finite difference or finite elements schemes, see, for instance, Hausenblas [2]. We propose using the method of Cubature
on Wiener space, introduced by Lyons and Victoir [3] in the finite dimensional setting, for weak approximation
of the stochastic partial differential equation (3). This means that we approximate
E(f(rt)) ≈
n�
λkf(rt(ωk)),
k=1
where f : H → R is a smooth functional, λ1, . . .,λn > 0 and ω1, . . . , ωn are functions of bounded variation [0, t] → R d .
rt(ωk) is the solution of equation (4) with Brownian motion replaced by ωk. Unlike Euler methods, cubature methods
in this sense fit naturally to the concept of mild solutions.
We present some theoretical results and a few numerical examples of the method.
References
[1] Da Prato, G., Zabczyk, J. (1992) Stochastic equations in infinite dimensions, Cambridge University Press.
[2] Hausenblas, E. (2003) Approximation for semilinear stochastic evolution equations, Potential Anal. 18(2), 141–
186.
[3] Lyons, T., Victoir, N. (2004) Cubature on Wiener space, Proc. R. Soc.. Lond. Ser. A 460, 169–198.
51

APPLICATION OF FILTERING IN LÉVY BASED
SPATIO-TEMPORAL POINT PROCESSES
BENEˇS, VIKTOR Charles University in Prague, Czech Republic, benesv@karlin.mff.cuni.cz
Frcalová, Blaˇzena Charles University in Prague, Czech Republic
Filtering, overdispersion, spatio-temporal point process:
A doubly stochastic point process is investigated with random driving intensity of type
�
Λ(ξ) = g(ξ, η)Z(dη)
where Z is a Lévy basis and g a suitable function [1]. The problem of non-linear filtering is solved, i.e. inference is
done on a parametric form of the driving intensity given the events of the point process. The jump character of the
intensity model based on a compound Poisson background driving field Z suggests to apply a hierarchical Bayesian
approach to filtering using the point process densities with respect to the unit Poisson process. Markov chain Monte
Carlo (MCMC) techniques [2] then enable simultaneous filtering and parameter estimation based on the posterior.
The presented model is related to an experiment monitoring the spiking activity of a place cell of hippocampus
of an experimental rat moving in a bounded arena. Since an overdispersion of events (spikes - electrical impulses)
was experimentally observed in previous studies, a doubly stochastic spatio-temporal point process on the rat’s path
is a relevant model [3]. Conditionally given the data of events (realization of a Cox point process) and given a path,
computing using the Metropolis within Gibbs algorithm enables to get estimators of any characteristics of the driving
intensity. The model selection is evaluated by means of posterior predictive distributions for a) counts of events in
spatio-temporal subregions, b) second-order characteristics (L-function), which allows for a comparison of the fit of
various models.
References
[1] Barndorff-Nielsen, O.E., Schmiegel J. (2004). Lévy based tempo-spatial modelling; with applications to turbulence.
Usp. Mat. Nauk 159, 63–90.
[2] Møller, J., Waagepetersen, R. (2003). Statistics and simulations of spatial point processes. Singapore, World Sci.
[3] Lánsk´y, P., Vaillant, J. (2000). Stochastic model of the overdispersion in the place cell discharge. BioSystems 58,
27–32.
52

THE QUINTUPLE LAW FOR SUMS OF DEPENDENT LÉVY
PROCESSES
EDER, IRMINGARD Munich University of Technology, Germany, eder@ma.tum.de
Clayton Lévy copula; fluctuation theory; insurance risk process; multivariate dependence:
We prove the quintuple law for a general Lévy process X = X 1 + X 2 for possibly dependent processes X 1 , X 2 . The
dependence between X 1 and X 2 is modeled by a Lévy copula. The quintuple law describes the ruin event of a Lévy
process by five quantities: the time of first passage relative to the time of the last maximum at first passage, the
time of the last maximum at first passage, the overshoot at first passage, the undershoot at first passage and the
undershoot of the last maximum at first passage. We calculate these quantities for some examples and present an
application in insurance risk theory. This is joint work with Claudia Klüppelberg.
References
[1] Doney, R., Kyprianou, A. (2006) Overshoot and undershoot of Lévy processes, Ann. Appl. Probab. 16, 91–106.
[2] Eder, I., Klüppelberg, C. (2007) The quintuple law for sums of dependent Lévy processes, In preparation.
[3] Kallsen, J., Tankov, P. (2006) Characterization of dependence of multidimensional Lévy processes using Lévy
copulas, Journal of Multivariate Analysis 97, 1551–1572.
53

PARAMETER ESTIMATION OF LÉVY COPULA
ESMAEILI, HABIB Munich University of Technology, Germany, esmaeili@ma.tum.de
Lévy copula, maximum likelihood estimation, dependence structure, compound Poisson process:
We review the concept of a Lévy copula to describe the dependence structure of a multidimensional Lévy process.
We consider parametric models for the Lévy copula and estimate the parameters based on a maximum likelihood
approach. For a bivariate compound Poisson model (i.e. finite Lévy measure) we derive the likelihood function based
on the three independent components of the process. Each component conveys the parameters of the Lévy copula
and some parameters of the Lévy measure depend on single jumps only in one component, or simultaneous jumps in
both components. For infinite Lévy measures we truncate the small jumps and base our statistical analysis on the
resulting compound Poisson model. We also present a simulation study to investigate the influence of the truncation.
Some references
[1] Bregman, Y. and Klüppelberg, C. (2005) Ruin estimation in multivariate models with Clayton dependence
structure., Scand. Act. J. 2005(6), 462-480.
[2] Cont, R. and Tankov, P. (2004) Financial Modelling with Jump Processes. Chapman & Hall/CRC, Boca Raton.
[3] Esmaeili, H. and Klüppelberg, C. (2007) Parameter estimation of multivariate Lévy measure . In preparation.
[4] Kallsen, J. and Tankov, P. (2006) Characterization of dependence of multidimensional Lévy processes using Lévy
copulas. J. Mult. Anal. 97, 1551-1572.
54

ALMOST SURE LIMIT THEOREMS FOR SEMI-SELFSIMILAR
PROCESSES
FAZEKAS, ISTVÁN University of Debrecen, Hungary, fazekasi@inf.unideb.hu
Almost sure limit theorem; Semi-selfsimilar process; Semistable process; Ergodic theorem; Functional limit theorem:
An integral analogue of the almost sure limit theorem is presented for semi-selfsimilar processes. In the theorem,
instead of a sequence of random elements, a continuous time random process is involved, moreover, instead of the
logarithmic average, the integral of delta-measures is considered.
X(u), u ≥ 0, is called a semi-selfsimilar process if there exists a c > 1 such that
� �
X(cu) d=
, u ≥ 0 {X(u), u ≥ 0},
c1/α for some α > 0. The sign d = means that the finite dimensional distributions are equal.
For t > 0 let Xt(u), u ∈ [0, 1], denote the following process
Xt(u) = X(tu)
, u ∈ [0, 1] .
t1/α Let µt denote the distribution of the process Xt on D[0, 1].
Theorem. Let X(u), u ≥ 0, be a semi-selfsimilar process with càdlàg trajectories. Let X(0) = 0 a.s. Assume
that the tail σ-algebra of the process X(u) is trivial, i.e. for any A ∈ F∞ P(A) is zero or one. Then for any bounded
measurable functional F : D[0, 1] → R
for almost all ω ∈ Ω where
lim
T →∞
� T
1 1
log T 1 t F[Xt(.,
�
ω)] dt = F[x] dµ(x)
D[0,1]
µ = 1
� c
1
log c 1 t µt dt
is a mixture of the distributions of the processes Xt.
Let µT, ω be the following random measure on the space D[0, 1]
Then
µT, ω(A) = 1
� T
1
log T 1 t IA(Xt(., ω)) dt .
lim
T →∞ µT, ω = µ
for almost all ω ∈ Ω.
Then the theorem is applied to obtain almost sure limit theorems for semistable processes. Discrete versions of
the above theorems are proved. In particular, an almost sure functional limit theorem is obtained for semistable
random variables.
References
[1] Fazekas, I., Rychlik, Z., (2005) Almost sure limit theorems for semi-selfsimilar processes, Probab. Math. Statist.
25, 241–255.
[2] Major, P. (1998) Almost sure functional limit theorems. Part I. The general case, Studia Sci. Math. Hungar. 34,
273–304.
55

ESTIMATES OF GREEN FUNCTION FOR SOME
PERTURBATIONS OF FRACTIONAL LAPLACIAN
GRZYWNY, TOMASZ Wroc̷law University of Technology, POLAND, tomasz.grzywny@pwr.wroc.pl
Ryznar, Micha̷l Wroc̷law University of Technology, POLAND
Green function; symmetric stable process; relativistic stable process; Lévy process:
Suppose that Yt is a d-dimensional symmetric Lévy process with its Lévy measure such that it differs from the Lévy
measure of the isotropic α-stable process (0 < α < 2) by a finite signed measure. For a bounded Lipschitz open
set D we compare the Green functions of the process Y and its stable counterpart. We prove a few comparability
results either one sided or two sided. Assuming an additional condition about the difference of the densities of the
Lévy measures, namely that it is of order of |x| −d+ϱ as |x| → 0, where ϱ > 0, we prove that the Green functions are
comparable, provided D is connected.
These results apply for example to the relativistic α-stable process. The bounds for its Green functions were
proved before for d > α and smooth sets. In the paper we also considered one-dimensional case for α ≥ 1 and proved
that the Green functions for a bounded open interval are comparable which is the case not treated in the literature
to our best knowledge.
References
[1] Grzywny, T., Ryznar, M., Estimates of Green function for some perturbations of fractional Laplacian, Illinois J.
Math. (to appear).
[2] Grzywny, T., Ryznar, M., Two-sided optimal bounds for Green functions of half-spaces for relativistic α-stable
process, preprint (2007).
56

APPROXIMATION OF JUMP PROCESSES ON FRACTALS
HINZ, MICHAEL University of Jena, Germany, mhinz@minet.uni-jena.de
Jump processes; fractals; Dirichlet forms; convergence:
We consider Markov pure jump processes on fractal sets in Euclidean space. Such processes have been studied
by several authors in a number of recent works. First we investigate jump processes given on a general d-set, usually
of zero Lebesgue measure. On closed parallel sets of positive Lebesgue measure decreasing to the d-set we can define
jump processes via Dirichlet forms. We show they converge in a reasonable way to the process on the d-set. Our
main idea is to use some suitable spatial averaging encoded in measures on the parallel sets and to prove the Mosco
convergence of the Dirichlet forms associated to the processes. This implies the convergence of the spectral structures
in the sense of Kuwae and Shioya and in particular the weak convergence of the finite dimensional distributions of
the processes under a canonical choice of initial distributions. For nice classes of self-similar sets, we also provide
approximations in terms of finite Markov chains. As usual, their state spaces are the vertices of the prefractal
graphs. The proofs are similar to the d-set case. Without much additional effort, we also obtain a result concerning
the convergence of the processes in the Skorohod spaces D in this case.
References
[1] Chen, Z.-Q., Kumagai, T. (2003) Heat kernel estimates for stable-like processes on d-sets, Stoch. Proc. and
their Appl. 108 , 27–62. [2] Ethier, St.N., Kurtz, Th.G. (1986) Markov processes, Wiley, New York. [3] Fukushima,
M., Oshima, Y., Takeda, M. (1994) Dirichlet forms and symmetric Markov processes, deGruyter, Berlin, New York.
[4] Hansen, W., Zähle, M. (2006) Restricting α-stable Lévy processes from R n to fractal sets, Forum Math. 18,
171–191. [5] Kumagai, T. (2001) Some remarks for jump processes on fractals, In: Grabner, Woess (Eds.), Trends
in mathematics: Fractals in Graz 2001, Birkhäuser, Basel. [6] Kuwae, K., Shioya, T. (2003) Convergence of spectral
structures, Comm. Anal. Geom. 11 (4), 599–673. [7] Stós, A. (2000) Symmetric α-stable processes on d-sets, Bull.
Polish Acad. Sci. Math. 48, 237–245.
57

ON TRACTABLE FINITE-ACTIVITY LÉVY LIBOR MARKET
MODELS
HUBALEK, FRIEDRICH Vienna University of Technology, Austria, fhubalek@fam.tuwien.ac.at
Hula, Andreas Vienna University of Technology, Austria
Lévy LIBOR market model; marked point process; interest rate models; jump processes in finance:
Two attractive features of classical LIBOR market models are (i) positive interest rates, and (ii) validity of the
Black 76 formula for caplets resp. floorlets. We investigate, how far those two properties can be realized in a Lévy
LIBOR market model along the lines of Eberlein and Özkan [1], either exactly, or approximately.
We study in particular models driven by finite-activity jump processes (marked point processes) with and without
Gaussian components. In the former case the model is conditionally lognormal, in the latter piecewise deterministic
on (small) time intervals inbetween jumps. This allows quite explicit exact simulation and calculation.
References
[1] Eberlein, E., Ózkan, F. (2005) The Lévy Libor Model, Finance and Stochastics 9 (3), 327–348.
[2] Eberlein, E., Kluge, W. (2007) Calibration of Lévy term structure models, In Advances in Mathematical Finance:
In Honor of Dilip Madan, M. Fu, R. A. Jarrow, J.-Y. Yen, and R. J. Elliott (Eds.), Birkhäuser, 155–180.
[3] Belomestny, D., Schoenmakers, J. (2006) A jump-diffusion Libor model and its robust calibration, SFB 649
Discussion Paper, number SFB649DP2006–037, Humboldt University, Berlin.
58

STOCHASTICALLY LIPSCHITZIAN FUNCTIONS AND LIMIT
THEOREMS FOR FUNCTIONALS OF SHOT NOISE PROCESSES
ILIENKO, ANDRII, National Technical University of Ukraine, Ukraine, a ilienko@ukr.net
Shot noise process; Non-linear function; Integrated process; Central limit theorem
Consider the stationary shot noise process θ of the form
�
θ(t) = g(t − s)dζ(s), t ∈ R, (5)
R
with a non-random function g ∈ L2(R) and a driving Lévy process ζ.
We focuse on limit theorems for non-linear functionals of the process (5). More precisely, we discuss the asymptotic
behavior of the integrated process ΘK of the form
ΘK(T) =
� T
where K : R → R is a non-random continuous function.
0
K(θ(t))dt, T > 0,
Denote by L2(Ω, R2 ) the space of all two-dimensional random vectors � ξ = (ξ1, ξ2) on the given probability space
with Eξ2 i < ∞, i = 1, 2, and let B(R) be the Borel σ-algebra on R.
Definition 1. Let M ⊂ L2(Ω, R 2 ). We say that a continuous function K : R → R is stochastically Lipschitzian
w.r.t. M if there exists L > 0 such that
for each � ξ ∈ M.
E(K(ξ1 + ξ2) − K(ξ1)) 2 ≤ L · Eξ 2 2
Definition 2. Let M ⊂ L2(Ω, R 2 ). We say that a continuous function K : R → R is stochastically locally
Lipschitzian w.r.t. M if there exist ɛ > 0, L > 0 such that
E � �2 2
(K(ξ1 + ξ2) − K(ξ1)) · I {|ξ2|≤ɛ} ≤ L · Eξ2 for each � ξ ∈ M. Here IA denotes the indicator function of the event A.
Definition 3. Let θ be a shot noise process with representation (5), and
�
Mθ = �ξ ∈ L2(Ω, R 2 �
�
): ξi = g(−s)dζ(s), i = 1, 2; A1, A2 ∈ B(R), A1 ∩ A2 = ∅ .
Ai
We say that a function K : R → R is stochastically Lipschitzian (resp., stochastically locally Lipschitzian) w.r.t. the
shot noise process θ (and write K ∈ SLθ or K ∈ SLLθ ) if K is stochastically Lipschitzian (resp., stochastically
locally Lipschitzian) w.r.t. the set Mθ.
We prove central limit theorems for ΘK(T) in both SLθ and SLLθ cases, and give some general examples of SLθ
and SLLθ functions.
59

COMPOSITION OF POISSON VARIABLES WITH
DISTRIBUTIONS
ISHIKAWA, YASUSHI Ehime University, Japan, slishi@math.sci.ehime-u.ac.jp
Composition; Poisson space; asymptotic expansion:
In the Wiener space, Watanabe theory for Sobolev spaces over Wiener space and the theory of stochastic calculus
constructed on these spaces have been aplied to finance by Yoshida [8], [9], especially by using asymptotic expansion.
On the Poisson space one may try to construct similar formulation. However the situation differs greatly due to the
lack of what plays the same role as the Ornstein-Uhlembeck operator on the Wiener space. In this article we try this
subject based on some tools for analysis developed by Picard, Kunita et al.
In expanding a function g
∞� 1
g(x) ∼ g(a) +
j! g(j) (a)(x − a) j ,
j=1
it happens that g (j) (.) is no longer a function for j ≥ j0, but an element of Schwarz distributions. This causes a
difficulity in studying the expansion g ◦F, where F is a random variable. For this reason we first study the composite
T ◦ F, where T is a distributon.
We define the composition by Fourier transform for a tempered distribution T with a “smooth” random variable
F. A sufficient condition so that the composition is weel defined is the non degeneracy condition (ND) of the first
order appearing below, as it often appears in the Malliavin calculus of jump type.
To this end we introduce a suitable function space (Sobolev spaces) over Poisson space, the dual spaces, and
define the composite. In this article we confine ourselves to tempered distribution for T due to the computational
convenience.
Arguments for the asymptotic expansion will be treated in Hayashi [2].
References
[1] Bichteler, K., Gravereaux, J.B., Jacod,J., Malliavin calculus for processes with jumps. Stochastic Monographs,
vol.2, M. Davis, ed., London, Gordon and Breach 1987.
[2] Hayashi, M., Asymptotic expansion for functionals of Poisson random measure, PhD Thesis (Preprint), 2007.
[3] Ishikawa, Y. and Kunuts, H., Malliavin calculus on the Wiener-Poisson space and its application to canonical
SDE with jumps, Stochastic processes and their applications 116 (2006) 1743–1769.
[4] Ishikawa, Y., Composition with distributions, Preprint, 2007.
[5] Kunita, H., Stochastic flows acting on Schwartz distributions, J. Theor. Prabab. 7 (1994), 247–278.
[6] Nualart, D., The Malliavin calculus and related topics, Springer, 1995.
[7] Picard, J., On the existence of smooth densities for jump processes, PTRF 105, 481-511 (1996).
[8] Yoshida, N., Asymptotic expansion for statistics related to small diffusions, J. Japan Statist. Soc. 22 (1992), no.
2, 139–159.
[9] Yoshida, N., Conditional expansions and their applications. (English summary) Stochastic Process. Appl. 107
(2003), no. 1, 53–81.
60

MULTIVARIATE IBNR CLAIMS RESERVING MODEL 1
IVANOVA, NATALIA Tver State University, Russia, natioanidi@tvcom.ru
Multidimensional Poisson process; collective risk model; IBNR reserves:
We consider the total ultimate multivariate claims random vector process U(t) of the m (commonly dependent)
lines of business as multidimensional compound Poisson [2], where claims arrival process has following special form
of component dependence:
�
�
N(t) = (N1(t), . . ., Nm(t)) = N (i) (t), �
N (i) (t), . . . , �
i∈I1
i∈I2
i∈Im
N (i) �
(t) ,
where i = (i1, . . .,im) is multivariate index, ik = 0, 1, k = 1, . . . , m; I is the set of all possible values of the index i;
Ik = {i ∈ I : ik = 1}; N (i) (t) are independent Poisson processes with parameters λ (i) ≥ 0, λ = �
i∈I λ(i) > 0. Here
each component Nk(t) shows the number of claims of the k-th type, that occur up to the moment t. The coordinates
of the vector N(t) are dependent.
For each index i let {X (i)
j } be a sequence of independent identically distributed m-dimensional vectors. These
variables represent the random claim sizes of the policies, whose payments has structure corresponding to index i.
This means that if the coordinate ik of the index i is equal 0, then the corresponding coordinate of the random vector
X (i)
j is equal 0 almost surely. If ik = 1, then the k-th component of the coordinate X (i,k)
j
i∈I
j=1
in the vector X (i)
j describes
the payment by a contract of k-th type for the policy with claim structure i. Corresponding total ultimate claim
process model has the form
U(t) = (U1(t), . . . , Um(t)) = � N (i) (t) �
I(εj = i)X (i)
j ,
where {εj, j ≥ 1} are i.i.d.r.v. with values from the index set I, whose distribution is given by the law P(εj = i) =
λ (i) /λ.
The total ultimate claims (TUC) incurred in a given period is defined as follows:
TUC = paid claims + outstanding claims reserve + IBNR (Incurred But Not Reported) claims reserve.
In [1] it was offered one dimensional claims reserving model with independent gamma distributed paid claims under
special type “homogeneous allocation principle”. It allocates the coefficient of variation (CoV ) of the TUC with
multiple underwriting periods to the CoV of the TUC of the single one.
We have following quantities for n underwriting periods, which are divided into the parts of each (i-th, i = 1, . . .,n)
underwriting period: V (Vi) — premium volume; U (Ui) — TUC; µ = E(U) (µi = E(Ui)) — mean of the TUC;
k = CoV (U) (ki = CoV (Ui)) — coefficient of variation; Sik, 1 ≤ i, k ≤ n — claims occurred in period i and reported
in period i + k − 1 (paid in period i + k); R = �n �n i=2 k=n−i+2 Sik — total INBR reserve.
In the paper [1] under the assumptions:
– V = (1 + θ)µ (Vi = (1 + θ)µi), where θ is the loading factor;
– Sik, 1 ≤ i, k ≤ n are independent, gamma distributed;
– E(Sik) = Vi/V xiyk (for some parameters xi, yk); CoV (Sik) = � �
V/Vi 1/α (for some parameters α > 0)
the (one-dimensional) estimation procedure of parameters (xi, yk, α) was offered; the IBNR claims reserves
parameters and distribution was restored. We consider multivariate generalization of offered method of IBNR claims
reserves modelling, based on the proposed multidimensional collective risk model.
References
[1] Hürlimann, W. (2004) A Gamma IBNR claims reserving model with dependent development period. Prepared
for the Actuarial Studies in Non-Life Insurance Colloquium, June 19-22 2007, Orlando.
[2] Ivanova, N.L., Khokhlov, Yu.S. (2005) On multivariate collective risk models, Mosc. Univ. Comput. Math.
Cybern., No. 3, 22–30.
1 This work was supported by the Russian Foundation for Basic Research, grant 05-01-00583, 06-01-626.
61

ROBUST WAVELET-DOMAIN ESTIMATION OF THE
FRACTIONAL DIFFERENCE PARAMETER IN HEAVY-TAILED
TIME SERIES
JACH, AGNIESZKA Universidad Carlos III de Madrid, Spain, ajach@est-econ.uc3m.es
Kokoszka, Piotr Utah State University, USA
Heavy tails; trend; wavelets; pseudo-likelihood:
We investigate the performance of several wavelet-based estimators of the fractional difference parameter. We consider
situations where, in addition to long-range dependence, the time series exhibit heavy tails and are perturbed by
polynomial and change-point trends. We study in greater detail a Wavelet-domain pseudo Maximum Likelihood
Estimator (WMLE), for which we provide an asymptotic and finite sample justification. Using numerical experiments,
we show that unlike the traditional time-domain estimators, the estimators based on the wavelet transform are robust
to additive trends and change points in mean, and produce accurate estimates even under significant departures from
normality. The WMLE appears to dominate a regression-based wavelet estimator in terms of smaller RMSE. The
techniques under consideration are applied to the Ethernet traffic traces.
References
[1] Craigmile, P., Guttorp, P., Percival, D. (2005) Wavelet-Based Parameter Estimation for Polynomial Contaminated
Fractionally Differenced Processes, IEEE Transactions on Signal Processing 53, 3151–61.
[2] Hsu, N-J. (2006) Long-memory wavelet models, Statistica Sinica 16, 1255–1272.
[3] Moulines, E., Roueff, F., Taqqu, M. (2006) A Wavelet Whittle estimator of the memory parameter of a nonstationary
Gaussian time series, Arxiv preprint.
[4] Park, C, Hernández-Campos, F., Marron J., Smith, F. (2005) Long-Range Dependence in a Changing Traffic Mix,
Computer Networks 48, 4010-422.
[5] Percival, D., Walden, A. (2000) Wavelet Methods for Time Series Analysis, Cambridge University Press.
[6] Stoev, S., Michailidis, G., Taqqu, M. (2006) Estimating heavy-tail exponents through max self-similarity, Tech.
Report 445 University of Michigan.
62

PERTURBATIONS OF FRACTIONAL LAPLACIAN BY
GRADIENT OPERATORS
JAKUBOWSKI, TOMASZ Wroc̷law University of Technology, Poland, Tomasz.Jakubowski@pwr.wroc.pl
Symmetric α-stable process; gradient perturbations, Green function; Harnack Principles:
We construct a continuous transition density of the semigroup generated by ∆α/2 + b(x) · ∇ for 1 < α < 2, d ≥ 1
and b in the Kato class K α−1
d on Rd . For small time the transition density is comparable with that of the symmetric
α-stable process. Then we study the potential theory of the perturbed process Yt. Green function of the process Yt
is comparable with the one of symmetric α-stable process. Harnack Inequality and Boundary Harnack Principle hold
for Yt.
References
[1] Bogdan, K., Jakubowski, T. (2007) Estimates of heat kernel of fractional Laplacian perturbed by gradient operators,
Commun. Math. Phys. 271 (1), 179–198.
[2] Jakubowski, T., (2007) Estimates of Green function for fractional Laplacian perturbed by gradient, Preprint.
63

EXOTIC OPTION PRICING ON SINGLE NAME CDS UNDER
JUMP MODELS
JÖNSSON, HENRIK EURANDOM, The Netherlands, jonsson@eurandom.tue.nl
Garcia, João Dexia Group, Belgium
Goossens, Serge Dexia Bank, Belgium
Schoutens, Wim K.U. Leuven, Belgium
Lévy processes; firm value model; Credit Default Swaps; swaptions:
Credit Default Swaps (CDSs) have become in the last decennium very important instruments to deal with credit
risk. These financial contracts are now available in quite liquid form on thousands of underlyers and are traded daily
in huge volume. A market has been formed dealing with options or derivatives on these CDSs. The market is for
the moment quite illiquid, but is expected to gain volume over the next years.
Credit risk modeling is about modeling losses. These losses are typically coming unexpectedly and triggered by
shocks. So any process modeling the stochastic nature of losses should reasonable include jumps. The presents of
jumps is even of greater importance if one deals with derivatives on CDSs because of the leveraging effects. Jump
processes have proven already their modeling abilities in other settings like equity and fixed income (see for example
[2]) and have recently found their way into credit risk modeling. In this paper we review a few jump driven models for
the valuation of CDSs and show how under these dynamic models also pricing of (exotic) derivatives on single name
CDSs is possible. More precisely, we set up fundamental firm value models that allow for fast pricing of the ’vanillas’
of the CDS derivative markets: payer and receiver swaptions. Moreover, we detail how a CDS spread simulator can
be set up under this framework and illustrate its use for the pricing of exotic derivatives on single name CDSs as
underlyers.
The starting point of the model is the approach originally presented by Black and Cox [1]. According to this
approach an event of default occurs when the asset value of the firm crosses a deterministic barrier. This barrier
corresponds to the recovery value of the firm’s debt. Black and Cox assumed a geometrical Brownian motion for the
firm’s value process.
We use the same methodology as Black and Cox but work under exponential Lévy models with positive drift and
allowing only for negative jumps. The pricing of a CDS depends fully on the default probability of the firm, that is,
the probability of jumping below the lower barrier. In our setting, only allowing for negative jumps, we can calculate
these probabilities using the double Laplace inversion approach based on the Wiener-Hopf factorization.
References
[1] Black, F., Cox, J. (1976) Valuing corporate securities: some effects on bond indenture provisions, J. Finance 31,
351–367.
[2] Schoutens, W. (2003) Lévy Processes in Finance: Pricing Financial Derivatives, Wiley.
64

TWO-SIDED EXIT PROBLEMS FOR A COMPOUND POISSON
PROCESS WITH EXPONENTIAL NEGATIVE JUMPS AND
ARBITRARY POSITIVE JUMPS
KADANKOVA, TETYANA Hasselt University, Belgium, tetyana.kadankova@uhasselt.be
Noël Veraverbeke Hasselt University, Belgium
Two-sided exit; process with reflecting barriers; asymptotic distribution; scale function;
Lévy processes reflected from the barriers are an object of many studies. Spectrally one-sided reflected Lévy processes,
for instance, were studied in [3], [9], [10] and [8]. Doney and Maller [5] considered random walks with curved barriers,
while reflected random walks with general barrier in context of molecular biology were studied in [6]. Asmussen and
Pihlsg˚ard studied loss rates of general Lévy processes with two reflecting barriers.
In this work we consider a special class of Lévy processes reflected at one (two) barriers. We will determine
several boundary characteristics for a compound Poisson process with arbitrary positive jumps and with negative
exponential jumps reflected at the barriers. Our motivation to study such problems stems from the fact that reflected
Lévy processes are widely applied as mathematical models in queueing theory [2] and financial mathematics [1], [4].
Some of the present results obtained are given in terms of the scale function [3] of the process. The limit
distribution of the first passage times and the first exit time is found. The joint distribution of the supremum,
infimum and the value of the process is also determined. The transient and asymptotic distribution of the process
reflected at two barriers is obtained. These results can be generalized to some other Lévy processes.
References
[1] Avram, F., Kyprianou, A.E. and Pistorius, M.R. (2004) Exit problems for spectrally negative Lévy processes and
applications to (Canadized) Russian options. Ann. Appl. Prob., 14, 215-235.
[2] Asmussen, S. (2003) Applied Probability and Queues, 2nd ed. Springer, New York.
[3] Bertoin, J. (1996). Lévy processes. Cambridge University Press.
[4] Cont, R., Tankov, P. (2004) Financial modelling with jump processes. Chapman & Hall, Boca, Raton.
[5] Doney, R.A., Maller, R.A. (2000) Random walks crossing curved boundaries: functional limit theorems, stability
and asymptotic distributions for exit times and positions. Adv. Appl. Prob. 32, 1117-1149.
[6] Hansen, N.R. (2006) The maximum of a random walk reflected at a general barrier. Ann. Appl. Prob. 16(1),
15-29.
[7] Kou, S.G., Wang H. (2003) First passage times of a jump diffusion process. Adv. Appl. Prob, 35, 504-531.
[8] Lambert, A. (2000) Completely asymmetric Lévy processes confined in a finite interval. Ann. Inst. H. Poincare
Prob. Stat. 36, 251–274.
[9] Pistorius, M.R. (2004) On exit and ergodicity of the completely asymmetric Lévy process reflected at its infimum.
J. Theor. Prob., 17(1), 183-220.
[10] Pistorius, M.R. (2003) On doubly reflected completely asymmetric Lévy processes. Stoch. Proc. and their
Appl., 107, 131-143.
[11] Kadankov, V. F., Kadankova, T. (2005) On the distribution of the first exit time from an interval and the value
of overshoot through the boundaries for processes with independent increments and random walks. Ukr. Math. J.
10(57), 1359–1384.
12] Kadankova T., Veraverbeke N. On several two-boundary problems for a particular class of Lévy processes. J.
Theor. Prob.(to appear).
65

APPROXIMATION OF SYMMETRIC JUMP PROCESSES
KASSMANN, MORITZ University of Bonn, Germany, kassmann@iam.uni-bonn.de
Husseini, Ryad University of Bonn, Germany
Jump processes; Markov chains; central-limit :
We study the problem of how to approximate symmetric jump processes with state dependent jump intensities
which may be quite irregular, e.g. only measurable. A similar result has been established for diffusions with generators
in non-divergence form by Stroock/Zheng in 1997 and has been extended by Bass/Kumagai in 2006.
More precisely, let Yn be for each n a continuous-time Markov chain whose state space is the grid n−1Zd . Specifying
for each chain a starting distribution gives rise to a sequence Qn of probability measures on the space of càdlàg paths
in Rd . We give criteria under which this sequence is tight and converges to a Hunt process associated to the regular
Dirichlet form
E(f, f) = 1
2
�
R d ×R d
� f(x) − f(y) � k(x, y)dxdy .
on L 2 (R d ) where k: R d × R d → R + is measurable, symmetric, bounded from above by c1|x − y| −d−α and satisfies
an additional lower bound. Our assumptions cover the case k(x, y) ≥ c2|x − y| −d−α . Here, we provide an explicit
construction of approximating Markov chains. We are also able to considerably weaken this lower bound, allowing,
for example, the jump kernel to vanish on certain open subsets touching the diagonal in R d × R d .
This joint project of R. Husseini and M. Kassmann is financially supported by the German Science Foundation
(DFG) via the collaborative research center SFB 611.
References
[1] Bass, R. F., Kumagai, T. (2006) Symmetric Markov chains on Z d with unbounded range Trans. Amer. Math.
Soc., in press.
[2] Husseini, R., Kassmann, M. (2007) Markov chain approximations for symmetric jump processes To appear in
Potential Analysis.
[3] Stroock, D. W., Zheng, W. (1997) Markov chain approximations to symmetric diffusions Ann. Inst. Henri
Poincaré, Probab. Statist. 33, 619–649.
66

YIELD CURVE SHAPES IN AFFINE ONE-FACTOR MODELS
KELLER-RESSEL, MARTIN Vienna University of Technology, Austria, mkeller@fam.tuwien.ac.at
Steiner, Thomas Vienna University of Technology, Austria, thomas@fam.tuwien.ac.at
Affine Process; Term Structure of Interest Rates; Ornstein-Uhlenbeck-Type Process; Yield Curve
We consider a model for interest rates, where the short rate is given by a time-homogenous, one-dimensional affine
process in the sense of Duffie, Filipović and Schachermayer. We show that in such a model yield curves can only be
normal, inverse or humped (i.e. endowed with a single local maximum). Each case can be characterized by simple
conditions on the present short rate rt. We give conditions under which the short rate process will converge to a limit
distribution and describe the limit distribution in terms of its cumulant generating function. We apply our results
to the Vasiček model, the CIR model, a CIR model with added jumps and a model of Ornstein-Uhlenbeck type.
References
[1] Duffie, D., Filipović, D., Schachermayer, W. (2003) Affine processes and applications in finance, The Annals of
Applied Probability, 13(3), 984–1053.
67

ASYMPTOTIC PROPERTIES OF RANDOM SUMS
KHOKHLOV, YURY Tver State University, Russia, yskhokhlov@yandex.ru
D’apice, Ciro Salerno University, Italy
Sidorova, Oksana Tver State University, Russia
Random sums; nonrandom centering; teletraffic modelling :
In many problems of probability theory and mathematical statistics and their applications we need to consider
of centered sums of independent random variables (r.v.) when the number of summunds is itself a random variables.
There are two types of centering of random sums: random centering and nonrandom centering. An important result
for random sums with random centering is transfer theorem of Gnedenko and Fahim ([1]). This result allows us
to reduce the problems for random sums to the analogous problems for nonrandom sums. In practical problems
nonrandom centering is more natural. The most results in theory of random summation have been proved in the
case when the summands are independent identically distributed random variables (i.i.d.r.v.). We consider the case
where the summands of random sums have the distributions from finite set of different distributions.
Let (X (k)
j , j ≥ 1) be a sequence of i.i.d.r.v. whose common distribution function (d.f.) Fk(x) = P(X (k)
j < x)
belongs to a finite set of different d.f. {F1, . . . , Fr}, and mk(n) be the number of summands with distribution function
Fk among the first ones. We assume also that these sequenses are independent for different k = 1, . . .,r, and there
exist finite µk = E(X (k)
j ). Let us assume that there exists a sequence of positive numbers Bn such that Bn → ∞,
n → ∞, and
S ∗ r� mk �
n := B−1 n ((X (k)
j − µk) ⇒ Y , (6)
k=1 j=1
where Y is some r.v. with nondegenerate distribution G. In what follows we consider only the case where limit
distribution G in (1) is a convolution of r stable distributions with different indices αk.
Now suppose {N (1)
n }, . . .,{N (r)
n } are some independent sequences of positive integer-valued r.v. (but may be
dependent on summands!) that are defined on the same probability space as (X (k)
j ). Denote
Zn = B −1
n
⎛
r�
N
⎝
(k)
n�
k=1
Tn = B −1
n
U (k)
n
X
j=1
(k)
j − mk · µk
r�
n�
N (k)
k=1 j=1
(X (k)
j − µk) ,
= B−1
n (N(k)
n − mk) · µk ,
Un = U (1)
n + . . . + U (r)
n .
It is evident that Zn = Tn + Un. The main result of our contribution is the following
Theorem. If µk �= 0 and U (k)
n
⎞
⎠ ,
P
→ U (k) , n → ∞, k = 1, . . .,r, U = U (1) + . . . + U (r) , then Zn ⇒ Z, n → ∞, and
r.v. Z has representation Z d = Y + U, where Y and U are independent.
We consider one application of this result to modelling of telecommunication traffic.
This investigation was supported by Russian Foundation for Basic Research, grants 05-01-00583, 06-01-00626.
References
[1] Gnedenko B.V., Fahim H (1969). On a transfer theorem, Soviet Math. Dokl. 10, 769–772.
68

PRICING EQUITY SWAPS IN AN ECONOMY DRIVEN BY GEOMETRIC
ITÔ-LÉVY PROCESSES
KONLACK, VIRGINIE University of Yaoundé I, Cameroon, ksognia@yahoo.fr
Itô-Lévy processes; martingale; Finance:
We consider the pricing of equity swaps and capped equity swaps. Since models driven by Itô-Lévy processes are more
attractive than classical diffusion models, we suppose that the market is driven by a geometric Itô-Lévy processes. We
use the martingale method and the technique of convexity correction of Hinnerich (2005). We extend the generalized
pricing formula for equity swaps to the case of geometric Itô-Lévy processes. Our results are extension of that of
Hinnerich (2005) where she derived the generalized formula for pricing equity swaps in an economy driven by a
diffusion and market point process. We also use the two-side Laplace transform to derive the generalized pricing
formula of capped equity swaps.
References
[1] Björk, T. (1998) Arbitrage Theory in Continuous Time, Oxford University Press.
[2] Chance, D. M. (July 2003) Equity swaps and equty investing.
[3] Cont, R. and Tankov, P. (2004) Financial Modelling With Jump Processes, Chapman & Hall/CRC Press.
[4] Eberlien, E., Liinev, L. (January 2006) The Lévy swap market model
[5] Hinnerich, M. (2005) Pricing Equity Swaps in and Economy with Jumps.
[6] Jacob, J., Shiryaev (1987) Limit Theorems for Stochastic Processes, Springer-Verlag.
[7] Liao, Wang (2003) Pricing Models of Equity Swaps, The Journal of Futures Markets.23, 8, 751–772.
[8] Musiela, M., Rutkowski, M. (1997) Martingale Methods in Financial Modelling, Springer, New York.
[9] ∅ksendal, B., Sulem, A. (2005) Aplied Stochastic Control of Jump Diffusions, Springer.
[10] Raible, S. (2000), Lévy processes in Finance: Theory, Numerics, and Empirical Facts. Ph.D. thesis, University
of Freiburg.
[11] Sato, K. I. (1999) Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press.
69

UNIFORM BOUNDARY HARNACK INEQUALITY AND MARTIN
REPRESENTATION FOR α-HARMONIC FUNCTIONS
KWA´SNICKI, MATEUSZ Wroc̷law University of Technology, Poland, mateusz.kwasnicki@pwr.wroc.pl
Bogdan, Krzysztof Wroc̷law University of Technology, Poland
Kulczycki, Tadeusz Wroc̷law University of Technology, Poland
Boundary Harnack inequality; Martin representation; Stable process:
Let Xt be the isotropic α-stable process in R d , α ∈ (0, 2), d ≥ 1. Consider an arbitrary open D ⊆ R d and define
the first exit time τD = inf{t ≥ 0 : Xt /∈ D}. A nonnegative function f : R d → [0, ∞) is called α-harmonic, if
f(x) = Exf(XτU) whenever x ∈ U and U is an open, precompact subset of D. If Xt was the Brownian motion (i.e.
the isotropic 2-stable process), we would only need to consider f defined on D, and 2-harmonicity would reduce to
classical harmonicity. In our case Xt have discontinuous paths and XτU ∈ D c with positive probability, so we need
f to be defined on whole R d . In fact we may extend the definition of α-harmonicity, allowing f to be a measure µ
on Dc , which will be referred to as the outer charge.
Let GD(x, y) denote the Green function of Xt on D, i.e. � GD(x, y)f(y)dy = Ex
point x0 ∈ D and define the Martin Kernel
whenever the limit exists. Furthermore we let
�
PD(x, z) =
� τD
0 f(Xt)dt. We fix a reference
GD(x, y)
MD(x, z) = lim , x ∈ D, z ∈ ∂D (7)
D∋y→z GD(x0, y)
D
GD(x, y)ν(z − y)dy , x ∈ D, z ∈ D c
be the Poisson kernel of Xt; here ν(z −y) = cd,α|y −z| −d−α is density of the Lévy measure of Xt. These are standard
definitions, see e.g. [2].
To simplify notation, in what follows we assume that D is bounded. The results can be easily extended to the
case of unbounded D be means of Kelvin transform.
Theorem. Let D ⊆ R d be open and bounded. The limit (7) exists for all z ∈ ∂D. Define the set of accessible points
∂MD = {z ∈ ∂D : PD(x, z) = ∞} ; (9)
this definition does not depend on x ∈ D. Then MD(x, z) is α-harmonic in x ∈ D with zero outer charge if and only
if z ∈ ∂MD. If z ∈ ∂D \ ∂MD, then MD(x, z) = PD(x, z)/PD(x0, z).
Theorem. (Boundary Harnack Inequality) Let D ⊆ B(0, 1) be open, x1, x2 ∈ B(0, 1
2 ) ∩ D, z1, z2 ∈ B(0, 1) c . Then:
PD(x1, z1)PD(x2, z2) ≤ cd,αPD(x1, z2)PD(x2, z1). (10)
Theorem. If f ≥ 0 is α-harmonic in bounded D ⊆ Rd with outer charge µ, then for some finite measure σ on ∂MD:
�
�
f(x) = PD(x, z)µ(dz) + MD(x, z)σ(dz) (11)
Above theorems extend earlier results in this area, see e.g. [1], [2], [3], [4], where Lipschitz and κ-fat domains are
considered. In these specific cases ∂MD = ∂D and MD(x, y) is always α-harmonic.
∂MD
References
[1] Bogdan, K., (1997) The boundary Harnack principle for the fractional Laplacian, Studia Math. 123, 43–80.
[2] Bogdan, K., (1999) Representation of α-harmonic functions in Lipschitz domains, Hiroshima Math. J. 29, 227–243.
[3] Chen, Z.-Q., Song, R., (1998) Martin Boundary and Integral Representation for Harmonic Functions of Symmetric
Stable Processes, J. Funct. Anal. 159, 267–294.
[4] Song, R., Wu, J.-M., (1999) Boundary Harnack Principle for Symmetric Stable Processes, J. Funct. Anal. 168,
403–427
70
(8)

ON MINIMAL β-HARMONIC FUNCTIONS OF RANDOM WALKS
LEMPA, JUKKA Turku School of Economics, Turku, jukka.lempa@tse.fi
Random walk; Minimal β-harmonic functions; Drifting Brownian motion:
We first consider the determination of the minimal β-harmonic functions of a general random walk on R. We
present a characterization of the minimal β-harmonic functions for the general random walks. This characterization
is a straightforward generalization of the one by Doob et al. [2] for spatially discrete random walks. Then we utilize
this characterization to determine the minimal β-harmonic functions for the random walk. In particular, we find
that there exist two of such functions and that they are of the same functional form as the ones of drifting Brownian
motion.
We consider also a class of transformations of the general random walk and their minimal β-harmonic functions.
We show that for this class of transformations the minimal β-harmonic functions can be obtained from the minimal
β-harmonic functions of the original random walk via functional transformations. This class of transformed random
walks contains, for example, geometric random walk on R+.
We discuss the potential applicability of the minimal β-harmonic functions in optimal stopping of random walks.
We address some key questions related to this potential connection.
References
[1] Borodin, A. and Salminen, P. (2002) Handbook on Brownian motion - Facts and formulae, Birkhauser
[2] Doob, J. L., Snell J. L. and Williamson R. E. (1960) Application of boundary theory to sums of random variables,
Contributions to probability and statistics,, Stanford University Press, 182 – 197
[3] Dynkin, E. B., (1969) Boundary theory of Markov processes (the discrete case), Russian Math. Surveys, 24:2,
1–42
[4] Peskir, G. and Shiryaev, A. (2006) Optimal stopping and free-boundary problems, Birkhauser
[5] Revuz, D., (1984) Markov Chains, North-Holland
71

MINIMAL Q-ENTROPY MARTINGALE MEASURES FOR
EXPONENTIAL LÉVY PROCESSES
LIEBMANN, THOMAS Ulm University, Germany, thomas.liebmann@uni-ulm.de
Kassberger, Stefan Ulm University, Germany
Exponential Lévy process; minimal generalized entropy; q-optimal equivalent martingale maesure:
Financial markets modeled by exponential Lévy processes usually are incomplete and infinitely many equivalent
martingale measures exist. Exceptions are models based on Brownian motion or a Poisson process, see for instance
Selivanov (2005). Choosing the martingale measure minimizing relative entropy with respect to the real-world
probability measure is considered e.g. in Fujiwara and Miyahara (2003). Generalizations of relative entropy are given
by monotonic functions of the q-th moment of the Radon-Nikod´ym derivative; the martingale measures resulting
from a minimization of these q-entropies are called q-optimal martingale measures. For q less than 0 or greater than
1, these measures are investigated by Jeanblanc et al. (2007). A related approach is minimization of the q-Hellinger
process, considered in a semimartingale setting by Choulli et al. (2007). If the support of the Lévy measure is too
large, however, relative q-entropy does not necessarily attain a minimum within the set of equivalent martingale
measures. Bender and Niethammer (2007) therefore consider minimization in the domain of signed measures in
order to show convergence of q-optimal measures as q decreases to 1 in a portfolio optimization context. Admitting
arbitrage, however, signed measures are not suitable for pricing.
Here, we consider a multidimensional exponential Lévy process setting. The structure of martingale measures
minimizing q-entropy is derived for every q in terms of the Radon-Nikod´ym derivative and of the characteristic triplet
of the Lévy process. Restricting the jump size of the Radon-Nikod´ym derivative, equivalent probability measures can
be obtained in cases where minimization of relative q-entropy does not assure equivalence. We derive the measure
transformation minimizing q-entropy subject to these restrictions and show that, except for the bounds introduced,
its structure is analogous to the unrestricted case. If they exist, all of these equivalent martingale measures preserve
the Lévy property. Further, their relation to the popular Esscher transform and an extension to time-changed Lévy
processes is investigated.
References
[1] Bender, C., Niethammer, C. (2007) On q-optimal signed martingale measures in exponential Lévy models, preprint.
[2] Choulli, T., Stricker, C., Li, J. (2007) Minimal Hellinger martingale measures of order q, Finance and Stochastics
11, 399–427.
[3] Fujiwara, T., Miyahara, Y. (2003) The minimal entropy martingale measure for geometric Lévy processes, Finance
and Stochastics 7, 509–531.
[4] Jeanblanc, M., Klöppel, S., Miyahara, Y. (2007) Minimal f q martingale measures for exponential Lévy processes,
Annals of Applied Probability, forthcoming.
[5] Selivanov, A. V. (2005) On the martingale measures in exponential Lévy models, Theory Probab. Appl. 49,
261–274.
72

HAUSDORFF-BESICOVITCH DIMENSION OF GRAPHS AND
P-VARIATION OF SOME LÉVY PROCESSES
MANSTAVIČIUS, MARTYNAS Vilnius University, Lithuania, martynas.manstavicius@mif.vu.lt
Hausdorff-Besicovitch dimension; Lévy process; Blumenthal-Getoor index; p-variation:
A connection between Hausdorff-Besicovitch dimension of graphs of trajectories and various indices of Blumenthal
and Getoor is well known for α-stable Lévy processes as well as for some stationary Gaussian processes possessing
Orey index. Here we show that the same relationship holds for several classes of Lévy processes that are popular in
financial mathematics models, in particular, the CGMY, the NIG, the GH, the GZ and the Meixner processes.
References
[1] Manstavičius, M., (2007) Hausdorff-Besicovitch dimension of graphs and p-variation of some Lévy processes,
Bernoulli 13(1), 40–53.
73

LÉVY BASE CORRELATION
MASOL, VIKTORIYA K.U.Leuven & EURANDOM, Belgium & The Netherlands,
masol@eurandom.tue.nl
Garcia, João Dexia Group, Belgium
Goossens, Serge Dexia Bank, Belgium
Schoutens, Wim K.U.Leuven, Belgium
Lévy CDO models; Base correlation; Pricing bespoke tranches:
Recently, a set of one-factor models that extend the classical Gaussian copula model for pricing synthetic CDOs
have been proposed in the literature. Albrecher, Ladoucette, and Schoutens (2007) unified these approaches and
proposed a one-factor Lévy model. In the talk, we introduce, investigate, and compare some of these Lévy models.
The proposed models are very tractable and perform significantly better than the classical Gaussian copula model.
Furthermore, we introduce the concept of Lévy base correlation. As shown in Garcia, Goossens, Masol, Schoutens
(2007), the obtained Lévy base correlation curve is much flatter than the corresponding Gaussian one. This indicates
that the Lévy models indeed do much better from a fitting point of view. Additionally, flat base correlation curves
allow to reduce the interpolation error and hence provide much more stable pricing of bespoke tranches.
In conclusion, we illustrate the application of Lévy base correlation to price non-standardized tranches of a
synthetic CDO, and compare the prices obtained under Gaussian and Lévy models.
References
[1] Albrecher, H., Ladoucette, S. and, Schoutens, W. (2007) A generic one-factor Lévy model for pricing synthetic
CDOs., Advances in Mathematical Finance, R.J. Elliott et al. (eds.), Birkhaeuser, to appear.
[2] Garcia J., Goossens, S., Masol, V., Schoutens, W. (2007) Lévy Base Correlation, EURANDOM Report 2007-038,
Technical University of Eindhoven, The Netherlands.
74

GENERALIZED FRACTIONAL ORNSTEIN-UHLENBECK
PROCESSES
MATSUI, MUNEYA Keio University, Japan, mmuneya@math.keio.ac.jp
Kotaro, Endo Keio University, Japan
Fractional Ornstein-Uhlenbeck processes; Generalized fractional Ornstein-Uhlenbeck processes; Stationarity; Long
memory:
An extended version of the fractional Ornstein-Uhlenbeck process of which integrand is replaced by the exponential
function of an independent Lévy process is considered. We call the process the generalized fractional
Ornstein-Uhlenbeck process. The process is also constructed by replacing the variable of integration of the generalized
Ornstein-Uhlenbeck process with an independent fractional Brownian motion (FBM). The stationary property
and the auto-covariance function of the process are studied. Consequently, some conditions of stationarity and the
long memory property of the process are obtained. Our underlying intention is to introduce the long memory property
into the generalized Ornstein-Uhlenbeck process which has the short memory property.
Definition
Let {ξt, t ∈ R} be a two sided Lévy process and a FBM {B H t } with index H ∈ (0, 1] which is independent of {ξt}.
Then, for λ > 0, σ > 0 and t ≥ 0 a generalized fractional Ornstein-Uhlenbeck process with initial value x ∈ R is
defined as
Y H,x
t
If the initial variable satisfies (if definable)
we can write Y H,x
t
as
It is shown that Y H
t is stationary.
The auto-covariance function
:= e −λξt
�
x + σ
� t
0
e λξs− dB H s
� 0
σ e
−∞
λξs− dB H s ,
Y H
� t
t := σ e
−∞
−λ(ξt−ξs−) dB H s .
Let H ∈ (0, 1
H
2 ) ∪ (1
2 , 1] and N = 0, 1, 2, . . .. Then, the generalized fractional Ornstein-Uhlenbeck process {Yt }
based on a two-sided Lévy process {ξt, t ∈ R} and an independent FBM {BH t , t ∈ R} has the following asymptotic
dependent structure under some assumptions (change of expectations, etc.). For fixed t ∈ R as s → ∞,
Cov � Y H
t , Y H � 1
t+s =
2 σ2
N�
θ −2n
� �
2n−1 �
1 (2H − k) s 2H−2n + O(s 2H−2N−2 ),
where θ1 := − logE[e −λξ1 ] > 0.
Thus, the process Y H
t
with H ∈ (1
2
n=1
k=0
�
.
, 1] can exhibit long-range dependence.
References
[1] Cheridito, P. and Kawaguchi, H. and Maejima, M. (2003) Fractional Ornstein-Uhlenbeck processes, Electron. J.
Probab. 8, 1–14.
[2] Endo, K. and Matsui, M. (2007) Generalized fractional Ornstein-Uhlenbeck Processes , Preprint.
[3] Lindner, A. and Maller, R.A. (2005) Lévy integrals and the stationarity of generalized Ornstein-Uhlenbeck
processes, Stochastic. Processes. Appl. 115, 1701–1722.
75

OCCUPATION TIME FLUCTUATIONS OF BRANCHING
PROCESSES
MI̷LO´S, PIOTR Institute of Mathematics, Polish Accademy of Sciences, Poland, pmilos@mimuw.edu.pl
Functional central limit theorem; Occupation time fluctuation; Branching particles system; generalised Wiener
process:
Branching particles systems form an area which receives a lot of research attention. I focus on a system that consists
of particles in R d evolving independently according to a symmetric α-stable Lévy motion and undergoing critical
finite variance branching after exponential time. The system starts off either from Poisson homogenous measure or
equilibrium measure.
The object of my interest is the occupation time fluctuation process
Xt =
� t
0
Ns − ENsds,
where Ns(A) denotes empirical process of the above system (i.e. for set A, Nt(A) is the number of particles of that
system in set A at time t). Functional central limit theorems can be obtained under rescaling of time and space of
Xt. Apart from strong (functional) type of convergence the results are interesting because the limit processes may
exhibit a long range dependence (depending on initial distribution and dimension d of the space). The theorems are
“classical” in a sense that the limit processes are Gaussian.
Another remarkable feature is the proof technique. The space-time method and Mitoma’s theorem are powerful tools
for proving the functional convergence in space of tempered distributions S ′ (R d ) (process Xt is measure-valued and
can be considered as S ′ (R d -valued).
My work is a part of a bigger program initiated by Bojdecki et al. Papers [4], [5] extend their previous results. Improving
results obtained for infinite variance branching law (where limits are stable processes) and adding immigration
to the system needs further investigation.
References
[1] Bojdecki T., Gorostiza L.G., Ramaswamy S, (1986), Convergence of S ′ -valued processes and space time random
fields, J. Funct. Anal. 66, pp. 21–41.
[2] Bojdecki T., Gorostiza L.G. , Talarczyk A, (2006) Limit theorems for occupation time fluctuations of branching
systems II: Critical and large dimensions Functional, Stoch. Proc. Appl. 116, 19–35
[3] Bojdecki T., Gorostiza L.G. , Talarczyk A, (2005) Occupation time fluctuations of an infinite variance branching
systems in large dimensions, arxiv:math.PR/0511745.
[4] Mi̷lo´s , P. (2006) Occupation time fluctuations of Poisson and equilibrium finite variance branching systems, to
appear in Probab. and Math. Stat.
[5] Mi̷lo´s , P. (2007) Occupation time fluctuations of Poisson and equilibrium branching systems in critical and large
dimensions, arXiv:0707.0316v1
76

GARCH MODELLING IN CONTINUOUS TIME FOR
IRREGULARLY SPACED TIME SERIES DATA
MÜLLER, GERNOT Munich University of Technology, Germany, cklu@ma.tum.de
Maller, Ross Australian National University, Canberra, Australia
Klüppelberg, Claudia Munich University of Technology, Germany
Szimayer, Alexander Fraunhofer Institut (ITWM) Kaiserslautern, Germany
COGARCH Process; Continuous Time GARCH Process; GARCH Process; Quasi-Maximum Likelihood Estimation;
Skorokhod Distance; Stochastic Volatility
Discrete time GARCH methodology has had a profound influence on the modelling of stochastic volatility in time
series. The GARCH model is intuitively well motivated in capturing many of the “stylised facts” concerning financial
series, in particular; and it is almost routine now to fit it in a wide range of situations.
Continuous time models are useful, especially, for the modelling of irregularly spaced data, and it is natural to
attempt to extend the successful GARCH paradigm to this arena. Probably the best known of these extensions is
the diffusion limit of Nelson (1990). Problems have arisen with the application of his result, however, among them
being that GARCH models and continuous time diffusion processes are not statistically equivalent (Wang 2002).
As an alternative, Klüppelberg, Lindner and Maller (2004) recently introduced a continuous time version of the
GARCH (the “COGARCH” process) which is constructed directly from a background driving Lévy process. One
of our aims is to show how to fit this model to irregularly spaced time series data using discrete time GARCH
methodology. To do this, the COGARCH is first approximated with an embedded sequence of discrete time GARCH
series which converges to the continuous time model in a strong sense (in probability, in the Skorohod metric), rather
than just in distribution. The way is then open to using, for the COGARCH, similar statistical techniques to those
already worked out for GARCH models, and we show how to implement a pseudo-maximum likelihood procedure to
estimate parameters from a given data set.
For illustration, an empirical investigation using ASX stock index data is given, and the quality of estimation is
investigated using simulations.
References
[1] Klüppelberg, C., A. Lindner, and R.A. Maller (2004) A Continuous Time GARCH Process Driven by a Lévy
Process: Stationarity and Second Order Behaviour, Journal of Applied Probability 41, 601–622.
[2] Nelson, D.B. (1990) ARCH Models as Diffusion Approximations, Journal of Econometrics 45, 7–38.
[3] Wang, Y. (2004) Asymptotic Nonequivalence of GARCH Models and Diffusions, Annals of Statistics 30, 754–783.
77

GENERALIZED HYPERBOLIC MODEL: EUROPEAN OPTION
PRICING IN DEVELOPED AND EMERGING MARKETS
MWANIKI, IVIVI JOSEPH University of Nairobi, Kenya,jimwaniki@uonbi.ac.ke
Virginie, S. Konlack University of Yaoundé I , Cameroon
Keyword Generalized Hyperbolic subclasses; Esscher Transform; Emerging Markets; Fourier Transform:
Generalized Hyperbolic Distribution and some of it subclasses like normal, hyperbolic and variance gamma distributions
are used to fit daily log returns of eight listed companies in Nairobi Stock Exchange (NSE) and Montréal
Exchange. We use EM-based ML estimation procedure to locate parameters of the model. Densities of Simulated
and Empirical data are used to measure how well model fits the data. We use goodness of fit statistics to compare the
selected distributions. Empirical results indicate that Generalized hyperbolic Distribution is capable of correcting
bias of Black-Scholes and Merton normality assumption both in Developed and Emerging markets. Moreover both
markets do have different stochastic time clock.
References
[1] Barndoff-Nielsen, O.E. (1977).Exponentially decreasing distributions for the logarithm of particle size. Proc. Roy.
Soc. London Ser. A, 353: 401-419.
[2] Black,F.,and M.Scholes,The Pricing of Options and Corporate Liabilities Journal of political economy 81,637-659.
[3] Carr, P., Madan, D. (1999). Option valuation using the fast Fourier transform. Journal of Computational Finance
2: 61-73.
[4] Cont, R., Tankov, P. (2004). Financial Modeling With Jump Processes.Chapman & Hall/CRC Press.
[5] Carr, P and Wu Liuren.(2004):Time -Changed Lévy process and option pricing. Journal of Financial Economics
71:113-141.
[6] Madan, D., Chang, C. & Carr, P. (1998). The Variance Gamma Process and Option Pricing. European Finance
Review. 2,79-105.
[7] Madan, D., Seneta, E. (1989).Chebyschev Polynomial Approximation for the Characteristic Function Estimation:Some
Theoretical Supplements. Journal of the Royal Statistic Society. Series B (Methodological). 51, 2: 281-285.
[8] Eberlein, E., Keller, U. (1995). Hyperbolic Distributions in Finance. Bernoulli 3: 281-299.
[9] Senata, E. (2004).Fitting the Variance-Gamma Model to Financial Data.Journal of Applied Probability. Special
Volume 41A
[10] Fajardo, J., Farias, A. (2003). Generalized Hyperbolic Distributions and Brazilian Data. Financelab working
paper.
[11] Hu, W. (2005). Calibration of Multivariate Generalized Hyperbolic Distributions Using the EM Algorithm, with
Applications in Risk Management, Portfolio Optimization and Portfolio Credit Risk. Ph.D. thesis, The Florida State
University.
[12] McNeil,A.,Frey,R.,Embrechts,P.(2005). Quantitaive risk management: Concepts Techniques and Tools. Priceton
University Press(2005).
[13] ∅ksendal, B. Sulem, A. (2005). Aplied Stochastic Control of Jump Diffusions. Springer.
[14] Prause, K. (1999).The Generalized Hyperbolic Model: Estimation, Financial Derivatives, and Risk Measures.
Ph.D. thesis, University of Freiburg.
[15] Predota, M. (2005).On European and Asian option pricing in the generalized hyperbolic model. European Jnl
of Applied Mathematics 11,111-144.
[16] Raible,S. (2000).Lévy processes in Finance: Theory, Numerics, and Empirical Facts. Ph.D. thesis, University of
Freiburg.
[17] Sato, K.I. (1999).Lévy Processes and Infinitely Divisible Distributions.Cambridge University Press.
[18] Shoutens,W.(2003). Lévy Processes in Finance : Pricing Financial derivatives John Wiley&Sons.
78

STOCHASTIC SIMULATION OF CORRELATION EFFECTS IN
CLOUDS OF ULTRA COLD ATOMS EXPOSED TO AN
ELECTROMAGNETIC FIELD
PIIL, RUNE University of Aarhus, Denmark, piil@phys.au.dk
Mølmer, Klaus University of Aarhus, Denmark
Cold atoms; Bose-Hubbard model; optical lattice:
In physics, the research of cold atoms has drawn a lot of attention during the last decade and recently, in 2001, the
Nobel prize was appointed to three key persons in this field. The physics of cold atoms has brought together atomic
physicists, opticians and solid state physicists; the cold atom clouds can possess the same coherence as laser light and
the clouds can be embedded in artificial crystals, i.e. periodic potentials, created by standing electromagnetic waves
generated by lasers to imitate solid state dynamics and thereby draw advantage of the rich world of possibilities in
periodic potentials.
One example of this overlap of the different fields of physics is the parametric generation and amplification of
ultra cold atom pairs in effectively one-dimensional periodic potentials. It has long been known that a coherent beam
of light shined onto a crystal can result in creation of two new highly correlated beams with different wavelengths.
The same is possible, if a moving cloud (beam) of atoms is exposed to a periodic potential. The potential will
modify the energy-momentum spectrum and create certain sets of degenerate momenta, which will allow for a similar
process as for the coherent light, where the cloud of atoms initially having one specific momentum will come out
as beams with two new momenta. It has been shown that in an effectively one-dimensional system a simple cosine
potential will allow for only one set of momenta in the outcome [1,2]. This process makes it possible to create
two macroscopically populated and correlated clouds of atoms, which might be an invaluable resource to quantum
computing and modern measurement theory.
Our present research is dedicated to give a precise description of this phenomenon, where we make use of the
Bose-Hubbard model and the Gutzwiller approximation, which both stem from solid state physics. When the
above mentioned process was first predicted in 2005 [1], a mean-field calculation was performed, which ignores all
correlations between the atoms. The correlation/entanglement of the outcome has yet to be proven. The Gutzwiller
approximation goes beyond mean-field, but still does not include correlations. The correlations, however, can be
introduced into this model by adding stochastic noise [3]. The stochastic approach has already proved its worth
when searching for the ground state of atomic clouds in periodic potentials, but appears to diverge heavily on short
timescales when doing real time simulations.
In order to describe the aforementioned process we have to use a modified Bose-Hubbard model, which again
needs a modified stochastic approach. This modified approach and hopefully some neat results on the correlations
will be presented.
References
[1] Hilligsøe, K. M. and Mølmer, K. Phase-matched four wave mixing and quantum beam splitting of matter waves
in a periodic potential, Physical Review A 71, (2005) 041602.
[2] Campbell, G. K. and Mun, J. and Boyd, M. and Streed, E. W. and Ketterle, W. and Pritchard, D. E. Parametric
Amplification of Scattered Atom Pairs, Phys. Rev. Lett. 96, (2006) 020406.
[3] Carusotto, I. and Castin, Y. An exact reformulation of the Bose–Hubbard model in terms of a stochastic Gutzwiller
ansatz, New Journal of Physics 5, (2003) 91.
79

HEAVY TRAFFIC SCALINGS AND LIMIT MODELS IN A
WIRELESS SYSTEM WITH LONG RANGE DEPENDENCE AND
HEAVY TAILS
PIPIRAS, VLADAS University of North Carolina, USA, pipiras@email.unc.edu
Buche, Robert North Carolina State University, USA
Ghosh, Arka University of Iowa, USA
High-speed wireless networks carrying applications with high capacity requirements (such as multimedia) are
becoming a reality where the transmitted data exhibit long range dependence and heavy-tailed properties. We
obtain heavy traffic limit models incorporating these properties, extending from previous results limited to short
range dependence and light-tailed cases. An infinite source Poisson arrival process is used and fundamental inequality
between the exponent in the power tail distribution of the data from source and the rate of channel variations in
the departure process is obtain. This inequality is important for determining the heavy traffic scaling in both the
“fast growth” and “slow growth” regimes for the arrival process, and along with the source rate, define the possible
queueing limit models—across the cases, they are reflected stochastic differential equations driven by Brownian
motion, fractional Brownian motion, or stable Lévy motion.
80

OPTIMAL DIVIDENDS IN PRESENCE OF DOWNSIDE RISK
RAKKOLAINEN, TEPPO Turku School of Economics, Finland, teppo.rakkolainen@tse.fi
Luis H. R. Alvarez E. Turku School of Economics, Finland
Dividend optimization; Downside risk; Impulse control; Jump diffusion; Optimal stopping; Singular stochastic
control:
We analyze the determination of a value maximizing dividend policy for a broad class of cash flow processes modeled as
spectrally negative jump diffusions. We extend previous results based on continuous diffusion models and characterize
the value of the optimal dividend policy explicitly. Utilizing this result, we also characterize explicitly the values
as well as the optimal dividend thresholds for a class of associated optimal stopping and sequential impulse control
problems. Our results indicate that both the value as well as the marginal value of the optimal policy are increasing
functions of policy flexibility in the discontinuous setting as well.
References
[1] Alvarez, L.H.R. (1996) Demand uncertainty and the value of supply opportunities, J. Econ. 64, 163–175.
[2] Alvarez, L.H.R. (2001) Reward functions, salvage values and optimal stopping, Math. Oper. Res. 54, 315–337.
[3] Alvarez, L.H.R. (2003) On the properties of r-excessive mappings for a class of diffusions, Ann. Appl. Probab.
13, 1517–1533.
[4] Alvarez, L.H.R. (2004) A class of solvable impulse control problems, Appl. Math. Opt. 49, 265–295.
[5] Alvarez, L.H.R., Rakkolainen, T. (2006) A class of solvable optimal stopping problems of spectrally negative jump
diffusions, Aboa Centre for Economics, Discussion Paper No. 9.
[6] Alvarez, L.H.R., Virtanen, J. (2006) A class of solvable stochastic dividend optimization problems: on the general
impact of flexibility on valuation, Econ. Theory 28, 373–398.
[7] Avram, F., Palmowski, Z., Pistorius, M. (2007) On the optimal dividend problem for a spectrally negative Lévy
process, Ann. Appl. Probab. 17, 1, 156–180.
[8] Bar-Ilan, A., Perry, D., Stadje W. (2004) A generalized impulse control model of cash management, J. Econ.
Dyn. Control 28, 1013–1033.
[9] Bertoin, J. (1996) Lévy processes, Cambridge University Press.
[10] Borodin, A., Salminen, P. (2002) Handbook on Brownian motion - facts and formulae (2nd edition), Birkhäuser.
[11] Chan, T., Kyprianou, A. (2006) Smoothness of scale functions for spectrally negative Lévy processes, preprint.
[12] Duffie, D., Pan, J., Singleton, K. (2000) Transform analysis and asset pricing for affine jump diffusions, Econometrica
68, 6, 1343–1376.
[13] Gerber, H., Shiu, E. (2004) Optimal dividends analysis with Brownian motion, N. Amer. Actuarial J. 8, 1, 1–20.
[14] Mordecki, E. (2002) Optimal stopping and perpetual options for Lévy processes, Financ. Stoch. 6:4, 473–493.
[15] Mordecki, E., Salminen, P. (2006) Optimal stopping of Hunt and Lévy processes, preprint.
[16] Perry, D., Stadje W. (2000) Risk analysis for a stochastic cash management model with two types of customers,
Ins.: Mathematics Econ. 26, 25–36.
[17] Protter, P. (2004) Stochastic integration and differential equations (2nd edition), Springer.
[18] Taksar, M. (2000) Optimal risk and dividend distribution control models for an insurance company, Math. Oper.
Res. 51, 1-42.
[19] Øksendal, B. (2003) Stochastic differential equations. An introduction with applications (6th edition), Springer.
[20] Øksendal, B., Sulem, A. (2005) Applied stochastic control of jump diffusions, Springer.
81

CHARACTERIZATIONS OF THE CLASS OF FREE SELF
DECOMPOSABLE DISTRIBUTIONS AND ITS SUBCLASSES
SAKUMA, NORIYOSHI Keio University, Japan, noriyosi@math.keio.ac.jp
Free probability; free selfdecomposable distribution; Voiculescu transform; Bercovici-Pata bijection:
In probability theory based on measure theory (classical probability theory), the key concept on relations among
random variables is independence. Free probability is non-commutative probability adding with free independence.
In 1980’s, D. Voiculescu introduced free independence and free convolution, which is distribution of sum of two free
independently random variables. Many free analogue of probabilistic concept based on measure theory was studied.
Especially, Voiculescu transform and free cumulant function, which are free analogue of characteristic function and
cumulant function respectively, open analytic and combinatoric study of free probability.
In Bercovici and Voiculescu [2], free infinitely divisible distributions and free stable distributions, which was
free analogue of (classical) infinitely divisible distributions and stable distributions, were introduced and Voiculescu
transform of free stable distributions were determined.
In Bercovici and Pata [3], Some free analogue of classical limit theorems are proved. They gave the relation
between the classical and free infinitely divisible distributions by the Bercovici-Pata bijection.
In Barndorff-Nielsen and Thorbjørnsen [1], free analogue of selfdecomposable distribution in classical probability
theory was introduced. They proved that their relations with some other subclasses of free infinitely divisible
distributions are the same as in the classical case.
References
[1] Barndorff-Nielsen, O. E. and Thorbjørnsen, S. (2002). Selfdecomposability and Lévy processes in free probability,
Bernoulli. 8, 323–366.
[2] Bercovici, H. and Voiculescu, D. (1993). Free convolution of measures with unbounded support, Indiana. J. math.
42, 733–773.
[3] Bercovici, H. and Pata, V. (1999). Stable laws and domains of attraction in free probability theory, Ann. Math.
149, 1023–1060.
82

STATISTICAL PHYSICS APPROACH TO HIGH-FREQUENCY
FINANCE
SCALAS, ENRICO East-Piedmont University, Italy, enrico.scalas@mfn.unipmn.it
Politi, Mauro Milan University, Italy
High-frequency finance; continuous-time random walks; stochastic integration; option pricing:
Based on the continuous-time random walk (CTRW) model for high-frequency financial data, we present some
preliminary results on the following issues:
• Analysis of the structure of waiting times between consecutive trades fit with Tsallis’ q-exponential as well as
Weibull distributions.
• We define stochastic integrals on CTRWs and we study the non-Markovian case of non-exponentially distributed
waiting times.
References
[1] Scalas, E. (2006) The application of continuous-time random walks in finance and economics, Physica A 362,
225–239.
83

NON-DANGEROUS RISKY INVESTMENTS FOR INSURANCE
COMPANIES
SCHAEL, MANFRED University of Bonn, Germany , Inst.Applied Math. schael@uni-bonn.de
jump processes; ruin probability ; control; financial market:
The control of ruin probabilities by investments in a financial market is studied. The insurance business and the
risk driver of the financial market are described by a joint jump model. An investment plan is non-dangerous if the
ruin probability has exponential decay under the plan, i.e., there exists an adjustment coefficient. It is known that
a plan investing a fixed fraction of capital leads to a polynomial decay and thus is dangerous. An investment plan
is profitable if its adjustment coefficient is larger than the classical Lundberg exponent defined for the uncontrolled
case. It is known that there exist profitable plans investing a fixed amount of capital in the stock independently of
the current level of capital. But they are not admissible when the insurance company is poor. Here we investigate
the existence of non-dangerous and profitable investment plans which are admissible as well.
References
[1] Schael, M. (2005) Control of ruin probabilities by discrete-time investments , Math. Meth. Oper. Res. 62,
141–158.
84

EXTENDING TIME-CHANGED LÉVY ASSET MODELS
THROUGH MULTIVARIATE SUBORDINATORS
SEMERARO, PATRIZIA University of Torino, Italy, semeraro@econ.unito.it
Luciano, Elisa University of Torino, Italy
Lévy processes; multivariate subordinators; multivariate asset modelling; multivariate time changed processes :
The technique of time change is a well established way to introduce Lévy processes at the univariate level: it has
proven to be theoretically helpful for financial applications, thanks to Monroe’s theorem. At the multivariate level,
however, time changing has been studied much less. The traditional multivariate Lévy process constructed by
subordinating a Brownian motion through a univariate subordinator presents a number of drawbacks, including
the lack of independence and a limited range of dependence. In order to face these, we investigate multivariate
subordination, with a common and an idiosyncratic component. Formally the time change can be written as
G(t) = (X1(t) + α1Z(t), X2(t) + α2Z(t), ..., Xn(t) + αnZ(t)) T , (12)
where Z = {Z(t), t ≥ 0}, Xj = {Xj(t), t ≥ 0}, j = 1, ..., n, are independent subordinators.
The multivariate log price process Y = {Y (t), t > 0} is defined by the following time change:
⎛
Y (t) = ⎝ Y1(t)
⎞ ⎛
... ⎠ = ⎝
Yn(t)
µ1G1(t) + σ1B1(G1(t))
....
⎞
⎠, (13)
µnGn(t) + σnBn(Gn(t))
where Bi, i = 1, ..., n are independent Brownian motions and G is a multivariate subordinator defined as above,
independent from the Brownian motions.
Specifing the subordinators, we introduce generalizations of some well known univariate Lévy processes for financial
applications: the multivariate compound Poisson, NIG, Variance Gamma and CGMY. In all these cases the
extension is parsimonious, in that one additional parameter only is needed.
First we characterize the subordinator, then the time changed processes via their Lévy triplet. Finally we study
the subordinator association, as well as the subordinated processes’ linear and non linear dependence. We show that
the processes generated with the proposed time change can include independence and that they span a wide range
of linear dependence. We provide some examples of simulated trajectories, scatter plots and both linear and non
linear dependence measures. The input data for these simulations are calibrated values of major stock indices.
References
[1] Barndorff-Nielsen, O.E., Pedersen, J. Sato, K.I. (2001). Multivariate Subordination, Self-Decomposability and
Stability. Adv. Appl. Prob. 33, 160-187.
[2] Barndorff-Nielsen, O.E.(1995). Normal inverse Gaussian distributions and the modeling of stock returns. Research
report no. 300, Department of Theoretical Statistics, Aarhus University. Adv. Appl. Prob. 33, 160-187.
[3] Carr, P., Geman, H., Madan, D. H. and Yor, M. (2002) The fine structure of asset returns: an empirical
investigation. Journal of Business 75, 305-332.
[4] Cont, R., Tankov, P. (2004) Financial modelling with jump processes. Chapman and hall-CRC financial mathematics
series.
[5] Geman, H., Madan, D.B., Yor, M. (2001) Time changes for Lévy processes. Mathematical Finance 11, 1, 79-96.
[6] Luciano, E., Schoutens, W. (2005). A multivariate Jump-Driven Financial Asset Model. Quantitative Finance ,
6 (5), 385-402.
[7] Samorodnitsky, G., Taqqu, M.S. (1994) Stable non-gaussian random processes. Stochastic Models with infinite
variance. Chapman & hall. New York.
[8] Sato, K.I. (2003) Lévy processes and Infinitely divisible distributions. Cambridge studies in advanced mathematics
Cambridge University Press.
[9] Semeraro, P. (2006) A multivariate time-changed Lévy model for financial application. Accepted for pubblication
Journal of Theoretical and Applied Finance.
85

MULTIFRACTALITY OF PRODUCTS OF GEOMETRIC
ORNSTEIN-UHLENBECK TYPE PROCESSES
SHIEH, NARN-RUEIH National Taiwan University, Taiwan, shiehnr@math.ntu.edu.tw
Anh, Vo V. Queensland University of Technology, Australia
Leonenko, Nikolai N. Cardiff University, UK
Multifractal products; geometric Ornstein-Uhlenbeck processes; Lévy processes.
We investigate the properties of multifractal products of geometric Ornstein-Uhlenbeck processes driven by Lévy
motion. The conditions on the mean, variance and covariance functions of the resulting cumulative processes are
interpreted in terms of the moment generating functions. We consider five cases of infinitely divisible distributions for
the background driving Lévy processes, namely, the gamma and variance gamma distributions, the inverse Gaussian
and normal inverse Gaussian distributions, and the z-distributions. We establish the corresponding scenarios for the
limiting processes, including their Rényi functions and dependence structure.
References
[1] Applebaum, D. Lévy Processes and Stochastic Calculus, Cambridge University Press, Cambridge, 2004.
[2] Anh, V.V. and Leonenko, N.N. Non-Gaussian scenarios for the heat equation with singular initial data, Stochastic
Processes and their Applications 84(1999), 91–114.
[3] Barndorff-Nielsen, O.E. Superpositions of Ornstein-Uhlenbeck type processes, Theory Probab. Appl. 45(2001),
175–194.
[4] Barndorff-Nielsen, O.E. and Leonenko, N.N. Spectral properties of superpositions of Ornstein-Uhlenbeck type
processes, Methodol. Comput. Appl. Probab. 7 (2005), 335–352.
[5] Kahane, J.-P. Sur la chaos multiplicatif , Ann. Sc. Math. Québec 9(1985), 105–150.
[6] Kahane, J.-P. Positive martingale and random measures, Chinese Annals of Mathematics 8B(1987), 1–12.
[7] Mannersalo, P., Norris, I. and Riedi, R. Multifractal products of stochastic processes: construction and some basic
properties , Adv. Appl. Probab. 34(2002), 888–903.
[8] Mörters, P. and Shieh, N.-R. On multifractal spectrum of the branching measure on a Galton-Watson tree , J.
Appl. Probab. 41(2004), 1223-1229.
[9] Shieh, N.-R. and Taylor, S.J. Multifractal spectra of branching measure on a Galton-Watson tree, J. Appl. Probab.
39(2002), 100-111.
86

QUESTIONABLE RESULTS ON CONVOLUTION EQUIVALENT
DISTRIBUTIONS
SHIMURA, TAKAAKI The Institute of Statistical Mathematics
T. Watanabe The University of Aizu
Cline [1] is one of the most well known and often referred paper in the study of exponential tail and convolution
equivalent distributions. However, unfortunately, it contains some mistakes and influences many other papers refer
it. We investigate this confusion. Cline [1] obtains several results from wrong lemmas. We can not regard something
based on uncertain basis as the truth. On the other hand, we can not declare that it is not true because its proof
is not perfect either. Therefore, what we should do is to investigate each doubtful statement minutely and to judge
whether it is true or not. We classify the uncertain statements into three cases including the unidentified. 1. The
statement is wrong. 2. The statement itself is true. 3. We are not sure whether the statement is true or not.
The classification is done mainly for the statements in Cline [1], but other influenced results are also mentioned.
In addition, we aim to put the situation in order by proper statements. This presentation will contribute toward
lessening the confusion.
References
[1] Cline, D.B.H. (1987). Convolutions of distributions with exponential and subexponential tails, J.Austral. Math.Soc.
Ser.A, 43, 347-365.
87

STOCHASTIC STABILIZATION
SIAKALLI, MICHAILINA University of Sheffield, UK, stp05ms@shef.ac.uk
Applebaum, D. University of Sheffield, UK
Stochastic stabilization; almost surely exponential stability; Lévy processes :
Consider a first order non-linear differential equation system dx(t)
dt
= f(x(t)).
In my poster I will present stochastic stabilization of the given non-linear system when is perturbed by
(a) a compensated Poisson noise that will represent the “small jumps” of a Lévy process and
(b) a Poisson random measure which represents the “large jumps” of a Lévy process.
We will show that the compensated Poisson noise and the Poisson noise can have a similar role to the Brownian
motion noise (as in [2]) in stabilizing dynamical systems.
References
[1] Applebaum, D. (2004) Lévy processes and Stochastic Calculus, 1st edition. Cambridge.
[2] Mao, X. (1994) Stochastic Stabilization and destabilization, Systems and Control Letters 23, 279–290 .
[3] Mao, X. (1997) Stochastic Differential Equations and Applications, Horwood.
88

MULTIVARIATE CONTINUOUS TIME LÉVY-DRIVEN GARCH
PROCESSES
STELZER, ROBERT Munich University of Technology, Germany, rstelzer@ma.tum.de
COGARCH; multivariate GARCH; second order moment structure; stationarity; stochastic differential equations;
stochastic volatility:
A multivariate extension of the COGARCH(1,1) process introduced in [2] is presented and shown to be well-defined.
The definition generalizes the idea of [1] for the definition of the univariate COGARCH(p, q) process and is in a
natural way related to multivariate discrete time GARCH processes as well as positive-definite Ornstein-Uhlenbeck
type processes.
Furthermore, we establish important Markovian properties and sufficient conditions for the existence of a stationary
distribution for the volatility process, which lives in the positive semi-definite matrices, by bounding it by a
univariate COGARCH(1,1) process in a special norm. Moreover, criteria ensuring the finiteness of moments of both
the multivariate COGARCH process as well as its volatility process are given. Under certain assumptions on the
moments of the driving Lévy process, explicit expressions for the first and second order moments and (asymptotic)
second order stationarity are obtained.
As a necessary prerequisite we study the existence of solutions and some other properties of stochastic differential
equations being only defined on a subset of R d and satisfying only local Lipschitz conditions.
References
[1] Brockwell, P., Chadraa, E., Lindner,A. (2006) Continuous-time GARCH Processes, Ann. Appl. Probab. 16,
790–826.
[2] Klüppelberg, C., Lindner, A., and Maller, R. (2004) A Continuous-Time GARCH Process Driven by a Lévy
Process: Stationarity and Second-order Behaviour, J. Appl. Probab. 41, 601–622.
89

REGULARITY OF HARMONIC FUNCTIONS FOR ANISOTROPIC
FRACTIONAL LAPLACIAN
SZTONYK, PAWE̷L Wroc̷law University of Technology, Poland and Philipps Universität Marburg,
Germany, sztonyk@pwr.wroc.pl
Potential kernel, Green function, harmonic function, Hölder continuity, stable process:
We prove (see [1,2]) that bounded harmonic functions of anisotropic fractional Laplacians are Hölder continuous
under mild regularity assumptions on the corresponding Lévy measure. Under some stronger assumptions the Green
function, Poisson kernel and the harmonic functions are even differentiable of order up to three.
References
[1] Sztonyk, P., Bogdan, K. (2007) Estimates of potential kernel and Harnack’s inequality for anisotropic fractional
Laplacian , to appeare in Studia Math.
[2] Sztonyk, P., (2007) Regularity of harmonic functions for anisotropic fractional Laplacian , preprint.
90

SERIES APPROXIMATION OF THE DISTRIBUTION OF LÉVY
PROCESS
TIKANMÄKI, HEIKKI Helsinki University of Technology, Finland, heikki.tikanmaki@tkk.fi
Lévy processes; asymptotic expansions; Cramér condition; cumulants; Edgeworth approximation :
The Lévy-Khinchine theorem gives the characteristic function of a Lévy process. In spite of this the distribution
function of a Lévy process is not analytically known, except in few special cases like Brownian motion, Poisson
process etc. For example, the distribution of the compound Poisson process is not known in general though the
popularity of the process as a risk process in insurance applications.
The normal approximation gives good asymptotic results when t → ∞, see for instance [1]. Many authors
have considered asymptotic expansions for the sums of independent random variables, see e.g. [2]. In insurance
mathematics and statistics this is often called the Edgeworth approximation [3,4]. Some approximation is introduced
also for Lévy processes in [5] as an analogue to the i.i.d case.
This work goes beyond this analogue and clarifies the connection between Lévy processes and classical approximation
results of sums of independent random variables.
If a Lévy process has all momens we can also prove with some extra assumptions exact series representation for
its distribution. This work also shows how the approximating functions scale along in time parameter. It is worth
noting that these series representations work fine for all t > 0.
References
[1] Valkeila, E. (1995) On normal approximation of a process with independent increments Russian Math. Surveys
50, 945-961.
[2] Petrov, V.V. (1995) Limit Theorems of Probability Theory, Oxford Science Publications.
[3] Beard, R.E., Pentikäinen, T., Pesonen, E. (1984), Risk Theory The Stochastic Basis of Insurance, Chapman and
Hall.
[4] Kolassa, J.E. (2006), Series Approximation Methods in Statistics, Springer.
[5] Cramér H. (1962) Random Variables and Probability Distributions
91

FEASIBLE INFERENCE FOR REALISED VARIANCE IN THE
PRESENCE OF JUMPS
VERAART, ALMUT University of Oxford, UK, veraart@stats.ox.ac.uk
Bipower variation; feasible inference; realised variance; semimartingale; stochastic volatility:
Inference on the variation of asset prices has been extensively studied in the last decade. Due to the fact that high
frequency asset price data have become widely available, one can now use nonparametric methods which exploit
the specific structure of high frequency data to learn about the price variation over a given period of time. While
logarithmic asset prices have often been modelled by Brownian semimartingales, the focus of research has recently
shifted towards more general models which allow for jumps in the price process. This paper follows this recent stream
of research by assuming that the logarithmic asset price is given by an Itô semimartingale of the form
dXt = btdt + σtdWt + dJt,
which consists of a Brownian semimartingale (btdt+σtdWt) and a jump component (dJt). The jump component can
be a Lévy process or an even more general jump process (where we assume some mild regularity conditions).
This paper is about inference on the quadratic variation process of the price process, which is given by [X]t =
[X] c t + [X] d t , where [X] c t = � t
0 σ2 sds, and [X] d t = �
0≤s≤t (∆Js) 2 denote the continuous and discontinuous (or jump)
part of the quadratic variation, respectively.
While inference on the continuous part of the quadratic variation has been studied in detail in the literature (see
e.g. Barndorff-Nielsen and Shephard (2002)), inference on the entire quadratic variation including the discontinuous
part has not been studied yet. A quantity called realised variance has become the focus of attention in this context.
Let us assume that we observe the price process over a time interval [0, t] at discrete times i∆n for i = 0, . . . , [t/∆n],
where ∆n > 0 and ∆n → 0 as n → ∞. We write ∆n i X = Xi∆n − X (i−1)∆n for the i-th return. The daily realised
variance is then defined as the sum of the squared returns over a day, i.e. � [t/∆n]
i=1 (∆n i X)2 . It can be shown that this
quantity is a consistent estimator for the accumulated daily variance.
In order to make inference we need a limit result for the volatility estimator. For this we use a very important
result by Jacod (2007) who has derived the asymptotic distribution of realised variance in the presence of jumps.
However, this limit result is infeasible since the variance of the limiting process is not observable. In this paper we
propose a new estimator for the asymptotic variance of the realised variance, which is based on a generalised version
of realised variance and locally averaged realised bipower variation. We prove the consistency of this estimator and
derive a feasible limit theorem for the realised variance. Monte Carlo studies show a good finite sample performance
of our proposed estimator. Finally, an empirical analysis of some high frequency equity data reveals the empirical
relevance of our theoretical results.
References
[1] Barndorff-Nielsen, O. E., Shephard, N. (2002) Econometric analysis of realised volatility and its use in estimating
stochastic volatility models, Journal of the Royal Statistical Society B 64, 253–280.
[2] Huang, X., Tauchen, G. (2005) The Relative Contribution of Jumps to Total Price Variance, Journal of Financial
Econometrics 3 (4), 456–499.
[3] Jacod, J. (2007) Asymptotic properties of realized power variations and related functionals of semimartingales,
Stochastic Processes and their Applications, Forthcoming.
[4] Lee, S. S., Mykland, P. A. (2006) Jumps in financial markets: A new nonparametric test and jump dynamics,
technical report 566, Dept of Statistics, The Univ. of Chicago.
92

ESTIMATION OF INTEGRATED VOLATILITY IN THE
PRESENCE OF NOISE AND JUMPS
VETTER, MATHIAS Ruhr-Universität Bochum, Germany, mathias.vetter@rub.de
Podolskij, Mark Ruhr-Universität Bochum, Germany
Bipower Variation; Jumps; High-Frequency Data; Integrated Volatility; Microstructure Noise :
We present a new concept of modulated bipower variation for diffusion models with microstructure noise and jumps.
This method provides simple estimates for such important quantities as integrated volatility or integrated quarticity
in the presence of microstructure noise. Under mild conditions the consistency of modulated bipower variation can
be proven. Under further assumptions we are able to prove stable convergence of our estimates with the optimal rate
n −1
4 .
Moreover, a generalisation of the concept gives estimates for the joint quadratic variation of the underlying process, if
further jumps are present. We are also able to construct estimates robust to jumps, and obtain therefore an estimate
for the jump part of the process. Both consistency and stable convergence of this quantity can be proven, which
enables us to test whether jumps are present or not.
References
[1] Podolskij, M., Vetter, M. (2006) Estimation of Volatility Functionals in the Simultaneous Presence of Microstructure
Noise and Jumps, Preprint.
93

STUDENT RANDOM WALKS AND RELATED PROBLEMS
VIGNAT, CHRISTOPHE Université de Marne la Vallée, France, vignat@univ-mlv.fr
Berg, Christian University of Copenhagen, Denmark, berg@math.ku.dk
Student t- distribution; random walk; asymptotics:
In a recent contribution [1], N. Cufaro-Petroni derived several results about the behaviour of non stable Lévy
processes. More precisely, he considered the random walk
N�
ZN =
where N ∈ N and each step Xi follows a Student t-distribution with f = 2n + 1 degrees of freedom. We recall that
the Student t-density with f = 2ν degrees of freedom - where ν is an abitrary positive number - is
Aν
fν (x) =
(1 + x2 1 ; Aν =
ν+ 2 ) Γ � ν + 1
�
2
Γ � �
1 .
2 Γ (ν)
Although Cufaro-Petroni obtained precise results about ZN in the case of f = 3 degrees of freedom only, he
expressed in his paper the following conjecture:
Conjecture 1: ∀N ∈ N and ∀f = 2n + 1, the distribution of N −1ZN is a convex combination of Student
t-distributions with odd degrees of freedom.
We show here that this conjecture holds true, and extend it to the case of a convex combination of independent
t-distributed variables Xi with different odd degrees of freedom, as a consequence of [2].
The next result by Cufaro-Petroni concerns the distribution of ZN for non-integer values of N: in this case, the
N−fold convolution power of distribution fν is defined, for any real positive N, as the inverse Fourier transform
f ∗N
ν (x) = 1
2π
� +∞
−∞
i=1
Xi
e iux [ϕν (u)] N du
where ϕν (u) is the characteristic function of the Student t-distribution. Cufaro-Petroni shows in [1] that
Theorem 1: for every N > 0, the asymptotic behaviour of the distribution of ZN scales as
f ∗N
3
2
We show that this result can be extended as follows
Theorem 2: : for every N ∈ N and for every ν > 0,
and for every N > 0 and ν = n + 1
2 ,
∼ A3
2 N
x 4 , |x| → +∞
f ∗N
ν ∼ AνN
, |x| → +∞
x2ν+1 f ∗N
n+ 1
2
∼ An+ 1
2 N
, |x| → +∞
x2n+2 Our last result is the following:
Theorem 3: If N /∈ N, the distribution f ∗N
ν can not be expanded as
f ∗N
ν (x) =
+∞�
k=0
βkf 1 k+ (x) .
2
In other words, Conjecture 1 holds for integer sampling times N only.
References
[1] Cufaro Petroni, N., (2007) Mixtures in nonstable Lévy processes, J. Phys. A: Math. Theor. 40, 2227–2250.
[2] Berg C., Vignat C., (2007) Linearization Coefficients of Bessel Polynomials and Properties of Student t-Distributions,
to appear in Constructive Approximation, DOI: 10.1007/s00365-006-0643-6
94

ORNSTEIN-UHLENBECK PROCESSES IN PHYSICS AND
ENGINEERING
WULFSOHN, AUBREY University of Warwick, UK, awu@maths.warwick.ac.uk
This is the first part of work in progress on the use of signal theory methods to simulate fractional noises. These
are usually assumed to be power-law Gaussian processes and lie between white and the so-called pink or hyperbolic
noises. Signal processes can be interpreted as electrical, acoustic, or optical and even hydrodynamical. Since most of
the processes dealt with are non-stationary we take a partially frequentist interpretation of probability and introduce
frequency and power-spectral methods to supplement the axiomatic approach.
We indicate the progress due by N.Wiener on signal theory and we identify the origins of fractional Brownian
motion, relating to J. Liouville, Riemann, K. Weierstrauss, H. Weyl, Kolmogorov and P.Lévy. We deal with both
Riemann-Liouville and Weierstrauss-Weyl fractional Brownian motions. Mathematicians favour Weierstrauss-Weyl.
Engineers usually consider a linear time-invariant situation and use the impulse- response-transfer function approach.
They favour Riemann-Liouville fractional Brownian motion as it has causal impulse response functions. However
this approach is restricted to linear time-invariant processes. For non time-invariant systems one can use Wigner’s
phase-space distribution. We describe Frequency/time phase space methods analogous to those of Wigner’s momentum/position
phase space in quantum theory. These were introduced by D. Gabor and J. Ville for communication
and signal theory. We discuss also the validity of Planck’s constant for the signal theory analogue of the Heisenberg
uncertainty principle.
We clarify the distinction between the Brownian motions of Wiener-Einstein-Smulochowski and of Ornstein-
Uhlenbeck (OU). We see that superpositions of independent OU velocity processes have have rational spectra so
are unsuitable for simulating fBm. We adapt these superpositions, using filters, to approach processes having the
spectrum of the required noise process. We use also an alternative method of approximation by random Fourier
series and in particular randomised Weierstrauss functions.
95

Agnieszka Jach
Universidad Carlos III de Madrid
calle Madrid 126
28903 Getafe-Madrid
Spain
ajach@est-econ.uc3m.es
Alexander Lindner
University of Marburg
Fachbereich Mathematik und Informatik
Hans-Meerwein-Str.
35032 Marburg
Germany
lindner@mathematik.uni-marburg.de
Almut Veraart
University of Oxford, Department of Statistics
St Anne’s College, Woodstock Road
OX2 6HS Oxford
UK
veraart@stats.ox.ac.uk
Andrea Karlova
Market Risk Methodology, Risk Management Dept.,
CSOB, KBC Group
Radlicka 333/150
150 57 Prague 5
Czech Republic
akarlova@csob.cz
Andreas Kyprianou
Dept. Mathematical Sciences, University of Bath
Claverton Down
BA1 2UU Bath
UK
a.kyprianou@bath.ac.uk
Astrid Hilbert
Växjö Universty
Vejdesplats 7
351 95 Växjö
Sweden
Astrid.Hilbert@vxu.se
List of Participants
97
Ahmadreza Azimifard
Technical University Munich
Dietlinden Strasse 16
80802 Munich
Germany
azimifard@yahoo.com
Alexander Schnurr
Philipps-Universitt Marburg
Hans-Meerwein-Strae
35032 Marburg
Germany
schnurr@mathematik.uni-marburg.de
Anders Tolver Jensen
Faculty of Life Sciences, University of Copenhagen
Thorvaldsensvej 40
1871 Frederiksberg C.
Denmark
tolver@life.ku.dk
Andreas Basse
Aarhus University
Institut for Matematiske Fag
Ny Munkegade, bygning 1530
8000 ˚Arhus
Denmark
basse@imf.au.dk
Andrii Ilienko
National Technical University of Ukraine, Kiev, and
Mathematical Institute of the University of Cologne
Marienburger Strasse, 8
50968 Cologne
Germany
a ilienko@ukr.net
Aubrey Wulfsohn
Mathematics Institute, Warwick University
CV4 7AL Coventry
UK
awu@maths.warwick.ac.uk

Bernt Øksendal
Center of Mathematics for Applications (CMA)
University of Oslo
CMA, University of Oslo, Box 1053 Blindern, N-0316
Oslo, Norway
N-0316 Oslo
Norway
oksendal@math.uio.no
Carlo Sgarra
Department of Mathematics-Politecnico di Milano-
Piazza Leonardo Da Vinci, 32
20133 Milano
Italy
carlo.sgarra@polimi.it
Christian Bayer
University of Technology, Vienna
Wiedner hauptstrasse 8 / 105-1
1040 Vienna
Austria
cbayer@fam.tuwien.ac.at
Christophe Vignat
Université de Marne la Vallée
5 Boulevard Descartes
77454 Marne la Valle cedex 2
france
vignat@univ-mlv.fr
Claudia Klüppelberg
Zentrum Mathematik
Technische Universitaet Muenchen
Boltzmannstrasse 3
85747 Garching b. Muenchen
Germany
cklu@ma.tum.de
Desislava Stoilova
Southwest University, Faculty of Economics
2 Krali Marko str.
2700 Blagoevgrad
Bulgaria
dstoilova@abv.bg
Edward Kao
Department of Mathematics, University of Houston
651 Philip G. Hoffman Hall
77204-3008 Houston
USA
edkao@math.uh.edu
98
Björn Böttcher
Uni Marburg
Philipps-Universität Marburg - Fachbereich Mathematik
und Informatik - Hans-Meerwein-Strae
35032 Marburg
Germany
boettcher@mathematik.uni-marburg.de
Cathrine Jessen
University of Copenhagen
Department of Mathematical Sciences
Universitetsparken 5
2100 Copenhagen
Denmark
catj@math.ku.dk
Christine Gruen
Institute for Applied Mathematics
Im Neuenheimer Feld 294
69120 Heidelberg
Germany
christine.gruen@web.de
Cindy Yu
Department of Statistics, Iowa State University
216A Snedecor Hall
IA 50010 Ames
USA
cindyyu@iastate.edu
Davar Khoshnevisan
University of Utah
Department of Mathematics, 155 S 1400 E
84105 Salt Lake City, UT
USA
davar@math.utah.edu
Donatas Surgailis
Institute of mathematics and informatics
Akademijos, 4
LT-08663 Vilnius
Lithuania
sdonatas@ktl.mii.lt
Ehsan Azmoodeh
Jämeräntaival 11 B 50
02150 Espoo
Finland
azmoodeh@cc.hut.fi

Eija Päivinen
University of Jyväskylä
P.O.Box 35
FI-40014 University of Jyväskylä
Finland
ekpaivin@maths.jyu.fi
Ely Merzbach
Bar Ilan University
Dept. of Mathematics, Bar-Ilan University
52900 Ramat Gan
Israel
merzbach@macs.biu.ac.il
Erik Baurdoux
Universiteit Utrecht
Mathematical Institute, Budapestlaan 6
3584 CD Utrecht
The Netherlands
baurdoux@math.uu.nl
Filip Lindskog
KTH, Matematik
10044 Stockholm
Sweden
lindskog@kth.se
Frederic Utzet
Universitat Autonoma de Barcelona
Departament de Matematiques, campus de Bellaterra
08193 Bellaterra (Barcelona)
Spain
utzet@mat.uab.cat
Gennady Samorodnitsky
Cornell University
School of ORIE, 220 Rhodes Hall, Cornell University
14850 Ithaca
USA
gennady@orie.cornell.edu
Gunnar Hellmund
University of Aarhus
Department of Mathematical Sciences
DK-8000 Aarhus C
Denmark
ghellmund@gmail.com
99
El Hadj Aly DIA
9B Bd Jourdan, college Franco-Britannique
75014 Paris
France
diaelhadjaly@yahoo.fr
Enrico Scalas
DISTA - Universita’ del Piemonte Orientale
via Bellini 25 g
15100 Alessandria
Italy
scalas@unipmn.it
Eva Vedel Jensen
Department of Mathematical Sciences
University of Aarhus
Ny Munkegade
DK-8000 Aarhus C
Denmark
eva@imf.au.dk
Francois Roueff
Telecom Paris
46 rue Barrault
75634 Paris Cedex 13
France
roueff@tsi.enst.fr
Friedrich Hubalek
Financial and Actuarial Mathematics
Vienna University of Technology
Wiedner Hauptstrasse 8 / 105-1
A-1040 Vienna
Austria
fhubalek@fam.tuwien.ac.at
Giulia Di Nunno
CMA - Department of Mathematics
University of Oslo
P.O. Box 1053 Blindern
0316 Oslo
Norway
giulian@math.uio.no
Gyula Pap
University of Debrecen, Faculty of Informatics
Pf. 12
4010 Debrecen
Hungary
papgy@inf.unideb.hu

Habib Esmaeili
TU Munich
Munich University of Technology
85747 Garching
Germany
esmaeili@ma.tum.de
Heikki Tikanmäki
Helsinki University of Technology (TKK) Institute
of Mathematics
P.O.Box 1100
02015 TKK
Finland
heikki.tikanmaki@tkk.fi
Henrik Jonsson
EURANDOM
P.O. Box 513
5600 MB Eindhoven
The Netherlands
jonsson@eurandom.tue.nl
Ingemar Kaj
Uppsala University
Box 480
SE 751 06 Uppsala
Sweden
ikaj@math.uu.se
Istvan Fazekas
University of Debrecen, Faculty of Informatics
B. O. Box 12
4010 Debrecen
Hungary
fazekasi@inf.unideb.hu
Ivivi Mwaniki
university of Nairobi
School of Mathematics
00200-30197 Nairobi
Kenya
jvivben@yahoo.com
Jamison Wolf
Tufts University
51 Cedar St. Apt. 4313
01801 Woburn, MA
USA
jamisonbwolf@yahoo.com
100
Haidar Al-Talibi
Växjö universitet
MSI, Matematiska och systemtekniska institutionen
351 95 Växjö
Sweden
Haidar.Al-Talibi@vxu.se
Henrik Hult
Brown University
Division of Applied Mathematics, Box F
02912 Providence, RI
United States
henrik hult@brown.edu
Holger Rootzén
Chalmers
Mathematical Sciences, Chalmers
SE 41296 Gteborg
Sweden
rootzen@math.chalmers.se
Irmingard Eder
Graduate Program Applied Algorithmic Mathematics,
TU Munich
Boltzmannstr.3
D-85747 Garching
Germany
eder@ma.tum.de
Iuliana Marchis
Babes-Bolyai University
Kogalniceanu 1
400084 Cluj-Napoca
Romania
marchis julianna@yahoo.com
Jacod Jean
UPMC-Paris 6, Institut de mathématqiues
175 rue du chevaleret
75013 Paris
France
jj@ccr.jussieu.fr
Jan Kallsen
TU München
Boltzmannstrae 3
85748 Garching
Germany
kallsen@ma.tum.de

Jan Rosinski
University of Tennessee
Department of Mathematics, 121 Ayres Hall, University
of Tennessee
37996-1300 Knoxville
USA
rosinski@math.utk.edu
Jay Rosen
College of Staten Island, CUNY
152 Pennington Ave.
07055 Passaic
United States
jrosen3@earthlink.net
Jean-François Le Gall
DMA - Ecole normale supé rieure de Paris
45, rue d’Ulm
75005 Paris
France
legall@dma.ens.fr
Jeffrey F. Collamore
University of Copenhagen
Universitetsparken 5
2100 Copenhagen
Denmark
collamore@math.ku.dk
John Joseph Hosking
Imperial College London
Department of Mathematics, South Kensington
Campus, Imperial College London
SW7 2AZ London
United Kingdom
john.hosking@imperial.ac.uk
Juan Carlos Pardo Millan
University of Bath
Mathematical Sciences,University of Bath
BA2 7AY Bath
United Kingdom
pardomil@ccr.jussieu.fr
101
Jang Schiltz
University of Luxembourg
162a, avenue de la Faencerie
1511 Luxembourg
Luxembourg
jang.schiltz@uni.lu
Jean Bertoin
Laboratoire de Probabilités Université Paris 6
175 rue du Chevaleret
F-75013 Paris
France
jbe@ccr.jussieu.fr
Jeannette Wörner
University of Goettingen
Institut fuer Mathematische Stochastik,
Maschmuehlenweg 8-10
D-37073 Goettingen
Germany
woerner@math.uni-goettingen.de
Jesper Lund Pedersen
University of Copenhagen
Department of Mathematical Sciences
Universitetsparken 5
2100 Copenhagen
Denmark
jesper@math.ku.dk
Josep Lluís Solé Clivillés
Universitat Autònoma de Barcelona.
Departament de Matemátiques. Facultat de
Ciències.
08193 bellaterra
Catalunya, Spain
jllsole@mat.uab.cat
Juergen Schmiegel
Aarhus University
Peder Skrams Gade 38
8200 Aarhus
Denmark
schmiegl@imf.au.dk

Jukka Lempa
Department of Economics
Turku School of Economics
Rehtorinpellonkatu 3
20500 Turku
Finland
jukka.lempa@tse.fi
Larbi Alili
Warwick University
Department of Statistics
The University of Warwick, Coventry
CV4 7AL Coventry
UK
L.Alili@warwick.ac.uk
Loïc Chaumont
Université d’Angers
Larema – 2, boulevard Lavoisier
49045 cedex 01 Angers
France
loic.chaumont@univ-angers.fr
Makoto Maejima
Keio University
Department of Mathematics, Keio University, 3-14-
1, Hiyoshi, Kohoku-ku
223-8522 Yokohama
Japan
maejima@math.keio.ac.jp
Mark Meerschaert
Department of Statistics and Probability
Michigan State University
48824 East Lansing
USA
mcubed@stt.msu.edu
Mark Veillette
Boston University
10 Emerson Pl. 18-k
02114 Boston
USA
mveillet@bu.edu
102
Katja Krol
Humboldt University Berlin
Unter den Linden 6
10099 Berlin
Germany
krol@math.hu-berlin.de
Lars N. Andersen
Aarhus Universitet
Naturvidenskabeligt Fakultet
Ny Munkegade, Bygning 1530
8000 rhus C
DK
lars@daimi.au.dk
Lukasz Delong
Warsaw School of Economics
Niepodleglosci 162
02-554 Warsaw
Poland
ldelong@wp.pl
Manfred Schael
Inst. Applied Math. University Bonn
Wegelerstr. 6
D 53315 Bonn
Germany
schael@uni-bonn.de
Mark Podolskij
Department of Mathematics, Ruhr-University of
Bochum
Mathematik III, NA 3 / 72 Universittsstrae 150
44780 Bochum
Germany
podolski@cityweb.de
Martin Jacobsen
Dept. of Mathematical Sciences
University of Copenhagen
5 Universitetsparken
2100 Copenhagen
Denmark
martin@math.ku.dk

Martin Keller-Ressel
1040 Vienna
Austria
mkeller@fam.tuwien.ac.at
Mateusz Kwasnicki
Wroclaw University of Technology
Wybrzeze Wyspianskiego 27
50-370 Wroclaw
Poland
mateusz.kwasnicki@pwr.wroc.pl
Matthieu Marouby
Laboratoire de Statistique et Probabilités
Université Paul Sabatier
LSP, Universit Paul Sabatier, 118 Route de Narbonne,
Batiment 1R1 Bureau 12
31062 Toulouse Cedex 9
France
marouby@cict.fr
Mauro Politi
Universita’ degli Studi di Milano
Dipartimento di Fisica, Via Celoria, 16
20133 Milano
Italy
politima@yahoo.it
Michael Hinz
FSU Jena
Friedrich-Schiller-Universitt Jena, Mathematisches
Institut, Fak. fr Mathematik und Informatik
Ernst-Abbe-Platz 2
07737 Jena
Germany
mhinz@minet.uni-jena.de
Michael Sørensen
University of Copenhagen
Universitetsparken 5
DK-2100 Copenhagen
Denmark
michael@math.ku.dk
103
Martynas Manstavicius
Vilnius University
Naugarduko str. 24
03225 Vilnius
Lithuania
martynas.manstavicius@mif.vu.lt
Mathias Vetter
Ruhr-Universitt Bochum
Stiepeler Str. 131
44801 Bochum
Germany
mathias.vetter@rub.de
Mattias Sunden
Chalmers Technical University
Matematiska Vetenskaper, Chalmers Tvärgata 3
41296 Göteborg
Sweden
mattib@math.chalmers.se
Meredith Brown
Tufts University
85 Frederick Ave.
02155 Medford
USA
meredith.brown@tufts.edu
Michael Marcus
CUNY
253 West 73rd. St., Apt. 2E
10023 New York, NY
USA
mbmarcus@optonline.net
Michaela Prokesova
Department of Mathematical Sciences
University of Aarhus
Ny Munkegade, Building 1530
DK - 8000 Aarhus C
Denmark
prokesov@imf.au.dk

Michailina Siakalli
University of Sheffield
49 Wellington Street, Devonshire Courtyard, Flat 47
S1 4HG SHEFFIELD
UK
stp05ms@shef.ac.uk
Mohammed Mikou
7, Bd Copernic, Pte 2g
77420 Champs Sur Marne
France
mmikou2004@yahoo.fr
Muneya Matsui
Department of Mathematics , Keio University
3-14-1 Hiyoshi Kohoku-ku, Yokohama-shi
Kanagawa-ken
223-8522 Yokohama
Japan
mmuneya@math.keio.ac.jp
Natalia Ivanova
Guseva srt., 7, 64, Russia, 170043, Tver
170043 Tver
Russia
natioanidi@tvcom.ru
Niels Jacob
University of Wales Swansea
Department of Mathematics
SA2 8PP Swansea
United Kingdom
N.Jacob@swansea.ac.uk
Oksana Sidorova
Tver State University
Zhelyabov str. 33
170100 Tver
Russia
Oksana.Sidorova@tversu.ru
Omar Rachedi
Universit di Pisa
via roma 96
57126 livorno
italy
omar.rachedi@yahoo.com
104
Mladen Savov
University Of Manchester
F5, 105 Hardy Lane, Chorlton-cum-Hardy
M21 8DP Manchester
UK
mladensavov@hotmail.com
Moritz Kassmann
University of Bonn
Institut für Angewandte Mathematik, Beringstrasse
6
53115 Bonn
Germany
kassmann@iam.uni-bonn.de
Narn-Rueih Shieh
Department of Mathematics
National Taiwan University
10617 Taipei City
Taiwan
shiehnr@math.ntu.edu.tw
Niels Hansen
University of Copenhagen
Department of Mathematical Sciences
Universitetsparken 5
2100 Copenhagen
Denmark
richard@math.ku.dk
Noriyoshi Sakuma
Keio University
Kohoku-ku Hiyoshi 7-2-1 Hiyoshi sun-hights 201
223-0061 Yokohama
Japan
noriyosi@math.keio.ac.jp
Ole Eiler Barndorff-Nielsen
Thiele Centre, Aarhus University
Department of Mathematical Sciences
8000 Aarhus
Denmark
oebn@imf.au.dk

Patrizia Semeraro
University of Turin
p.za Arbarello 8
10100 Turin
Italy
semeraro@econ.unito.it
Pawel Sztonyk
Wroclaw University of Technology
FB 12 - Mathematik, Universitt Marburg,
D-35032 Marburg
Germany
sztonyk@pwr.wroc.pl
Peter Becker-Kern
University of Dortmund
Fachbereich Mathematik, Universität Dortmund
D-44221 Dortmund
Germany
pbk@math.uni-dortmund.de
Piotr Milos
Institute of Mathematics of the
Polish Academy of Sciences
Klaudyny 32/247
01-684 Warsaw
Poland
pmilos@mimuw.edu.pl
Rene Schilling
Uni Marburg
FB 12 - Mathematik
D-35032 Marburg
Germany
schilling@mathematik.uni-marburg.de
Robert Stelzer
Centre for Mathematical Sciences
Munich University of Technology
Boltzmannstrae 3
85747 Garching bei Mnchen
Germany
rstelzer@ma.tum.de
Ronnie Loeffen
University of Bath
Clevelands Building Room 3.2.1 Sydney Wharf
BA2 4EP Bath
United Kingdom
rll22@maths.bath.ac.uk
105
Pauline Sculli
London School of Economics
Department of Statistics, Houghton Street
WC2A 2AE London
United Kingdom
p.k.sculli@lse.ac.uk
Perez-Abreu Victor
CIMAT-Guanajuato-Mexico
Apdo Postal 402
36000 Guanajuato
Mexico
pabreu@cimat.mx
Peter Scheffler
University of Siegen, Department of Mathematics
Walter-Flex-Str. 3
57068 Siegen
Germany
scheffler@mathematik.uni-siegen.de
Rama Cont
Columbia University and CNRS
500 W120th St, Office 313
10027 New York
USA
Rama.Cont@columbia.edu
Riedle Markus
Humboldt University of Berlin
Department of Mathematics, Unter den Linden 6
10099 Berlin
Germany
riedle@mathematik.hu-berlin.de
Ronald Doney
Manchester University
School of Mathematics, PO Box 88, Sackville Street
M60 1QD Manchester
UK
rad@ma.man.ac.uk
Rune Piil Hansen
Department of Physics and Astronomy
University of Aarhus
Ny Munkegade, Bygn. 1520
8210 Aarhus
Denmark
piil@phys.au.dk

Ryad Husseini
Institute of Applied Mathematics
Poppelsdorfer Allee 82
53115 Bonn
Germany
ryad@uni-bonn.de
Sándor Baran
Faculty of Informatics
University of Debrecen, Hungary
Egyetem square 1.
H-4032 Debrecen
Hungary
barans@inf.unideb.hu
Serge Cohen
Institut de Mathmatique Universit Paul Sabatier
serge.cohen@math.ups-tlse.fr
31062 Toulouse
France
serge.cohen@math.ups-tlse.fr
Sidney Resnick
Cornell ORIE
Rhodes 284, Ithaca, NY
14853 Ithaca
USA
sir1@cornell.edu
Suzanne Cawston
LAREMA, Université d’Angers
Département de Mathématiques, Université
d’Angers, 2 Bld Lavoisier
49000 Angers
FRANCE
cawston@tonton.univ-angers.fr
Takahiro Aoyama
Department of mathematics, Keio University
3-14-1 Hiyoshi, Kouhoku-ku, Yokohama
223-0052 Yokohama
Japan
taoyama@math.keio.ac.jp
Tetyana Kadankova
Hasselt University
Center for Statistics, Hasselt University
Agoralaan, building D, 3590 Diepenbeek
Belgium
tetyana.kadankova@uhasselt.be
106
Saeid Rezakhah
Amirkabir University of Technology
rezakhah@aut.ac.ir
15914 Tehran
Iran
rezakhah@aut.ac.ir
Seiji Hiraba
Tokyo University of Science
2641, Yamazaki
278-8510 Noda
Japan
hiraba seiji@ma.noda.tus.ac.jp
Sergio Bianchi
DIMET, University of Cassino
Via S. Angelo
03043 Cassino
Italy
sbianchi@eco.unicas.it
Sören Christensen
Christian-Albrechts-Universitt, Kiel
Ahlmannstrae 19
24118 Kiel
Germany
s.christensen@gmx.net
Takaaki Shimura
The Institute of Statistical Mathematics
4-6-7 Minami-Azabu Minato-ku Tokyo Japan
108-8569 Tokyo
Japan
shimura@ism.ac.jp
Teppo Rakkolainen
Turku School of Economics
Dept. of Economics, Rehtorinpellonkatu 3
20500 Turku
Finland
teppo.rakkolainen@tse.fi
Thomas Liebmann
Ulm University
Mörikeweg 5
88339 Bad Waldsee
Germany
thomas.liebmann@uni-ulm.de

Thomas Mikosch
University of Copenhagen
Laboratory of Actuarial Mathematics
Universitetsparken 5
2100 Copenhagen
Denmark
mikosch@math.ku.dk
Thomas Steiner
PhD student at TU Vienna
Wiedner Hauptstrae 8-10/105-1
A-1040 Vienna
Austria
thomas@fam.tuwien.ac.at
Tomasz Jakubowski
Wroclaw University of Technology
ul. Janiszewskiego 14a
Wroclaw
Poland
Tomasz.Jakubowski@pwr.wroc.pl
Vicky Fasen
Munich University of Technology
Centre for Mathematical Sciences
Boltzmannstrasse 3
85747 Munich
Germany
fasen@ma.tum.de
Viktor Benes
Charles University
Faculty of Mathematics and Physics
Dept. of Probability and Mathematical Statistics
Sokolovska 83
18675 Praha 8
Czech Republic
benesv@karlin.mff.cuni.cz
Violetta Bernyk
Institut de Mathématiques
Ecole Polytechnique
Fédérale de Lausanne, Station 8
1015 Lausanne
Switzerland
violetta.bernyk@epfl.ch
107
Thomas Simon
Departement de Mathematiques
Universite d’Evry-Val d’Essonne
Cours Monseigneur Romero
91025 EVRY
France
tsimon@univ-evry.fr
Tomasz Grzywny
Wroclaw University of Technology
Institute of Mathematics and Computer Science
ul. Wybrzeze Wyspianskiego 27
50-370 Wroclaw
Poland
tomasz.grzywny@pwr.wroc.pl
Tomasz Mostowski
Warsaw University
Dluga 44/55
00-241 Warszawa
Poland
tmostowski@wne.uw.edu.pl
Victor Rivero
Centro de Investigacion en Matematicas A. C.
Calle Jalisco s/n col. Mineral de Valenciana, AP 420
36240 Guanajuato
Mexico
rivero@cimat.mx
Viktoriya Masol
K.U. Leuven and EURANDOM
P.O.Box 513
5600 MB Eindhoven
The Netherlands
masol@eurandom.tue.nl
Virginie Konlack
University of Yaounde I
Department of Mathematics
Box 812 Yaounde
Cameroon
ksognia@yahoo.fr

Vladas Pipiras
University of North Carolina
Dept. of Statistics & OR
UNC-CH, Smith Bldg, CB#3260
NC 27599 Chapel Hill
USA
pipiras@email.unc.edu
Xiaowen Zhou
Concordia University
1455 de Maisonneuve Blvd. W.
H3G 1M8 Montreal
Canada
zhou@alcor.concordia.ca
Yi Shen
Ecole Polytechnique
X-2005 Cie.10, Palaiseau, France
F-91128 Palaiseau
France
yi.shen@polytechnique.edu
Yury Khokhlov
Tver State University
Boulevard Nogina b. 6, ap. 103
170001 Tver
Russia
YSkhokhlov@yandex.ru
Zenghu Li
School of Mathematical Sciences
Beijing Normal University
100875 Beijing
China
lizh@bnu.edu.cn
108
Wei Liu
Universitaet Bielefeld, Germany
App.107, Jakob-Kaiser-Strasse 16
D-33615 Bielefeld
Germany
weiliu0402@yahoo.com.cn
Yasushi Ishikawa
Dept. Math., Ehime University
5 Bunkyocho-2 chome
7908577 Matsuyama
Japan
slishi@math.sci.ehime-u.ac.jp
Yimin Xiao
Michigan State University
Department of Statistics and Probability
A-413 Wells Hall
MI 48824 East Lansing
U.S.A.
xiao@stt.msu.edu
Zbigniew Jurek
Institute of Mathematics
University of Wroclaw,
Pl. Grunwaldzki 2/4
50-384 Wroclaw
Poland
zjjurek@math.uni.wroc.pl
Zoran Vondracek
University of Zagreb
Department of Mathematics, Bijenicka 30
10000 Zagreb
Croatia
vondra@math.hr