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Centre for Mathematical Physics and Stochastics
Funded by The Danish National Research Foundation
Centre for Analytical Finance
Funded by The Danish Social Science Foundation
Statistical Methods for Dynamical Stochastic Models
EU Research Training

Summer School organized jointly by CAF, DYNSTOCH and MaPhySto

From Lévy Processes to Semimartingales - Recent Theoretical Developments and Applications to Finance

Tuesday 20 - Tuesday 27 August, 2002
Aud. F, Building 534
Department of Mathematical Sciences, University of Aarhus


In addition to the lecture series there will be several invited talks, and some tutorial classes.

Contents of Lecture Series:

Introduction to Semimartingales and their Basic Algebra (A.N. Shiryaev):

  1. Martingale approach to the study of stochastic processes with discrete time (martingales, local martingales, Doob's decomposition, stochastic exponential and stochastic logarithms, jump measures, compensators)
  2. Martingales, local martingales, sigma-martingales in the continuous time case. Basic properties and theorems
  3. Semimartingales: basic definitions and properties. Characteristics of semimartingales, the canonical representation. Semimartingales and Lévy processes
  4. Sample function properties of the Lévy processes from the point of view of the semimartingale canonical representation
  5. Transformations of the semimartingale predictable characteristics
  6. Semimartingales in the mathematical finance (martingale measures and their construction, fundamental theorems of the 'Arbitrage theory')

A.S. Cherny:

  1. Vector stochastic integrals and stochastic integrals up to infinity
  2. Itô's formula and its extensions

Change of Time and Measure (A.N. Shiryaev):

  1. Change of time (basic definitions, constructions, properties)
  2. Time-change representations of stochastic processes (in the strong and the weak sense)
  3. Change of time in the stochastic integrals
  4. Change of measure (absolute continuity and singularity, Lebesgue decomposition, Girsanov type theorems)
  5. Fourier and Laplace cumulant processes. The Esscher change of measure
  6. Change of measure and pricing and hedging

Change of Time and Measure for Lévy Processes (A.S. Cherny):

  1. Some financial models
  2. Change of time for a Brownian motion
  3. Change of measure for Lévy processes
  4. Change of time for Lévy processes

Stochastic Volatility Models and some of their Applications (N. Shephard and O.E. Barndorff-Nielsen):

The lectures will partly be based on draft material for a book with the working title Lévy Processes, Stochastic Volatility, Power Variation and Financial Econometrics. The chapter headings for the book are:

  1. Introduction
  2. Basics of Lévy Processes
  3. Simulation and Inference for Lévy Processes
  4. Time Deformation and Chronometers
  5. Mathematics of Lévy processes
  6. Stochastic Volatility
  7. Realised Variation and Covariation
  8. Power Variation

Additional topics will be pricing of derivatives under stochastic volatility models and multivariate extensions. Much of the research work to be discussed is also available in a series of recent paper that may be accessed at the web address http://www.levyprocess.org/

Additional Lectures:

Hans-Jürgen Engelbert (Friedrich-Schiller-Universität Jena): Stochastic equations driven by symmetric stable processes

Martin Jacobsen (University of Copenhagen): Multiscaling n-dimensional markov processes and their Lamperti representation (joint work with Marc Yor, Paris)

Jan Pedersen (University of Aarhus): Periodic Ornstein-Uhlenbeck processes

Michael Sørensen (University of Copenhagen): Diffusion type models with given marginals and autocorrelation

Peter Tankov (Ecole Polytechnique, Palaiseau): Inverse problems for Lévy processes and nonparametric calibration of jump-diffusion models (joint work with Rama Cont)

Jeannette H.C. Woerner (University of Oxford): Power variation: some insight in model selection and estimation for semimartingale models

Time Schedule:

AS = Albert N. Shiryaev
NS = Neil Shephard
FH = Freidrich Hubalek
EN = Elisa Nicolato
OBN = Ole E. Barndorff-Nielsen
AC = Alexander S. Cherny
JW = Jeannette H.C. Woerner
MJ = Martin Jacobsen
PT = Peter Tankov
JP = Jan Pedersen
HE = Hans-Juergen Engelbert
MS = Michael Sørensen

Tue. 20/8 Wed. 21/8 Thu. 22/8 Fri. 23/8 Sat. 24/8 Mon. 26/8 Tue. 27/8

Social Events:

Monday 19 August 19.00-20.30: Registration and get-together at the Department of Mathematical Sciences, Building 530
Saturday 24 August 12.15: Excursion to Ebeltoft
Monday 26 August 19.00: Dinner in the canteen of the Department of Mathematical Sciences


No financial support is available. Participants are expected to have their expenses covered by their home institutions. Note, however, that there is no registration fee.


Ole E. Barndorf-Nielsen, University of Aarhus
Bent Jesper Christensen, University of Aarhus
Albert N. Shiryaev, Steklov Institute, Moscow
Michael Sørensen, University of Copenhagen

List of Participants

Here you can download the papers from the lectures given by Alexander Cherny and Albert Shiryaev

More Information:

Do not hesitate to contact the MaPhySto secretariat (at maphysto@maphysto.dk) for more information.

(This announcement in [ postscript-format | pdf-format ])

This document, http://www.maphysto.dk/oldpages/events/LPS2002/index.html, was last modified January 19, 2004. Questions or comments to the contents of this document should be directed to maphysto@maphysto.dk.