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Centre for Mathematical Physics and Stochastics
Department of Mathematical Sciences, University of Aarhus

Funded by The Danish National Research Foundation

Concentrated Advanced Course on

Lévy Processes


Branching Processes

Lectures by

Jean Bertoin (UniversitÚ Pierre et Marie Curie)


Jean-Franšois Le Gall (ENS, Paris)

Monday August 28 - Friday September 1, 2000

Auditorium G1, Department of Mathematical Sciences
University of Aarhus

In the above-mentioned week, MaPhySto organized a Concentrated Advanced Course on LÚvy Processes and Branching processes. The course took place at the Department of Mathematical Sciences, University of Aarhus. Each day there were 2-4 hours of lectures plus exercise sessions.

The course was part of a longer thematic period on LÚvy Processes and their applications - the first event in this respect was the Conference on LÚvy Processes: Theory and Applications held by MaPhySto in January 1999; see the Conference mini-proceedings. Furthermore, Ken-iti Sato gave in the week January 24-28, 2000 a Concentrated Advanced Course on LÚvy Processes.

Content of main course

The main goal of this series of lectures was to present some connections between Lévy processes with no negative jumps and branching processes or random trees. The lectures by J. Bertoin described some of the basic theory of Lévy processes, including subordinators, connections with Markov processes and fluctuation theory in the case of processes with no negative jumps.

The lectures by J.-F. Le Gall dealt more specifically with the coding of the genealogy of continuous branching processes, including applications to limit theorems for discrete Galton-Watson trees and to the construction of superprocesses.

Part A (J. Bertoin)

  1. Subordinators. (3 hours)
    1. Preliminaries on Poisson measure, compensation and exponential formulas.
    2. Construction of subordinators (Lévy-It\^o), Laplace exponent and Lévy-Khintchine formula.
    3. Examples, connections with excursions of Markov processes, regenerative sets and local times.
    4. Passage times, renewal theory and Dynkin-Lamperti theorem.
  2. Lévy processes with no negative jumps. (3 hours)
    1. Elementary study of subordinators with a negative drift.
    2. General case, Lévy-Khintchine formula and the subordinator of passage times.
    3. Fluctuation theory: Markov property of the reflected process, duality lemma and calculation of the exponents.
    4. The scale function and its applications (two-sided exit problem, points of increase)
  3. Continuous state branching processes. (2 hours)
    1. Branching property and connection with subordination.
    2. Lamperti's construction and applications.

Part B (J.-F. Le Gall)

  1. Discrete and continuous branching processes. (2 hours)
    1. Galton-Watson processes and their scaling limits.
    2. The construction of superprocesses and their Laplace functionals.
  2. The quadratic branching case. (2 hours)
    1. Coding the genealogy of Feller's diffusion by Brownian excursions.
    2. Aldous' continuum random tree, ISE and connections with statistical mechanics.
    3. The Brownian snake approach to quadratic superprocesses.
  3. LÚvy processes and the general branching case. (4 hours)
    1. The height process of a LÚvy process with no negative jump.
    2. The connection with branching processes: A generalized Ray-Knight theorem.
    3. Limit theorems for Galton-Watson trees and applications to reduced trees.
    4. The genealogical structure of general continuous trees.
    5. Applications to superprocesses.

The following books were references:

Jean Bertoin:
LÚvy processes. Cambridge Tracts in Mathematics, 121. Cambridge University Press, 1996. ISBN: 0-521-56243-0
Jean-Franšois Le Gall:
Spatial branching processes, random snakes and partial differential equations. Lectures in Mathematics. Birkhńuser, 1999. ISBN: 3-7643-6126-3

Additional lectures

In addition to the main lectures there was a mini-course (4 hours) with lectures by Ole E. Barndorff-Nielsen (MaPhySto) on LÚvy Processes from a modelling perspective. This mini-course will, in particular, treat applications to financial economics.

Furthermore there were the following two survey lectures:

Sergei Levendorskii (Rostov-on-Don):
Option pricing under Levy processes and boundary problems for pseudo-differential operators.
Jan Rosinski (University of Tennessee):
Continuity and extrema of stochastic integrals with respect to Lévy processes.


Note: 10-11 really means 10.15-11.00, and so forth.

      |	Monday		Tuesday		Wednesday	Thursday	Friday
9-10  |	registration	OEBN		OEBN		OEBN		OEBN
10-11 |	Bertoin		Le Gall		Le Gall		Le Gall		Le Gall
11-12 |	Bertoin		Le Gall		Le Gall		Le Gall		Le Gall
14-15 |	Bertoin		Bertoin		Bertoin		Bertoin
15-16 |	Levendorskii 	Bertoin		Rosinski	Bertoin	
16-17 |	Exercises 	Exercises	Exercises	Exercises

Informal Workshop

A very informal workshop took place on Monday and Tuesday in the week after the course on Lévy Processes and Branching Processes.

The main idea of the workshop was to have some of our many guests during that time (see 'People at MaPhySto') to give a talk on their current research. So some of the talks may be related to the subject of the course, whereas others may focus on other areas of stochastics.


Monday (4 September); in Auditorium G2

11.15-12.00: Fred Espen Benth (University of Oslo):
Portfolio optimization in Levy markets.
ABSTRACT: We consider optimal portfolio selection and consumption in a market where the logreturns of the risky assets are not normally distributed.
Recent empirical studies show that the normal inverse Gaussian distribution is a very good and flexible model for logreturns capturing non-Gaussian effects like, e.g., heavy tails and skewness. This distribution leads to a geometric (pure-jump) Levy process dynamics of the stock prices.
We treat a portfolio optimization problem where the investor derives her utility from present and past consumption through Hindy-Huang-Kreps preferences. The value function of the (singular) stochastic control problem is shown to be the unique (constrained) viscosity solution of the Hamilton-Jacobi-Bellman equation, which will be an integro-differential equation subject to gradient constraints in our case. We study markets both with and without transaction costs, and derive explicit solutions in some special cases.

13.00-13.45: Elisa Nicolato (MaPhySto & CAF, University of Aarhus) and Emmanuel Venardos (Nuffield College):
Option pricing in stochastic volatility models of the Ornstein-Uhlenbeck type with a leverage effect.
ABSTRACT: In order to capture key features of stock returns Barndorff-Nielsen and Shephard (1999) have introduced a new class of stochastic volatility models characterized by the use of processes of the Ornstein-Uhlenbeck type and allowing for a leverage effect. In this work we discuss these models from the viewpoint of derivative asset analysis. Instead of selecting a priori one particular equivalent martingale measure (EMM), we study the subset of EMMs M' under which the Lévy property of the process driving the volatility is preserved. For each of these possible pricing measures Q, a closed form formula for the price p(Q) of a European call option is determined. Moreover M' is rich enough for allowing the pricing function p(Q) to span the interval of values that the option might take and is sufficiently tractable for implementing calibration procedures. Finally, for several concrete examples we discuss different numerical approaches to the actual computation of option values.

14.00-14.45: Alexander Cherny (Moscow State University):
Qualitative behaviour of solutions of Stochastic Differential Equations with singular coefficients.
ABSTRACT: We consider a one-dimensional homogeneous stochastic differential equation of the form
   dX_t = b(X_t)dt + \sigma(X_t) dB_t,    X_0 = x,
where b and \sigma are supposed to be measurable functions and \sigma is non-zero. No assumptions of boundedness (or boundedness away from zero) are imposed
We introduce a class of points which will be called isolated singular points, and investigate the weak existence as well as the uniqueness of the solution in the neighborhood of such a point. A complete qualitative classification if these points is presented: there are 63 different types.
It has been found that, for 59 types, there exists a unique solution in the neighborhood of an isolated singular point. (This solution is defined up to the moment it leaves some interval.) Moreover, the solution is a strong Markov process.
The remaining 4 types of isolated singular points (we call them branch types) disturb the uniqueness. One can construct various 'bad' solutions in the neighborhood of a branch point. In particular, there exist non-Markov solutions.
As an application of the obtained results, we consider the equations of the form
   dX_t = \mu |X_t|^\alpha dt + \nu |X_t|^\beta dB_t
and present the classification for this case.

15.15-16.00: Steen Thorbj°rnsen (University of Southern Denmark, Odense):
Selfdecomposability in Free Probability.
ABSTRACT: Free Probability was founded in the 1980's by D.V. Voiculescu (U.C. Berkeley). The theory is a non-commutative analog to ``classical'' probability, in which random variables are replaced by linear operators on an infinite dimensional Hilbert space, and the notion of independence is replaced by a new concept: Freeness. This concept may be encountered as the asymptotic relation between independent (Gaussian) random matrices, as the size of the matrices increase to infinity, and thus free probability provides a concrete model for the joint asymptotic behavior of independent random matrices.
The first part of the talk will be a short introduction to free probability, and subsequently I shall focus on the free version of the theory of self-decomposability.
The talk is on joint work with O.E. Barndorff-Nielsen.

16.15-17.00: Albert N. Shiryaev (Steklov Mathematical Institute):
Cumulant's algebra for semimartingales and Esscher's change of measures.

Tuesday (5 September); in Auditorium G2 before lunch; in D2 after lunch

9.15-9.45: Jan Pedersen (University of Aarhus):
Selfdecomposability and stability in multivariate subordination I.

10.15-11.00: Ken-iti Sato (Nagoya University):
Selfdecomposability and stability in multivariate subordination II.

15.15-15.45: Francesco Mainardi (University of Bologna):
Fractional Diffusion Processes I: analytical properties and special functions.
ABSTRACT: The general one-dimensional diffusion equation, fractional both in space (of order \alpha) and in time (of order \beta), is discussed. Its fundamental solution is the probability density (evolving in time) governing the modelled stochastic process. For a wide range of parameter this fundamental solution is shown to be representable by aid of a Fox H-function with the similarity variable x/t^{\alpha/\beta} in the argument. Convergent and asymptotic series for its approximation are given.

15.45-16.15: Rudolf Gorenflo (Free University of Berlin):
Fractional Diffusion Processes II: types of random walk models and transition to the limit of vanishing step-sizes.
ABSTRACT: Four types of random walk models of Markov type (in one space-dimension) are considered and their interrelations via passages to the limit of vanishing space or time step (separately or in a correctly scaled manner simultaneously in space and time) are considered. The transition probabilities are chosen in the domain of attraction of Levy stable probability distributions so that these random walks approximate Levy-Feller diffusion processes that are governed by a pseudo-differential evolution equation generalizing the classical diffusion equation. The four types of random walk are distinguished via being discrete OR continuous IN space OR time. Finally, a sketch is presented how to generalize the theory to random walks with memory, thus approximating diffusion processes that are fractional also in time.

16.30-17.15: Goran Peskir (University of Aarhus):
Newtonian Finance.


The following notes were used for the course:

Jean Bertoin:
Subordinators, Lévy processes with no negative jumps, and branching processes.
Download in [ gzipped postscript-format | pdf-format ]
Jean-Franšois Le Gall:
Random Trees and Spatial Branching Processes.
Download in [ gzipped postscript-format | pdf-format ]

The Notes will appear in the MaPhySto Lecture Notes Series.


More Information

Do not hesitate to contact the MaPhySto secretariat (at maphysto@maphysto.dk) or the local organizer Goran Peskir for more information.

This document, http://www.maphysto.dk/oldpages/events/LevyBranch2000/index.html, was last modified January 19, 2004. Questions or comments to the contents of this document should be directed to maphysto@maphysto.dk.