Zürich Spring School on Lévy Processes

Sunday 29 March – Thursday 2 April 2015

In the spring of 2015 the University of Zürich and ETH Zürich will host a spring school on Lévy processes, with a diverse range of courses from the fundamental theory to applications in finance.

The school aims to be accessible to a wide audience, from doctoral students onwards.

Lectures

Introduction to Lévy processes
Victor Rivero (CIMAT) [slides]
Lévy processes in mathematical finance
Dilip Madan (Smith - Maryland) [slides: 1 2 3 4 5] Abstract. The following topics will be discussed: 1. The general case for discontinuous processes; 2. Specific examples of processes, their calibration and what we learn from this; 3. The extension to Sato processes; 4. Hunt Processes and their calibration; 5. Two price economies, Nonlinear Martingales and Nonlinear PIDE's.
Lévy processes and random trees
Bénédicte Haas (Paris Dauphine) [slides] Abstract. We will present some of the recent progress on the description of large-scale structure of random trees, using Lévy processes. We will in particular focus on sequences of trees that satisfy a certain Markov property, which appears naturally in a large set of models, including e.g. conditioned Galton-Watson trees. Due to this property, we will see that a typical path in the tree is asymptotically proportional to a positive self-similar Markov process, which will be a crucial point to describe the behavior of the whole tree. This approach leads to a new proof of the convergence of conditioned Galton-Watson trees towards the Brownian continuum tree of Aldous and the stable Lévy trees of Duquesne, Le Gall and Le Jan. Several other applications will be discussed, in particular to dynamical models of randomly-growing trees.
Lévy processes and random planar maps
Nicolas Curien (Paris Orsay)Abstract. The peeling process is a discrete algorithm that discovers step by step a random planar map by revealing one face at a time. This probabilistic structure is directly linked with a discrete version of a 3/2-stable Lévy process with no positive jumps. We will carefully introduce the basics on random planar triangulations and the peeling algorithm to show how the theory of Lévy processes gives crucial information about the geometric structure of random triangulations.
Igor Kortchemski (CNRS & École polytechnique) [slides: part 1 42MB, part 2 38MB] Abstract. We will introduce a one-parameter family of random compact metric spaces called "stable looptrees", which are coded by stable Lévy processes. They are made of a collection of random loops glued together along a tree structure, and can informally be viewed as dual graphs of alpha-stable Lévy trees. By using codings of discrete plane trees by random walks, we will see that stable looptrees are universal scaling limits, for the Gromov-Hausdorff topology, of various combinatorial models. Finally, a connection will be made with random planar maps: we will see the stable looptree of parameter 3/2 is the scaling limit of cluster boundaries in critical site-percolation on large random triangulations. In this lecture, we shall only assume basic background on Lévy processes.
Lévy processes and self-similar Markov processes
Andreas Kyprianou (Bath) [slides]
Statistical inference for Lévy processes
Mark Podolskij (Aarhus)Abstract. In this lecture we will present modern estimation methods for Levy processes and related models. After highlighting standard estimators of Levy characteristics, we will learn how these principles transfer to more general processes such as e.g. Ito semimartingales. One of the basic principles will be the extension of limit theorems for a simple driving motion (e.g. Brownian motion) to integrals with respect to this motion. This will be done in the context of high frequency observations and the concept of stable convergence will be crucial for our purpose. Finally, we will also discuss some recent limit and estimation theory for certain non-semimartingale processes.
Poster session for participants
Abstracts booklet. Photos from the poster session.

Group photograph in UZH Lichthof

Location

The school will take place at the Zentrum campus of the University of Zurich, as follows.

Both rooms have stepless access. The closest public transport connections are: Tram stop ETH/Universitätsspital (lines 6, 9, 10), and Polybahn station Polyterrasse ETH (Mon – Sat only). Routes can be planned with ZVV or Google Maps.

Programme

Registration

Registration is now closed.

Contact

The organising committee is:

If you have any questions, please get in touch at levy15@math.uzh.ch.

This school was made possible thanks to funding and organisational help from ETH Zürich, the University of Zürich, and the Zürich Graduate School for Mathematics.

Picture credits / Paradeplatz: Raymond Teodo, CC BY-NC-ND