These special lecture series in probability and mathematical finance are made possible by the generous support of the Minerva Research Foundation and are organized by Ioannis Karatzas and Julien Dubédat. Time and location vary.

March 22-25

Walter Schachermayer, Vienna

The asymptotic theory of transaction costs

School of Social Work 903

Mon Mar 22: 4-5:30pm

Tue Mar 23: 3-4:30pm

Thu Mar 25: 3-4:30pm

March 25- April 2

Geoffrey Grimmett, Cambridge

Lattice models in probability

Thu Mar 25: 9:20-10:50 am, Math 203

Fri Mar 26: 9:20-10:50am, Math 203

Mon Mar 29: 3-4:30 pm, Math 520 (note time, location)

Tue Mar 30: 9:20-10:50am, Math 203Mon Mar 29: 3-4:30 pm, Math 520 (note time, location)

Thu Apr 1: 9:20-10:50am, Math 203

April 12- 22

Ofer Zeitouni, UMN & Weizmann Institute

Random Walks in Random Environments

Mon Apr 12

Fri Apr 16

Mon Apr 19

Wed Apr 21

Thu Apr 22

9:20-10:50 am, Math 622

Lecture Notes (check for updates)

Mar 22-25: Walter Schachermayer, Vienna

The asymptotic theory of transaction costs

Following the pioneering work by Cvitanic/Karatzas (1996) we investigate the duality theory of portfolio optimisation under transaction costs. Special emphasis will be given to asymptotic results when the proportional transaction costs tend to zero.

The approach of utility indifference pricing under transaction costs makes it - at least in principle - possible to also deal with market models which fail to be semi-martingales, such as fractional Brownian motion, and to formulate a consistent theory of pricing and hedging derivative securities.

Mar 25- Apr 2: Geoffrey Grimmett, Cambridge

Lattice models in probability

This course is an introduction to the menagerie of stochastic processes at the boundary of probability and statistical physics. It is designed to be widely accessible to PhD students in cognate areas of mathematics and perhaps physics. Topics to be reviewed include percolation, uniform spanning trees and aspects of random walk, the classical and quantum Ising/Potts models, and the Lorentz gas model. The methodology includes stochastic inequalities, and the theory of influence and concentration. The course notes may be found at www.statslab.cam.ac.uk/~grg/

Apr 12- Apr 22: Ofer Zeitouni, UMN & Weizmann Institute

Random Walks in Random Environments

The (physically motivated) model of random walks in random

environments (RWRE) is extremely simple to state, yet presents numerous

challenges for analysis. Fundamental questions (e.g., concerning

transience/recurrence and homogenization) are still open, in spite

of rapid progress in the last decade.

In this lecture series I will introduce the model of RWRE in the d-dimensional lattice and then state the main results and introduce the many different techniques that have been used in RWRE analysis. Starting from the (solvable) nearest-neighbor one-dimensional case and its immediate extensions, I will move to 0-1 laws in higher dimension, law of large numbers statements, regeneration times and CLT's, homogenization techniques (also for walks on trees), and multiscale methods.

A preliminary plan of the lectures is the following:

L1: Introduction to the RWRE model. The solvable one dimensional nearest neighbor model: transience/recurrence, law of large numbers and limit laws via recursions and branching structures. Annealed vs. quenched behavior. Extensions: non nearest neighbor models, reinforced walks.

L2: RWRE for higher dimensional lattices: Kalikow's 0-1 law and counter examples. The Merkl-Zerner theorem for dimension 2. Implications of 0-1 laws: the law of large numbers. Possible limiting velocities. Zero velocity in the isotropic case.

L3: Regeneration times and their implications: Sznitman-Zerner's regeneration times, annealed central limit theorems. Quenched CLT's as consequence of annealed ones. Non nestling example. Kalikow's condition and estimates on regenerations. Sznitman's conditions and criterion (review of multiscale analysis, Berger bounds). Mixing environments. Cut times.

L4: Homogenization techniques: symmetric models (Lawler's approach and Alexandrov-Bakelman-Pucci estimates). The non-uniformly elliptic case. Large deviations (if time permits). An introduction to walks on random trees.

L5: Multiscale methods in the isotropic case (dimension 3 or larger):

the Bricmont-Kupiainen model. Invariance principles and exit measures.