Minerva Foundation Lectures
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.
Spring Semester 2010
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
Thu Mar 25: 9:20-10:50 am, Math 203
Fri Mar 26: 9:20-10:50am, Math 203
Tue Mar 30: 9:20-10:50am, Math 203
Mon Mar 29: 3-4:30 pm, Math 520 (note
Thu Apr 1: 9:20-10:50am, Math 203
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
Notes (check for updates)
22-25: Walter Schachermayer,
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
Mar 25- Apr
2: Geoffrey Grimmett, Cambridge
Lattice models in
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/books/pgs.html
Apr 12- Apr
22: Ofer Zeitouni, UMN &
Random Walks in
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
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
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.
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
L5: Multiscale methods in the isotropic case (dimension 3 or larger):
the Bricmont-Kupiainen model. Invariance principles and exit measures.