Introduction to Gibbs measures - Einzelansicht

References

1. Friedli, S. and Velenik Y. Statistical Mechanics of Lattice Systems: a Concrete Mathematical Introduction, Cambridge University Press, 2017 [link]
2. Prum, T. and Fort, J.-C. Stochastic Processes on a Lattice and Gibbs Measures, Springer 1991
3. S.E. Shreve, Stochastic Calculus for Finance I, The binomial asset pricing model.Springer 2004

Important remarks:

1- The first lecture takes place on Wednesday April 20th, at 10:15 am in the Room 0.11, Building  5 Campus Golm.

2- Latest updates are available on the Moodle-plattform https://moodle2.uni-potsdam.de/course/view.php?id=32797

3- In addition to this course we strongly encourage to attend the Seminar: Simulation of Stochastic Processes hold by Dr. Peter Keller about applications in numerical mathematics.

Prerequisite : A first course on Stochastics with measure Theory knowledge.

No knowledge in Physics is expected.

There will be an oral exam.

In the first part of this course we aim to introduce a probabilistic approach to (rigorous) equilibrium statistical mechanics. The focus will be done on Gibbs measures and in particular on the Ising model, which is possibly the simplest "realistic" model which exhibits a non-trivial collective behavior. As such, it has played, and continues to play, a central role in mathematics. We will introduce several central notions (e.g. infinite-volume Gibbs states), precise definitions and main results. The analysis makes use of convexity properties of functionals, correlation inequalities, cluster expansions and Peierls' argument, all of which are discussed in detail. This part concludes with a list of open questions covering a wide range of topics.

In a second part of this course, a visiting Lecturer, Dr. Myriam Fradon, will propose a short introduction to discrete time random models in financial mathematics. Probabilistic tools for the pricing and hedging of european style stock options - self-financing portfolio, risk-neutral probability, binomial model, Harrison-Pliska and Cox-Ross-Rubinstein theorems - will be presented.
The celebrated Black-Scholes formula will be obtained as a limit of the option premium when the discretization period tends to zero.

This lecture is appropriate for Master students in Mathematics and is part of both profiles "Mathematical modeling and data analysis" and "Structures of Mathematics with physical background".

This lecture is appropriate for advanced Bachelor students too, as a natural application / extension of the course Aufbaumodul Stochastik.

The lecture also addresses to students of physics.