Il Prof. Jan Swart [ Institute of Information Theory and Automation - Czech Republic ] sarà ospite del Dipartimento di Informatica dell’Università di Verona [ Strada le Grazie, 15 - Vr ] nei giorni dal 20 al 24 Ottobre 2014 e terrà un mini corso [8 ore] intitolato

” *Interacting Particle Systems with Applications in Finance* “

ed articolato in tre lezioni, che si terranno nell'edificio Ca' Vignal 2 - Aula M, nei giorni

- Lunedì 20 Ottobre [ 13:30-16:30 ] - Mercoledì 22 Ottobre [ 15:30-18:30 ] - Giovedì 24 Ottobre [ 13:30-15:30 ]

con il seguente programma

Interacting particle systems are mathematical models for systems consisting of a large number of components that interact with each other in a random way. While the behavior of the individual components is governed by simple rules, the behavior of the whole system can be quite complicated due to the interaction. Often, one observes a phase transition between a regime of weak interaction where the components behave more or less as if they were independent and a regime of strong interaction where collective phenomena such as multiple stable states occur. Particle systems have been used in finance to model phenomena such as collective decision making or contagion of credit risk, and also in fields like mathematical physics, population biology, and sociology to model lots of other phenomena. In these lectures, I will demonstrate on the basis of examples some of the basic mathematical tools for analyzing interacting particle systems.

*On Monday*, we will look at a number of different interacting particle systems that will serve as motivating examples during the lectures, such as stochastic Ising and Potts models, (biased) voter models, systems of branching and coalescing particles, and more. We will analyze the mean-field versions of these models and use numerical simulations to look at phenomena for spatial models like multiple invariant laws, first and second order phase transitions and critical exponents. We will make a start with the rigorous theory by looking at graphical representations and use these to give sufficient conditions for uniqueness of the invariant law.

*On Wednesday*, the main focus will be on the regime where multiple invariant laws occur. The main techniques on this day are duality and comparison with oriented percolation, which are especially suitable for variations of the voter model and branching and coalescing particle systems.

*On Thursday*, we will look at the Ising and Potts models to which the techiques of Wednesday do not apply but which are reversible with Gibbs invariant law. Through the random cluster representation, percolation again appears as the unifying principle behind collective behavior.

Per informazioni contattare: luca.dipersio@univr luca.dipersio@univr.it

__ Luca Di Persio - PhD assistant professor of Probability and Mathematical Finance

Dept. Informatics University of Verona strada le Grazie 15 - 37134 Verona - Italy Tel : +39 045 802 7968

Dept. Math University of Trento V. Sommarive, 14 - 38123 Povo - Italy Tel : +39 0461 281686