Interacting Particle Systems with Applications in Finance

 Abstract

Interacting particle systems are mathematical models for systems consisting of a large number of components that interact with each other in a randomway. 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