Probability Colloquium

Master equation for mean-fi eld games

Speaker(s): 
François Delarue (Université Nice Sophia-Antipolis, CNRS)
Date: 
Wednesday, June 11, 2014 - 6:00pm
Location: 
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

Mean field games are part of large population stochastic control. The point is to describe asymptotic Nash equilibria within a large (asymptotically infinite) population of controlled players interacting one with another in a mean-field way. The purpose of the talk is to make rigorous the notion of master equation. The master equation is a PDE built on the space of measures. It encapsulates all the necessary information to describe an equilibrium. Part of talk will be dedicated to the construction of a solution, based on the analysis of the flow of a suitable forward-backward SDE system.

Noise-Induced Strong Stability

Speaker(s): 
Matti Leimbach (TU Berlin)
Date: 
Wednesday, June 11, 2014 - 5:00pm
Location: 
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

We examine a 2-dimensional ODE which exhibits explosion in finite time. Considered as an SDE with additive white noise, it is known to be stable - in the sense that for each initial condition there is almost surely no explosion. Furthermore, the associated Markov process even admits an invariant probability measure. The question of interest is whether the noise also induces a stronger concept of stability, namely the existence of an attractor. We give two examples which answer the question in different ways.

Joint work with M. Scheutzow.

Interacting particle systems, nonlinear Markov processes and SDEs driven by nonlinear Levy noise

Speaker(s): 
Vassili Kolokoltsov (University of Warwick)
Date: 
Wednesday, May 28, 2014 - 6:00pm
Location: 
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

We shall discuss general approaches to the analysis of the dynamics of interacting particle systems, their dynamic law of large numbers and the dynamic central limit for the fluctuations, stressing both analytic aspects (nonlinear Markov semigoups and processes) and probablistic (weak SDEs driven by nonlinear Levy noise). Recent developments include the application to stochastic control (e.g. mean field games) with various applications in economics and finances.

Spectral estimation of volatility coefficient of scalar diffusion

Speaker(s): 
Jakub Chorowski (HU Berlin)
Date: 
Wednesday, May 28, 2014 - 5:00pm
Location: 
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

We consider one dimensional diffusion process observed at regularly sampled time points, with fixed time between observations. We give example of optimal estimator of diffusion coefficients, based on spectral properties of the transition semi group. We investigate the behavior of this estimator when time between observations decreases to zero and discuss its effectiveness in high frequency regime. Joint work with M. Reiß.

Minicourse on Lévy Processes and Optimal Stopping

Speaker(s): 
Erik Baurdoux (LSE)
Date: 
Monday, May 26, 2014 - 3:00pm to Friday, May 30, 2014 - 4:00pm
Location: 
HU Berlin, TU Berlin, WIAS Berlin

We start with the classical secretary problem, where our aim is to choose the best candidate out of a number of applicants appearing in front of us in a random order, without having the option of going back to a previously rejected applicant. This example of an optimal stopping problem has been well studied (and still is!), and it will illustrate in a rather simple setting some important features of optimal stopping problems. We will then move our attention to Levy processes, which form a surprisingly rich class and for example include Brownian motion and (compound) Poisson processes.

Interplay Between the Nonlinear and Nonlocal Components of Diffusions Driven by Levy Processes

Speaker(s): 
Wojbor A. Woyczynski (Case Western Reserve University, USA)
Date: 
Wednesday, May 21, 2014 - 5:30pm
Location: 
WIAS, Erhard-Schmidt-Hörsaal, Erdgeschoss, Mohrenstraße 39, 10117 Berlin

One of the motivations of our program was to develop understanding of the interplay between the nonlinear and nonlocal components in evolution equation driven by the infinitesimal generators of stochastic processes with jumps, such as Levy processes and flights.

Overview on results and open problems related to heavy-tailed distributions

Speaker(s): 
Sergey Foss (Heriot-Watt University Edinburgh)
Date: 
Wednesday, May 14, 2014 - 6:00pm
Location: 
TU Berlin, Room MA041, Straße des 17. Juni 136, 10623 Berlin

I will speak about various topics related to heavy tails, formulate some results and hypotheses, and introduce a number of open problems. I will start with asymptotic analysis of tail probabilities for the supremum M = supn Sn of a random walk S0 = 0, Sn = X1 + ... + Xn given M is finite a.s., and review the five main cases, two cases for heavy-tailed distributions and three cases for light-tailed distributions.

A weak mutation - strong selection model for experimental evolution

Speaker(s): 
Adrian Gonzalez (TU Berlin)
Date: 
Wednesday, May 14, 2014 - 5:00pm
Location: 
TU Berlin, Room MA041, Straße des 17. Juni 136, 10623 Berlin

Inspired by the Lenski experiment for the evolution of E. coli (http://en.wikipedia.org/wiki/E._coli_long-term_evolution_experiment), we discuss a model with random reproduction that, under a suitable rescaling, leads to a stochastic differential equation. We quantify assumptions which lead to a separation of timescales for the effects of mutation and selection. This makes the model tractable and gives some explanation of the form of the fitness curve observed in the long term experiment. Joint work with N.

Macroscpic and microscopic structures of the family tree for decomposable branching processes

Speaker(s): 
Vladimir Vatutin (Steklov Mathematical Institute RAS)
Date: 
Wednesday, April 30, 2014 - 6:00pm
Location: 
WIAS, Room 406, Mohrenstraße 39, 10117 Berlin

A decomposable strongly critical Galton-Watson branching process with N types of particles labelled 1,2,...,N is considered in which a type i parent may produce individuals of types j >= i only. This model may be viewed as a stochastic model for the sizes of a geographically structured population occupying N islands, the location of a particle being considered as its type. The newborn particles of island i <=N-1 either stay at the same island or migrate, just after their birth to the islands i+1,i+2, ...,N.

Discrete, Non Probabilistic Market Models. Arbitrage and Pricing Intervals

Speaker(s): 
Sebastian Ferrando (Ryerson University, Toronto)
Date: 
Wednesday, April 30, 2014 - 5:00pm
Location: 
WIAS, Room 406, Mohrenstraße 39, 10117 Berlin

In the simplest discrete setting of a stock and a bank account with 0 interest rates, we describe a trajectory based approach to pricing options.
The approach does not make use of probabilities. We describe no arbitrage results and a minmax pricing interval. Connections with the standard martingale based approach will also be explained. If time permits, we will describe a dynamic programming approach to evaluate the minmax price interval.

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