Probability Colloquium

Optimal Liquidation in a Stochastic Market Impact Model

Speaker(s): 
Peter Frentrup (HU Berlin)
Date: 
Wednesday, November 26, 2014 - 5:00pm
Location: 
HU Berlin, Rudower Chaussee 25, 12489 Berlin, Room 1.115

We describe a market model for trading a single risky asset, in which a large investor seeks to liquidate his position in an infinite time horizon, while maximizing expected proceeds. Trading large orders has an adverse effect on the asset's price, which is determined by the investor's current volume impact and is multiplicative in relation to the current price. The volume impact follows a mean-reverting Ornstein-Uhlenbeck process whenever no trade occurs.

Time Homogeneous Processes with Given Marginal

Speaker(s): 
John M. Noble (University of Warsaw)
Date: 
Wednesday, November 12, 2014 - 6:00pm
Location: 
HU Berlin, Rudower Chaussee 25, 12489 Berlin, Room 1.115

In this talk, I consider the following problem: given a probability measure \mu over R with well defined expected value and given (deterministic) time, does there exist a gap diffusion with the prescribed law at the prescribed time?

This is answered in the affirmative and it is shown that, at least for an atomised space, that a diffusion satisfying the property may be approximated by solutions to fixed point problems.

A primal-dual algorithm for backward SDEs

Speaker(s): 
Christian Bender (Universität des Saarlandes)
Date: 
Wednesday, November 12, 2014 - 5:00pm
Location: 
HU Berlin, Rudower Chaussee 25, 12489 Berlin, Room 1.115

Numerical methods for backward stochastic differential equations (BSDEs) typically consist of two steps. In a first step a time discretization is performed, which leads to a backward dynamic programming equation. In the second step this dynamic program has to be solved numerically. This second step requires to approximate high order nestings of conditional expectations, which is a challenging problem in particular when the BSDE is driven by a high-dimensional Brownian motion.

Anomalous random walks and their scaling limits: From fractals to random media

Speaker(s): 
Takashi Kumagai (Kyoto University)
Date: 
Wednesday, October 29, 2014 - 6:00pm
Location: 
HU Berlin, Rudower Chaussee 25, 12489 Berlin, Room 1.115

In this talk, I present results concerning the behavior of random walks and diffusions on disordered media. Examples treated include fractals and various models of random graphs, such as percolation clusters, trees generated by branching processes, Erdos-Rényi random graphs and uniform spanning trees. As a consequence of the inhomogeneity of the underlying spaces, we observe anomalous behavior of the corresponding random walks and diffusions. The main focus is to estimate the long time behavior of the heat kernel and to obtain a scaling limit of the random walk.

Diffusive limits for stochastic kinetic equations

Speaker(s): 
Arnoud Debussche (ENS Rennes)
Date: 
Wednesday, October 29, 2014 - 5:00pm
Location: 
HU Berlin, Rudower Chaussee 25, 12489 Berlin, Room 1.115

In this talk, we consider kinetic equations containing random terms. The kinetic models contain a small parameter and it is well known that, after scaling, when this parameter goes to zero the limit problem is a diffusion equation in the PDE sense, i.e. a parabolic equation of second order. A smooth noise is added, accounting for external perturbation. It scales also with the small parameter. It is expected that the limit equation is then a stochastic parabolic equation where the noise is in Stratonovitch form.
Our aim is to justify in this way several SPDEs commonly used.

Einstein relation for random walks in random environment

Speaker(s): 
Xiaoqin Guo (TU München)
Date: 
Wednesday, July 9, 2014 - 6:00pm
Location: 
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

The Einstein relation describes the relation between the response of a system to a perturbation and its diffusivity at equilibrium. It states that the derivative of the velocity (with respect to the strength of the perturbation) equals the diffusivity. We consider random walks in an iid random environment (RWRE) under perturbation. We obtain the derivative of the speed of the RWRE assuming one of the following:

(i) the environment has no drift and the perturbation satisfies a ballisticity condition;

The KPZ equation: universality and well-posedness

Speaker(s): 
Giuseppe Cannizzaro (TU Berlin)
Date: 
Wednesday, July 9, 2014 - 5:00pm
Location: 
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

The KPZ equation is presumed to be a universal object describing random growing surfaces. Nevertheless, it is dramatically ill-posed and various attempts have been made to make sense of it. The Cole-Hopf solution proposed by Bertini and Giacomin was shown to be the physically correct one but it is unfortunately unable to fully capture such a universality. In this talk, after presenting some discrete models, we will give a brief introduction about the new notions of solution established in 2013 independently by Martin Hairer and Massimiliano Gubinelli.

Poisson and Compound Poisson Asymptotics in Conventional And Nonconventional Setups

Speaker(s): 
Yuri Kifer (Hebrew University Jerusalem)
Date: 
Wednesday, June 25, 2014 - 6:00pm
Location: 
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

The Poisson limit theorem which appeared in 1837 seems to be the first law of rare events in probability. Various generalizations of it and estimates of errors of Poisson approximations were obtained in probability and more recently this became a popular topic in dynamics in the form of study of asymptotics of numbers of arrivals at small (shrinking) sets by a stochastic process or by a dynamical system. I will describe recent results on Poisson and compound Poisson asymptotics in a nonconventional setup, i.e.

Disorder and Homogenization in the Parabolic Anderson Model

Speaker(s): 
Benedikt Rehle (TU Berlin)
Date: 
Wednesday, June 25, 2014 - 5:00pm
Location: 
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

The Parabolic Anderson Model is a paradigmatic toy model for transport in disordered media. Concretely, one considers the discrete heat equation with a random potential term representing a disordered environment of sources and sinks. The homogenizing effect of the heat flow competes with the potentially localizing effect of the random environment. This is reflected in the behavior of solutions but also - more abstractly - in the spectrum of the corresponding random Schrödinger operator.

Minicourse on Random Interlacements

Speaker(s): 
Alexander Drewitz (Columbia University)
Date: 
Tuesday, June 24, 2014 - 4:00pm to Friday, June 27, 2014 - 12:00pm
Location: 
TU Berlin: Room MA748, HU Berlin: Room 1.023
In these lectures we will give an introduction to the model of random interlacements that has been introduced by Sznitman in 2007.

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