Probability Colloquium

Stochastic nonlinear Schrödinger equations with linear multiplicative noise: the rescaling approach

Speaker(s): 
Michael Röckner (University of Bielefeld)
Date: 
Wednesday, December 18, 2013 - 6:00pm
Location: 
TU Berlin, Raum MA 041, Straße des 17. Juni 136, 10623 Berlin

We present well-posedness results for stochastic nonlinear Schrödinger equations with linear multiplicative Wiener noise including the non-conservative case. Our approach is different from the standard literature on stochastic nonlinear Schrödinger equations. By a rescaling transformation we reduce the stochastic equation to a random nonlinear Schrödinger equation with lower order terms and treat the resulting equation by a fixed point argument, based on generalizations of Strichartz estimates proved by J. Marzuola, J. Metcalfe and D. Tataru in 2008.

A Stochastic Free Boundary Problem and Limit Order Book Model

Speaker(s): 
Marvin Müller (TU Berlin)
Date: 
Wednesday, December 18, 2013 - 5:00pm
Location: 
TU Berlin, Raum MA 041, Straße des 17. Juni 136, 10623 Berlin

Free boundary problems allow for modeling of multi-phase systems with separating boundaries evolving in time. We want to model a stock market with a large amount of transactions in rather short time and consider buy- and sell-side of the limit order book as a price-time-continuous two-phase system. Doing so, we introduce an infinite dimensional model based on a generalized stochastic Stefan problem and analyze the resulting second order SPDE with free boundary.

Linear Rough Differential Equations

Speaker(s): 
Antoine Lejay (Université de Lorraine, Nancy, Frankreich)
Date: 
Wednesday, December 11, 2013 - 5:00pm
Location: 
Erhard-Schmidt-Hörsaal, Weierstrass Institute, Mohrenstrasse 39, 10117 Berlin

Although linear Rough Differential Equations (LRDE) could be seen as a particular case of Rough Differential Equations (RDE), they are worth to be considered as objects with their specific properties. Following the work of D. Feyel, A. de la Pradelles and G. Mokobodzki, we present in this talk a coherent theory of LRDE by considering rough resolvent (or semi-group, propagators) in Banach algebra. A particular well-known case is the one of rough paths, by specializing the underlying space to tensor algebra.

Large Deviations and Subexponential Random Variables and application to a System of Interacting Particles

Speaker(s): 
Michail Loulakis (National Technical University Athens)
Date: 
Wednesday, December 4, 2013 - 6:00pm
Location: 
TU Berlin, Raum MA 041, Straße des 17. Juni 136, 10623 Berlin

We will discuss a Gibbs Conditioning Principle for subexponential random variables. We will then use this result in the context of Zero Range Processes to explore the bulk fluctuations and the fluctuations of the size of the condensate in equilibrium, as well as the onset of condensation as we move from subcritical to supercritical densities.
(Joint work with Ines Armendariz and Stefan Grosskinsky.)

On regularizing properties of non-degenerate Brownian noise in coupled FBSDE

Speaker(s): 
Alexander Fromm (HU Berlin)
Date: 
Wednesday, December 4, 2013 - 5:00pm
Location: 
TU Berlin, Raum MA 041, Straße des 17. Juni 136, 10623 Berlin

We apply the theory of decoupling fields, which was developed to study general forward-backward systems in a Brownian setting, to construct global solutions for a special class of coupled problems characterized by a non-degenerate noise component (in the martingale part of the forward equation) and some additional conditions. The non-degeneracy "regularizes" this problem, which is actually ill-posed if the noise component vanishes. We study this effect with purely stochatic methods, i.e. without making use of PDE theory.

On some recent advancements in embedding problems and their interactions with stochastic control, variational inequalities and optimal transportation

Speaker(s): 
Jan Obloj (University of Oxford)
Date: 
Wednesday, November 20, 2013 - 6:00pm
Location: 
TU Berlin, Raum MA 041, Straße des 17. Juni 136, 10623 Berlin

In this talk I will discuss recent advancements in the field of Skorokhod embeddings. The focus coming from mathematical finance meant that new methodologies were applied recently to better understand existing results and approach previously unsolved problems. I will aim to showcase some of these. I will discuss Root’s solution, its computation through solutions to a free boundary problem and a probabilistic representation, and proof of the embedding, via optimal stopping techniques.

The Predictable Representation Property of Compensated-Covariation Stable Families of Martingales

Speaker(s): 
Paolo Di Tella (Humboldt-Universität zu Berlin)
Date: 
Wednesday, November 20, 2013 - 5:00pm
Location: 
TU Berlin, Raum MA 041, Straße des 17. Juni 136, 10623 Berlin

We investigate the predictable representation property for compensated-covariation stable families of martingales in the Hilbert space of square integrable martingales. To give a general definition of the predictable representation property we use the theory of stable subspaces. The main result is that any compensated-covariation stable family of martingales which satisfies some further conditions possesses the predictable representation property. We apply the theory to the special case of Lévy processes.

Mean Field Games and Systemic Risk

Speaker(s): 
Jean-Pierre Fouque (University of California Santa Barbara)
Date: 
Wednesday, October 23, 2013 - 6:00pm
Location: 
TU Berlin, Raum MA 041, Straße des 17. Juni 136, 10623 Berlin

We propose a simple model of inter-bank borrowing and lending where the evolution of the log-monetary reserves of N banks is described by a system of diffusion processes coupled through their drifts in such a way that stability of the system depends on the rate of inter-bank borrowing and lending. Systemic risk is characterized by a large number of banks reaching a default threshold by a given time horizon, and computed using large deviation estimates.

Toward a coherent Monte Carlo simulation of CVA

Speaker(s): 
Lokman Abbas-Turki (TU Berlin)
Date: 
Wednesday, October 23, 2013 - 5:00pm
Location: 
TU Berlin, Raum MA 041, Straße des 17. Juni 136, 10623 Berlin

This work is devoted to the simulation of the Credit Valuation Adjustment (CVA) using a pure Monte Carlo technique with Malliavin Calculus (MCM). The procedure presented is based on a general theoretical framework that includes a large number of models as well as various contracts, and allows both the computation of CVA and its sensitivity with respect to the different assets. Moreover, we provide the expression of the backward conditional density of assets vector that can be simulated off-line in order to reduce the variance of the CVA estimator.

On large deviations for the empirical measures of weakly interacting systems

Speaker(s): 
Markus Fischer (University of Padua)
Date: 
Wednesday, July 11, 2012 - 5:00pm
Location: 
TU Berlin, MA041, Strasse des 17. Juni 136, 10623 Berlin

One of the basic results of large deviations theory is Sanov's theorem, which states that the sequence of empirical measures of independent and identically distributed samples satisfies a large deviation principle with rate function given by relative entropy with respect to the common sample distribution. Large deviation principles for the empirical measures are known to hold also for broad classes of weakly interacting systems (or mean field systems).

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