Probability Colloquium

Maximal regularity of SPDEs

Speaker(s): 
Mark Veraar (TU Delft)
Date: 
Wednesday, July 11, 2012 - 4:00pm
Location: 
TU Berlin, MA041 Strasse des 17. Juni 136, 10623 Berlin

The space-time regularity of paths of solutions of SPDEs plays an important in both nonlinear SPDEs and numerical approximation of solutions. In this talk I present an overview on optimal regularity results for a large class of parabolic SPDEs. The talk is based on joint works with van Neerven and Weis.

Optimal estimation of quarticity and other functionals of the volatility; Nonparametrics for Lévy measures and copulas

Speaker(s): 
Jean Jacod, Mathias Vetter
Date: 
Tuesday, July 10, 2012 - 3:15pm
Location: 
HU Berlin, Rudower Chaussee 25, building 1, room 1.115

Abstract for ‘Nonparametrics for Lévy measures and copulas’: In this talk nonparametric methods to assess the multivariate Lévy measure are introduced. Starting from high frequency observations of a Lévy process X, we construct estimators for its tail integrals and the Pareto Lévy copula and prove weak convergence of these estimators in certain function spaces.

A Frechet law and an Erdös-Philipp law for Maximal Cuspidal Windings

Speaker(s): 
Marc Keßeböhmer (Universität Bremen)
Date: 
Wednesday, July 4, 2012 - 5:15pm
Location: 
TU Berlin, MA041 Strasse des 17. Juni 136, 10623 Berlin

We establish a Frechet law for maximal cuspidal windings of the geodesic flow on a Riemannian surface associated with an arbitrary finitely generated, essentially free Fuchsian group with parabolic elements. This result extends previous work by Galambos and is obtained by applying Extreme Value Theory. Subsequently, we show that this law gives rise to an Erdös-Philipp law and to various generalised Khintchine-type results for maximal cuspidal windings.

Applications of controlled paths

Speaker(s): 
Massimiliano Gubinelli (Paris)
Date: 
Wednesday, June 27, 2012 - 5:00pm
Location: 
TU Berlin, MA041 Strasse des 17. Juni 136, 10623 Berlin

Controlled paths have been introduced to provide an alternative formulation of the rough path theory of Lyons. I will illustrate some applications of the idea of controlled paths in contexts unrelated to the integration theory for stochastic processes. In particular in (S)PDEs and in the phenomenon of regularization by noise for ordinary differential equations.

Obstacle problem for quasilinear SPDE's

Speaker(s): 
Laurent Denis (Évry)
Date: 
Wednesday, June 27, 2012 - 4:00pm
Location: 
TU Berlin, MA041, Strasse des 17. Juni 136, 10623 Berlin

We prove existence and uniqueness of the solution of quasilinear stochastic PDEs with obstacle. Our method is based on analytical technics coming from the parabolic potential theory. The solution is expressed as a pair made of a predictable continuous process which takes values in a proper Sobolev space and a random regular measure satisfying minimal Skohorod condition.

Conformal Loop Ensembles

Speaker(s): 
Wendelin Werner (Paris - Orsay)
Date: 
Wednesday, June 13, 2012 - 5:00pm
Location: 
TU Berlin, MA041 Strasse des 17. Juni 136, 10623 Berlin

We will survey recent results concerning various definitions of these random collections of loops, their properties and their relations to the Gaussian Free Field.

Predicting the ultimate maximum of a Lévy process

Speaker(s): 
Erik Baurdoux (LSE, London)
Date: 
Wednesday, June 13, 2012 - 4:15pm
Location: 
TU Berlin, MA041 Strasse des 17. Juni 136, 10623 Berlin

Optimal prediction of the ultimate maximum is a non-standard optimal stopping problem in the sense that the pay-off function depends on a process which is not adapted to the given filtration, in this case the ultimate maximum. For a finite time horizon, this problem has been studied in various papers including Du Toit, J. and Peskir, G. (2009 AAP 2009) and Bernyk, V., Dalang, R.C. and Peskir, G. (2011 Ann. Probab.).

Condensation effects in preferential attachment models with fitness

Speaker(s): 
Steffen Dereich (Philipps-Universität Marburg)
Date: 
Wednesday, May 30, 2012 - 5:00pm
Location: 
TU Berlin, MA041 Strasse des 17. Juni 136, 10623 Berlin

A preferential attachment network model is a sequence of random graphs that is built according to a simple dynamic rule. In each step a new vertex is added and linked by a random or deterministic number of edges to the vertices already present in the system. In this process, links to vertices with high degree are preferred. A variant of the model, additionally, assigns each vertex a random positive fitness (say a $\mu$-distributed value) which has a linear impact on its attractiveness in the network formation. Such network models show an intriguing phase transition.

Consistent maximal displacements for branching Brownian motion

Speaker(s): 
Matt Roberts (McGill University)
Date: 
Monday, May 21, 2012 - 5:00pm
Location: 
TU Berlin, MA041 Strasse des 17. Juni 136, 10623 Berlin

There has been much recent progress on understanding the system of particles near the frontier of branching Brownian motion, which is a model sharing universality properties with several processes from statistical physics. We consider the question: how close can particles stay to the critical line sqrt{2}t?

Viscosity Solutions of Fully Nonlinear Path-Dependent PDEs

Speaker(s): 
Jianfeng Zhang (USC, Los Angeles)
Date: 
Monday, May 21, 2012 - 4:00pm
Location: 
TU Berlin, MA041, Strasse des 17. Juni 136, 10623 Berlin

In this talk we introduce a notion of viscosity solutions for Path Dependent PDEs (PPDEs for short). Such new type of PDEs include, among others: Backward SDEs (semilinear PPDE), $G$-martingales and Second Order BSDEs (Path dependent HJB), Path dependent HJB-Issac equation (corresponding to zero-sum stochastic differential game), Backward SPDEs, and SPDEs.

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