Probability Colloquium

PDEs with non-Markovian random noise

Speaker(s): 
Bohdan Maslowski (Charles University)
Date: 
Wednesday, January 4, 2017 - 6:00pm
Location: 
TU Berlin, Straße des 17. Juni 136, 10623 Berlin, Raum MA 004

SPDEs in which the noise is (not necessarily Gaussian) Volterra process are discussed. Examples of such processes are cylindrical fractional Brownian motion or more generally, cylindrical mutifractional Brownian motion (in the Gaussian case) or Rosenblatt process (in the non-Gaussian case). For linear equations (and a large class of equations with additive noise) we distinguish two levels of regularity of kernels of the driving processes.

Modified Arratia Flow and Wasserstein Diffussion

Speaker(s): 
Max von Renesse (Universität Leipzig)
Date: 
Wednesday, December 7, 2016 - 6:00pm
Location: 
TU Berlin, Straße des 17. Juni 136, 10623 Berlin, Raum MA 004

We introduce a modification of a system of coalescing 1D Brownian motions starting from every point of the unit interval.
In contrast to previous models like Arratia flow or Brownian Web in our model each particle carries a mass which is aggregated upon coalescence and which determines the particle's diffusivity in an inverse proportional way.

Harnack inequalities for symmetric non-local Dirichlet forms and their stability

Speaker(s): 
Zhen-Qing Chen (University of Washington)
Date: 
Wednesday, November 23, 2016 - 6:00pm
Location: 
TU Berlin, Straße des 17. Juni 136, 10623 Berlin, Raum MA 004

In this talk, we will discuss parabolic and elliptic Harnack inequalities for symmetric non-local Dirichlet forms on metric measure spaces under general volume doubling condition. We will present stable equivalent characterizations of parabolic Harnack inequalities in terms of the jumping kernels, variants of cutoff Sobolev inequalities, and Poincare inequalities.

Stability of the elliptic Harnack Inequality

Speaker(s): 
Martin Barlow (University of British Columbia)
Date: 
Wednesday, November 23, 2016 - 5:00pm
Location: 
TU Berlin, Straße des 17. Juni 136, 10623 Berlin, Raum MA 004

Following the work of Moser, as well as de Giorgi and Nash, Harnack inequalities have proved to be a powerful tool in PDE as well as in the study of the geometry of spaces. In the early 1990s Grigor'yan and Saloff-Coste gave a characterisation of the parabolic Harnack inequality (PHI). This characterisation implies that the PHI is stable under bounded perturbation of weights, as well as rough isometries. In this talk I will discuss the proof of the stability of the elliptic Harnack inequality.

This is joint work with Mathav Murugan (UBC).

Rough reflected RDEs

Speaker(s): 
Massimiliano Gubinelli (Universität Bonn)
Date: 
Wednesday, October 26, 2016 - 6:00pm
Location: 
TU Berlin, Straße des 17. Juni 136, 10623 Berlin, Raum MA 004

I will talk about differential equations driven by irregular signals and reflected inside a certain domain. Following and simplifying a bit a recent work of Aida we will use tools from rough path there to define a suitable notion of solution and obtain existence of solutions. Moreover, in a joint work with Deya, Hofmanova and Tindel we recently obtained certain uniqueness results for such equations.

Making high order methods effective

Speaker(s): 
Terry Lyons (University of Oxford)
Date: 
Wednesday, June 29, 2016 - 6:15pm
Location: 
HU Berlin, Rudower Chaussee 25, Room 1.115

High order methods allow the inclusion of more information. How're there can be a horrendous price. Methods, like recombination and meoization can be very effective in counterbalancing the complexity that otherwise arises rather rapidly.

Joint work with Maria Tchernychova and Wei Pan.

Gaussian and Self-Similar Stochastic Volatility Models

Speaker(s): 
Archil Gulisashvili (Ohio University, Athens)
Date: 
Wednesday, June 15, 2016 - 6:00pm
Location: 
HU Berlin, Rudower Chaussee 25, Room 1.115

The results discussed in the talk are joint with F. Viens and X. Zhang (Purdue University). The talk is devoted to uncorrelated Gaussian stochastic volatility models. The volatility of an asset in such a model is described by the absolute value of a Gaussian process. We find sharp asymptotic formulas with error estimates for the realized volatility and the asset price density in a general Gaussian model, and also characterize the wing behavior of the implied volatility.

Random Sierpinski gaskets. Geometric, analytic and stochastic properties

Speaker(s): 
Uta Freiberg (Universität Stuttgart)
Date: 
Wednesday, June 1, 2016 - 6:15pm
Location: 
HU Berlin, Rudower Chaussee 25, Room 1.115

Self similar fractals are often used in modeling porous media. Hence, defining a Laplacian and a Brownian motion on such sets describes transport through such materials. However, the assumption of strict self similarity could be too restricting. So, we present several models of random fractals which could be used instead. After recalling the classical approaches of random homogenous and recursive random fractals, we show how to interpolate between these two model classes with the help of so called V-variable fractals.

Minicourse "Dependence, Risk bounds, Optimal Allocations and Portfolios"

Speaker(s): 
Ludger Rüschendorf (Universität Freiburg)
Date: 
Tuesday, May 10, 2016 - 10:30am to Thursday, May 12, 2016 - 12:00pm
Location: 
TU Berlin (room MA748) and HU Berlin (room 1.115)

The main focus in this course is on the description of the influence of dependence in multivariate stochastic models for risk vectors. In particular we are interested in the description of the impact of dependence on the formulation of risk bounds, on the range of portfolio risk measures on problems of optimal risk allocation (diversification), and the construction of optimal portfolios.
In more detail:

Fluctuations of random surfaces

Speaker(s): 
Ron Peled (Tel Aviv University)
Date: 
Wednesday, April 27, 2016 - 6:15pm
Location: 
HU Berlin, Rudower Chaussee 25, Room 1.115

We study the fluctuations of random surfaces on a two-dimensional discrete torus. The random surfaces we consider are defined via a nearest-neighbor pair potential which we require to be twice continuously differentiable on a (possibly infinite) interval and infinity outside of this interval. This includes the case of the so-called hammock potential, when the random surface is uniformly chosen from the set of all surfaces satisfying a Lipschitz constraint.

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