PL2US 2011

Program

Talks are 45 minutes long + 5 minutes for questions.

February 10 (Room 48-210)

16:00 Andreas Neuenkirch (Kaiserslautern) and Christian Wille (UGR): ``Introduction to the workshop''

16:10 Henryk Zaehle (Saarbrücken): ``Limit theorems and robustness for tail-dependent statistical functionals''

17:00 Coffee break

17:30 Felix Heidenreich (Kaiserslautern): ``A Multilevel Monte Carlo Algorithm for Lévy Driven Stochastic Differential Equations''

18:20 Sophie Pénisson (Nancy): ``Conditioned multitype branching processes and illustration in epidemiology''

19:10 End of the first day, followed by a dinner in Sommerhaus Restaurant all together

February 11 (Room 48-208)

9:00 Guest Speaker: Francesco Russo (ENSTA Paris-Tech): ``Stochastic calculus via regularizations in Banach spaces and applications''

9:50 Dejun Luo (Luxembourg): ``Asymptotic estimates on the time derivative of entropy on a Riemannian manifold''

10:40 Coffee break

11:10 Octavio Arizmendi (Saarbrücken): ``Non-classical infinite divisibility and power semicircle disitributions''

12:00 Thanh Tan Mai (Kaiserslautern): ``Existence and uniqueness of weak solutions for SDEs in Hilbert spaces with time-dependent parameters and an application to an SPDE describing a fiber dynamics''

12:50 Lunch

14:00 Alexandra Chronopoulou (Nancy): ``Maximum Likelihood Estimation for Fractional Diffusion Equations''

14:50 Raphaël Lachièze-Rey (Luxembourg): ``Asymptotic rearrangements of Gaussian fields''

15:40 End of the workshop

Abstracts

Octavio Arizmendi (Saarbrücken)

Non-classical infinite divisibility and power semicircle disitributions

Beyond the classical independence, new types of independences such as boolean, monotone, antimonotone and free have arisen from non commutative probability. In the spirit of trying to understand all the independences together, Power Semicircle Distributions appear naturally as an interpolating family between the "gaussians" with respect to this new type of independences. In this talk we will explain the role of power semicircle laws in this different types of independences. In particular, how some members of this class of distributions come from central limit type theorems with respect to the different additive convolutions in non-commutative probability. We also derive some properties of this measures such as moments, Cauchy transforms and, using a very easy criteria involving kurtosis, the infinite divisibility with respect to different convolutions in non-commutative probability.

Alexandra Chronopoulou (Nancy)

Maximum Likelihood Estimation for Fractional Diffusion Equations

We consider the parameter estimation problem for a multidimensional stochastic differential equation with non-linear random drift and diffusion coefficients, driven by a fractional Brownian motion with Hurst parameter H > 1/2. More specifically, assuming that H is known, we propose an approach for the computation of the maximum likelihood estimators, based on the use of Malliavin calculus. Although the model is in continuous time, we assume that we only have access to discrete observations of the process, for which we derive a novel expression for the log-likelihood as a fraction of two expectations. For the maximum likelihood estimators to be computed in practice, we propose a pathwise discretization for the kernels of the expectations, for which we also study the rate of convergence. Furthermore, we use Monte Carlo simulations of the discretized kernels and we study the efficiency of our numerical scheme.

Felix Heidenreich (Kaiserslautern)

A Multilevel Monte Carlo Algorithm for Lévy Driven Stochastic Differential Equations

Dejun Luo (Luxembourg)

Asymptotic estimates on the time derivative of entropy on a Riemannian manifold

We consider the entropy of the solution to the heat equation on a Riemannian manifold. When the manifold is compact, we provide two estimates on the rate of change of the entropy in terms of the lower bound on the Ricci curvature and the spectral gap respectively. Our explicit computation for the three dimensional hyperbolic space shows that the time derivative of the entropy is asymptotically bounded by two positive constants.

Thanh Tan Mai (Kaiserslautern)

Existence and uniqueness of weak solutions for SDEs in Hilbert spaces with time-dependent parameters and an application to an SPDE describing a fiber dynamics

We generalize the techniques of DaPrato and Zabczyk for solving linear equations with additive noise to the case of SDEs in Hilbert spaces with time-dependent parameters. The concepts of two-parameter operator semi-groups are used as the main analytic tool. Furthermore, we consider an SPDE from industrial mathematics, describing a fiber dynamics, as an application.

Raphaël Lachièze-Rey (Luxembourg)

Asymptotic rearrangements of Gaussian fields

We consider simplicial regularisations X_n of a Gaussian field X, and more particularly their renormalised gradient fields Y_n. Under regularity hypotheses on the covariance function, the image of Lebesgue measure under Y_n converges a.s. towards a deterministic limit. This allows for instance to extend the convex and monotone rearrangements of univariate processes, used in econometrics to measure inequalities, to higher dimensions. The obtention of a limit deterministic object for each sample-path can be also used in medical imagery, for the estimations of some parameters related to the irregularity and the anisotropy of the field.

Sophie Pénisson (Nancy)

Conditioned multitype branching processes and illustration in epidemiology

In the context of multitype branching processes, we present several kinds of conditioning on non-extinction. We make the connection between these different methods, as well as between the different mathematical objects involved (Feller diffusion and Bienaymé-Galton-Watson processes). In addition, we present an illustration in epidemiological risk analysis, where such conditioned processes enable to study the worst-case scenario.

Francesco Russo (ENSTA Paris-Tech)

Stochastic calculus via regularizations in Banach spaces and applications

See the pdf file.

Henryk Zaehle (Saarbrücken)

Limit theorems and robustness for tail-dependent statistical functionals

In the context of nonparametric statistics, we will address central and noncentral limit theorems, Marcinkiewicz-Zygmund LLNs and qualitative robustness for tail-dependent statistical functionals. These properties will be derived for plug-in estimates based on strictly stationary time series exhibiting weak dependence (e.g., mixing) or strong dependence (long-memory). The key tools are the new concepts of quasi-Hadamard differentiability and quasi-Hölder continuity as well as results on weighted empirical processes. The theoretical results will be illustrated by means of fairly general L- and V-functionals as well as distribution-invariant risk measures.