Gemeinsames Kolloquium der Arbeitsgruppen Stochastik

TU Darmstadt / Goethe-Universität Frankfurt / Gutenberg-Universität Mainz

Termine im Wintersemester 2018/2019

Freitag 16.11.2018, Universität Mainz, Institut für Mathematik, Hilbertraum (Raum 05-432), Staudingerweg 9, 55128 Mainz

15:15 Prof. Alexander Schnurr (Siegen): The fourth characteristic of a semimartingale

We extend the class of semimartingales in a natural way. This allows us to incorporate processes having paths that leave the state space R^d. By carefully distinguishing between two killing states, we are able to introduce a fourth semimartingale characteristic which generalizes the fourth part of the Levy quadruple. Using the probabilistic symbol, we analyze the close relationship between the generators of certain Markov processes with killing and their (now four) semimartingale characteristics.

16:45 Prof. Markus Bibinger (Marburg): Statistical analysis of path properties of volatility

In this talk, we review recent contributions on statistical theory to infer path properties of volatility. The interest is in the latent volatility of an Itô semimartingale, the latter being discretely observed over a fixed time horizon.

We consider tests to discriminate continuous paths from paths with volatility jumps. Both a local test for jumps at specified times and a global test for jumps over the whole observation interval are discussed.

We establish consistency and optimality properties under infill asymptotics, also for observations with additional additive noise.

Recently, there is high interest in the smoothness regularity of the volatility process as conflicting models are proposed in the literature.

To address this point, we consider inference on the Hurst exponent of fractional stochastic volatility processes. Even though the regularity of the volatility determines optimal spot volatility estimation methods, forecasting techniques and the volatility persistence, its identifiability is one of the few unsolved questions in high-frequency statistics. We discuss a first approach which can reveal if path properties are stable over time or changing. Eventually, we discuss some recent considerations and conjectures on this open question.

Freitag, 07.12.2018, TU Darmstadt, Physik Institut S2|07 Raum 167, Hochschulstraße 6, 64289 Darmstadt

15:15 Jakob Björnberg (Universität Göteborg): Random permutations and the Heisenberg model

Abstract:We discuss models for random permutations which are closely linked to quantum spin systems from statistical physics.

The cycle structure of the random permutations is intimately connected with the correlation structure in the spin-system, and it is expected that this cycle structure converges to a distribution known as Poisson--Dirichlet, in the limit of large systems. This problem is still open but we present some partial progress.

16:45 Dr. Piotr Miłoś (Universität Warschau):Phase transition for the interchange and quantum Heisenberg models on the Hamming graph

Abstract: In my talk I will present a family of random permutation models on the 2-dimensional Hamming graph H(2,n), containing the interchange process and the cycle-weighted interchange process with parameter θ>0. This family contains the random representation of the quantum Heisenberg ferromagnet. The main result is that in these models the cycle structure of permutations undergoes a phase transition -- when the number of transpositions defining the permutation is <cn^2, for small enough c>0, all cycles are microscopic, while for more than >Cn^2 transpositions, for large enough C>0, macroscopic cycles emerge with high probability. For the quantum Heisenberg ferromagnet on H(2,n) this implies that for low enough temperatures spontaneous magnetization occurs, while it is not the case for high temperatures. At the core of our approach is a novel application of the cyclic random walk, which might be of independent interest. By analyzing explorations of the cyclic random walk, we show that sufficiently long cycles of a random permutation are uniformly spread on the graph, which makes it possible to compare our models to the mean-field case, i.e., the interchange process on the complete graph, extending the approach used earlier by Schramm (joint work with Radosław Adamczak, Michał Kotowski).

Freitag 25.01.2019, Universität Frankfurt,Goethe-Universität Frankfurt,

Institut für Mathematik, Robert-Mayer-Str. 10, 60486 Frankfurt

Raum 711 (groß), 7. Stock

15:15 Uhr: Antti Knowles (Universität Genf)
Eigenvalues and eigenvectors of supercritical Erdos-Renyi graphs

Abstract: I review some recent results on Erdos-Renyi graphs G(N,p) near and above the critical scale pN = log N, where the graph undergoes a connectivity crossover. For pN >> log N, the graph G(N,p) is with high probability connected, while for pN << log N it has with high probability isolated vertices. In the supercritical regime pN >> log N, the eigenvalues stick to the bulk spectrum, a local law holds down to optimal scales, and the eigenvectors are completely delocalized. All three statements are false in the subcritical regime pN << log N. Based on joint work with F. Benaych-Georges, C. Bordenave, Y. He, and M. Marcozzi.

16:15 Uhr: Kaffee und Tee

16:45 Uhr: Aernout van Enter (Universität Groningen)
One-sided versus two-sided stochastic descriptions

Abstract: Stochastic systems can be parametrised by time (like Markov chains), in which conditioning is one-sided(the past) or by one-dimensional space (like Markov fields), where conditioning is two-sided (right and left). I will discuss some examples, in particular generalising this to g-measures versus Gibbs measures, when the two descriptions are the same and when they are different. We show the role one-dimensional entropic repulsion plays in this setting. Joint work with R. Bissacot, E. Endo and A. Le Ny

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