Seminar on Probability Theory - summer term 2013

Time: Tuesday, 17:00-18:30
Room: Seminarraum SR12 (Mathematik), Oskar-Morgenstern-Platz 1
Organizer: Jiří Černý
First meeting: Tuesday, 8 October 2013, 17:00-18:00

Seminar will have two parts:

Working group on the topics Markov processes and related functional inequalities
Research talks on various topics that will (mostly) take place on Thursday, jointly with the Seminar on Mathematical Finance

Date:	Speaker:	Title :
Tue, 8 October 17:00-18:00	Jiří Černý	Working group: "Markov processes and related functional inequalities" - Introduction 1

Tue, 15 October 17:00-18:30	Jiří Černý	Working group: "Markov processes and related functional inequalities" - Introduction 2

Tue, 22 October 17:00-18:00	Patrick Streif (Master-Student Mathematics Vienna)	The Connection of Dual and Primal Solutions to Expected Utility Optimization with Transaction Costs in BMS-Setting
	Abstract: Roughly speaking expected utility optimization is a search for the trading strategy expecting the biggest return in a mathematically definedfinancial market based on stochastic processes. In contrast to classical model where trading was free of charge in recent time this theory was also developed with transactions accumulating proportional transaction costs, where there is more mathematical theory involved. Through some considerations and technical transformations it is possible to connect knowledge of the recent results to the primal ones and in particular have an explicit representation of the optimal discounted value function describing the long-time behavior of the optimal strategy.

Tue, 5 November 17:00-18:00	Wolfgang Woess (TU Graz)	Stochastic dynamical systems with weak contractivity properties
	Abstract: (collaboration with Marc Peigne', Tours) Consider a proper metric space X and a sequence (F_n) of i.i.d. random continuous mappings of X onto itself. It induces the stochastic dynamical system (SDS) X_n^x = F_n(X_{n-1}^x)) starting at X_0 = x in X. We study existence and uniqueness of invariant measures, as well as recurrence and ergodicity of this process. In the first part, we elaborate, improve and complete the unpublished work of Martin Benda on local contractivity, which merits publicity and provides an important tool for studying stochastic iterations. In the second part, we consider the case where the F_n are Lipschitz mappings. The main results concern the critical case when the associated Lipschitz constants are log-centered. Prinicpal tools are the Chacon-Ornstein theorem and a hyperbolic extension of the space X as well as of the process (X_n^x). An example where the results apply is the reflected affine stochastic recursion given by X_n^x = \|A_nX_{n-1}^x - B_n\|, where (A_n,B_n) is a sequence of two-dimensional i.i.d. real random variables with A_n > 0.

Tue, 12 November 17:00-18:30	Tobias Wassmer	Working group: "Markov processes and related functional inequalities" - Poincaré inequality

Tue, 19 November 17:00-18:30	Miryana Grigorova	Choquet integrals, stochastic dominance with respect to a capacity and risk measures
	Abstract: Please click on the above title for the PDF

Thu, 9 January 17:00-18:30	Sebastian Andres (University of Bonn)	Invariance Principle for the Random Conductance Model in a degenerate ergodic environment
	Abstract: In this talk we consider a continuous time random walk X on Z^d in an environment of random conductances taking values in [0,∞). We will discuss recent results on a quenched functional central limit theorem for this random walk. Assuming that the law of the conductances is i.i.d. or - more general - stationary ergodic with respect to space shifts, we present such an invariance principle for X under some moment conditions on the environment. Under the same conditions we also obtain a local limit theorem. This is joint work with J.-D. Deuschel and M. Slowik.

Tue, 14 January 17:00-18:30	Alexander Drewitz (Columbia)	Asymptotics of the critical parameter for level set percolation of the Gaussian free field
	Abstract: We consider the Gaussian free field in Z^d, d≥3. It is known that there exists a non-trivial phase transition for its level set percolation; i.e., there exists a critical parameter h_(d)⋲ [0,∞) such that for h < h_(d) the excursion set above level h does have a unique infinite connected component, whereas for h > h_(d) it consists of finite connected components only. We investigate the asymptotic behavior of h_(d) as d→∞ and give some ideas on the proof of this asymptotics. (Joint work with P.-F. Rodriguez)

Tue, 28 January 17:00-18:30	Walter Schachermayer	Working group: "Markov processes and related functional inequalities"