PL2US 2014

- Registration is closed. The list of participants can be found here

- The conference will take place in the seminar room of the Institut Elie Cartan (see here for the direction in french). Here are some explanations on how to reach the conference room from the central railway station.

- The conference dinner will be held at the restaurant L’Excelsior (50 rue Henri Poincaré, Nancy) on Thursday evening and is offered to all the participants. Note that the restaurant is located in a street which is less than 2 minutes walk from the central railway station.

- Here are a few hotels suggested by the local organizers:

- Hotel Jean Jaurès (website)

- Hotel Coeur de City (website)

- Ibis Hotel (website)

- Hotel Best Western (website)

- Hotel des Prélats (website)

- Hotel La Résidence (website)

- The poster of the workshop can be downloaded here.

New! Some pictures of the conference dinner (taken by Rola Zintout)

Click on it to enlarge!

Program

Talks are 30 minutes long (except for the guest speaker) + 5 minutes for questions.

January 30th

15:00 A welcome coffee break will be served in the 2nd floor of the Institut Elie Cartan de Lorraine

16:00 Céline Lacaux and Ivan Nourdin (Université de Lorraine): ``Introduction to the workshop''

16:10 Guillaume Poly (Luxembourg University): ``Stein's method and iterated logarithm''

16:45 Florian Jahnert (University of Kaiserslautern): ``An infinite dimensional analysis with respect to non-Gaussian measures of Mittag-Leffler type''

17:20 Coffee break

17:50 Carlos Vargas (Saarland University): ``Relations between cumulants in non-commutative probability''

18:25 End of the first day, followed by a dinner in L’Excelsior (50 rue Henri Poincaré, Nancy) all together

January 31st

9:30 [Guest Speaker] Matthias Reitzner (University of Osnabrück): ``Poisson U-statistics: Limit theorems and applications''

10:20 Laura Vinckenbosch (Université de Lorraine): ``Monte Carlo methods for light propagation in biological tissues''

10:55 Coffee break

11:30 Erwan Hillion (Luxembourg University): ``Benamou-Brenier curves on graphs''

12:05 Robert Voßhall (University of Kaiserslautern ) : ``Ergodicity and L^p-strong Feller properties of a sticky reflected distorted Brownian motion''

12:40 Lunch

14:00 Christian Bender (Saarland University): ``A first-order backward SPDE for swing option pricing''

14:35 René Schott (Université de Lorraine): ``Some Applications of Dynamic Random Walks in Computer Science and Quantum Probability''

15:10 End of the workshop

Abstracts

Christian Bender (Saarland University): ``A first-order backward SPDE for swing option pricing''

Abstract: We study an optimal control problem related to swing option pricing in a general non-Markovian setting in continuous time. As a main result we uniquely characterize the value process in terms of a first-order non-linear backward stochastic partial differential equation and a (pathwise) differential inclusion. Based on this result we also determine the set of optimal controls. When the payoff of the option is leftcontinuous in expectation we finally represent the derivative of the value process in the space variable via some non-standard optimal stopping problems over appropriate subsets of predictable stopping times. This talk is based on joint work with Nikolai Dokuchaev (Curtin University, Perth).

Florian Jahnert (University of Kaiserslautern): ``An infinite dimensional analysis with respect to non-Gaussian measures of Mittag-Leffler type''

Abstract: We consider certain non-Gaussian measures on an infinite dimensional space defined via Mittag-Leffler functions. The fundamental role of Brownian motion in Gaussian analysis is taken over by the so-called Grey Brownian motion (gBm) -- a self-similar stochastic process with stationary increments, which possesses fractional Brownian motion as a special case. It turns out that the well-known Wick ordered polynomials in Gaussian analysis cannot be generalized to the non-Gaussian case. Using a system of biorthogonal polynomials, called Appell system, we are able to introduce and characterize spaces of test functions and distributions and we can construct grey Donsker's Delta (composition of Dirac delta with gBM). Applications to the time-fractional heat equation are presented. Based on joint work with Martin Grothaus and Felix Riemann.

Erwan Hillion (Luxembourg University): ``Benamou-Brenier curves on graphs''

Abstract: Given two probability distributions $f_0$, $f_1$ on a graph, we construct an interpolating curve $(f_t)_{t \in [0,1]}$ and sharing some properties with Wasserstein geodesics on continuous spaces. We then study the entropy $H(t)$ of $f_t$ as a function of $t$ and show that the convexity properties of this function are linked with the geometry of the underlying graph.

Guillaume Poly (Luxembourg University): ``Stein's method and iterated logarithm''

Abstract: By using Stein's method, we provide a general method enabling to prove the law of the iterated logarithm for a sequence of random variables converging to a Gaussian distribution. This methods applies to functional of Gaussian fields and generalizes previous works of Arcones and Ho. Applications will be provided to p-variation of fBm (including critical case), iterated logarithm for fBm, Polyà urn, U-statistics, etc. Based on joint work with E. Azmoodeh and G. Peccati.

Matthias Reitzner (University of Osnabrück): ``Poisson U-statistics: Limit theorems and applications''

Abstract: Assume that $\eta$ is a Poisson point process. A Poisson U-statistic with kernel $f$ is the sum of $f(x_1, \dots, x_k)$ over all $k$-tuples of $\eta$. Poisson U-statistics play an important role in Stochastic Geometry. Many functionals of interest can be written as U-statistics. We investigate some elementary properties of Poisson U-statistics, and the limit behaviour of Poisson U-statistics, central limit theorems and concentration inequalities.

René Schott (Université de Lorraine): ``Some Applications of Dynamic Random Walks in Computer Science and Quantum Probability"

Abstract: We consider a model of random walks where the transition probabilities are time-dependent. We state some limit theorems (central limit theorem and large deviation principle) for these random walks and show their usefulness in the probabilistic analysis of distributed algorithms and dynamic data structures as well as in quantum probability (as illustration, we will focus on dynamic random walks on Heisenberg groups and dynamic quantum Bernoulli random walks). Based on joint works with Nadine Guillotin-Plantard.

Carlos Vargas (Saarland University): ``Relations between cumulants in non-commutative probability''

Abstract: Cumulants are helpful to understand independence as they are additive with respect to the convolution of measures. The connection between moments and cumulants can be described using set partitions. In non-commutative probability there exist several notions of independence: tensor (classical), free, Boolean and monotone. The corresponding cumulants can be defined for each case by considering special subsets of partitions. Lehner, Belinschi and Nica, and Josuat-Vergues have obtained some formulas relating some types of cumulants in terms of others. In this talk we complete the picture obtaining (almost all) relations between different cumulants. During the process, the coefficients we obtain come from various interesting combinatorial objects. This is joint work with Arizmendi, Lehner and Hasebe.

Laura Vinckenbosh (Université de Lorraine): ``Monte Carlo methods for light propagation in biological tissues"

Abstract: Light propagation in turbid media is driven by the equation of radiative transfer. We give a formal probabilistic representation of its solution in the framework of biological tissues and we implement algorithms based on Monte Carlo methods in order to estimate the quantity of light that is received by an homogeneous tissue when emitted by an optic fiber. A variance reduction method is studied and implemented, as well as a Markov chain Monte Carlo method based on the Metropolis-Hastings algorithm. The resulting estimating methods are then compared to the so-called Wang-Prahl (or Wang) method. Finally, the formal representation allows to derive a non-linear optimization algorithm close to Levenberg-Marquardt that is used for the estimation of the scattering and absorption coefficients of the tissue from measurements.

Robert Voßhall (University of Kaiserslautern ) : ``Ergodicity and L^p-strong Feller properties of a sticky reflected distorted Brownian motion''

Abstract: We consider a distorted Browian motion in $[0,\infty)^n$, where the behavior on the boundary is determined by the competing effects of reflection from and pinning at the boundary. Construction of the process takes place via a Dirichlet form approach on a suitable $L^2$-space. First, we are concerned with the ergodicity of the constructed process. In particular, a sticky boundary behavior of the process is proved, i.e., the occupation time on specified parts of the boundary is positive for large times. Secondly, we present first results regarding the regularity of the associated $L^p$-resolvent of the underlying Dirichlet form. The resolvent maps $\mathcal{L}^p$-functions to locally weakly differentiable and locally bounded functions which solve some elliptic PDE (in a weak sense) with Wentzell boundary condition. Our studies are motivated by and can be applied to the dynamical wetting model (also known as Ginzburg-Landau dynamics) on a bounded subset of $\mathbb{Z}^d$, $d \geq 1$. Based on joint work with Torben Fattler and Martin Grothaus.