PL2US 2012

Program

Talks are 45 minutes long + 5 minutes for questions.

March 1st (Room 3.03, in the third floor of the "Bâtiment des Sciences").

14:30 Welcome buffet

16:00 Giovanni Peccati and Anton Thalmaier (Luxembourg University): ``Introduction to the workshop''

16:10 Philippe Chassaing (Nancy University): ``Cellular Automata & Branching ballistic annihilation''

17:00 Coffee break

17:30 Renaud Marty (Nancy University): ``Time-splitting for a random nonlinear Schrodinger equation''

18:20 Wolfgang Bock (Kaiserslautern University): ``Hamiltonian path integrands as distributions of white noise''

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

March 2nd (Room 2.01, in the second floor of the "Bâtiment des Sciences").

9:00 Guest Speaker: Peter Imkeller (Humboldt University at Berlin): ``A Fourier approach of Young integration''

9:50 Andreas Neuenkirch (Kaiserslautern University): ``Strong Approximation of Stochastic Differential Equations under Non-Lipschitz Assumptions''

10:40 Coffee break

11:10 Jessica Steiner (Saarbrücken University): ``Enhancing least squares Monte Carlo for BSDEs by choosing martingale basis functions''

12:00 Roland Speicher (Saarbrücken University) : ``Asymptotic eigenvalue distribution of random matrices and free stochastic analysis''

12:50 Lunch

14:00 Simon Campese (Luxembourg University): ``Optimal Rates of Convergence and One-Term Generalized Edgeworth Expansions for Multidimensional Functions of Gaussian Fields''

14:50 Robert Philipowski (Luxembourg University): ``Ricci flow, coupling of Brownian motions and Perelman L-functional''

15:40 End of the workshop

Abstracts

Wolfgang Bock (Kaiserslautern University)

Hamiltonian path integrands as distributions of white noise

Feynman integrals were introduced as an alternative approach to quantum mechanics and are used in many branches of theoretical physics. In this talk concepts for the construction of the Feynman integrand in phase space (also called the Hamiltonian integrand) will be discussed. Additionally ideas of an extension of the complex scaling (of Doss type) approach known for classical Feynman integrals to phase space are presented. The latter gives hope to enlarge the class of admissible potentials beyond perturbation theory.

Simon Campese (Luxembourg University)

Optimal Rates of Convergence and One-Term Generalized Edgeworth Expansions for Multidimensional Functions of Gaussian Fields

We will derive exact Berry-Esseen type asymptotics for the multidimensional normal approximation of functions of Gaussian fields with fluctuating covariance structures. In a second part, these results will be used to show that a generalized one-term Edgeworth expansion can yield faster rates of convergence.

Philippe Chassaing (Nancy University)

Cellular Automata & Branching ballistic annihilation

We exhibit a Probabilistic Cellular Automaton (PCA) on the integers with an alphabet and a neighborhood of size 2 which is non-ergodic although it has a unique invariant measure. This answers by the negative an old open question on whether uniqueness of the invariant measure implies ergodicity for a PCA. Along the way, Catalan numbers and vicious walkers occurs.

Peter Imkeller (Humboldt University at Berlin)

A Fourier approach of Young integration

In 1961, Ciesielski established a remarkable isomorphism of spaces of Hölder continuous functions and Banach spaces of real valued sequences. This isomorphism leads to wavelet decompositions of Gaussian processes giving access for instance to a precise study of their large deviations, as shown by Baldi and Roynette. We will use Schauder representations for a pathwise approach of Young integrals, using Ciesielski's isomorphism. This talk is based on work in progress with N. Perkowski (HU Berlin).

Renaud Marty (Nancy University)

Time-splitting for a random nonlinear Schrodinger equation

In this talk we consider a stochastic nonlinear Schrodinger (SNLS) equation. In the deterministic case it is usual to solve numerically nonlinear evolution equations by using time-splitting schemes. Here we analyze a time-splitting scheme to solve SNLS equation. More precisely we establish the order of convergence of the scheme.

Andreas Neuenkirch (Kaiserslautern University)

Strong Approximation of Stochastic Differential Equations under Non-Lipschitz Assumptions

The traditional convergence analysis for the strong approximation of stochastic differential equations (SDEs) relies on the global Lipschitz assumption. However, in applications as molecular dynamics or mathematical finance, this assumption is often violated, so the standard results can not be applied here. A prototype example is the Heston model, which is a popular stochastic volatility model in mathematical finance and contains square-root coefficients.

Starting with a seminal work by Higham, Mao and Stuart in 2002, the strong approximation of SDEs with non-globally Lipschitz coefficients has been an active field of research in the last years. In this talk, I will give an overview of the recent developments, which include the divergence of the Euler method for SDEs with superlinear coefficients and the derivation of strong convergence rates for discretization schemes for the Heston model.

Robert Philipowski (Luxembourg University)

Ricci flow, coupling of Brownian motions and Perelman L-functional

In this talk I will show that on a manifold whose Riemannian metric evolves under backwards Ricci flow two Brownian motions can be coupled in such a way that their normalized L-distance is a supermartingale. As a corollary, one obtains a new proof and a generalization of a recent result of Peter Topping concerning L-optimal transport. This is joint work with Kazumasa Kuwada.

Roland Speicher (Saarbrücken University)

Asymptotic eigenvalue distribution of random matrices and free stochastic analysis

I will address some questions about the asymptotic eigenvalue distribution of polynomials in independent Gaussian random matrices and will present the basics of free stochastic analysis, which might in the long run help to answer those questions.

Jessica Steiner (Saarbrücken University)

Enhancing least squares Monte Carlo for BSDEs by choosing martingale basis functions

Many option pricing and portfolio selection problems in mathematical finance can be reformulated in terms of backward SDEs (BSDEs). As the corresponding BSDE can rarely be solved in closed form, simulation of BSDEs is of prime importance. In our talk we first give a short review on the least squares Monte Carlo (LSMC) approach suggested by Gobet, Lemor and Warin (Ann. Appl. Prob. 15, 2005). Then, we suggest the choice of basis functions that form a system of martingales, and explain how the LSMC scheme can be enhanced by exploiting the martingale property. We also compare the convergence behaviour of both the original and the simplified LSMC approach and illustrate the advantages of our modified LSMC scheme by a numerical example on option pricing problems under different interest rates for borrowing and lending. We also derive a-posteriori estimates for the L2-error between a generic numerical solution and the true solution, which can be computed in terms of the numerical solution only. These estimates are applied for the comparison of the numerical solutions arising from the original and the modified LSMC approach. This talk is based on joint work with Christian Bender.