Events, Fall 2009

Schedule

Sunday, August 2, 2009, 06:30 PM
Public Event, Purdue University Friends and Alumni Reception

Tuesday, September 29, 2009,
Stochastic Analysis at Purdue 2009 Workshop

Tuesday, September 29, 2009, 11:30 AM in HAAS 111
Carlo Marinelli, Institute for Applied Mathematics, Universität Bonn, Germany
Ergodicity for nonlinear stochastic evolution equations with multiplicative Poisson noise

We study the asymptotic behavior of solutions to stochastic evolution equations with monotone drift and multiplicative Poisson noise in the variational setting, thus covering a large class of (fully) nonlinear partial differential equations perturbed by jump noise. In particular, we provide sufficient conditions for the existence, ergodicity, and uniqueness of invariant measures. Furthermore, under mild additional assumptions, we prove that the Kolmogorov equation associated to the stochastic equation with additive noise is solvable in L1 spaces with respect to an invariant measure.

Tuesday, September 29, 2009, 02:30 PM in HAAS 111
Jason Swanson, Department of Mathematics, University of Central Florida
Fluctuations of the empirical quantiles of independent Brownian motions

We consider n independent, identically distributed one-dimensional Brownian motions, Bj(t), where Bj(0)has a rapidly decreasing, smooth density function f. The empirical quantiles, or pointwise order statistics, are denoted by Bj:n(t), and we are interested in a sequence of quantiles Qn (t) = Bj(n):n , (t) where j(n)|n → ∈ (0,1). This sequence converges in probability in C [0,∞) to q(t), the α-quantile of the law of Bj (t). Our main result establishes the convergence in law in C [0,∞) of the fluctuation processes Fn = n1/2 (Qn - q). The limit process F is a centered Gaussian process and we derive an explicit formula for its covariance function. We also show that F has many of the same local properties as B1/4, the fractional Brownian motion with Hurst parameter H = 1/4. For example, it is a quartic variation process, it has Hölder continuous paths with any exponent γ > 1/4, and (at least locally) it has increments whose correlation is negative and of the same order of magnitude as those of B1/4.

Tuesday, September 29, 2009, 03:30 PM in HAAS 111
Alexandra Chronopoulou, Department of Statistics, Purdue University
Variations and Hurst Index Estimation for non-Gaussian Hermite processes

Using multiple stochastic integrals and the Malliavin calculus, we analyze the asymptotic behavior of quadratic variations for a class of non-Gaussian self-similar processes, the Hermite processes (Z(q,H), q > 1). The process Z(q,H) has stationary, H-self-similar increments that exhibit long-memory, identical to that of the fractional Brownian motion (fBm). For q = 1, Z(1,H), is fBm, which is Gaussian; for q = 2, Z(2,H), is the Rosenblatt process, which lives in the second Wiener chaos; for any q > Z(q,H), is a process in the qth Wiener chaos. We prove a reproduction property for this class of processes in the sense that the terms appearing in the chaotic decomposition of their variations give rise to other Hermite processes of different orders and with different Hurst parameters. We also study the behavior of the variations of the Roseblatt process using longer filters. We apply our results to construct a strongly consistent estimator for the self-similarity parameter H from discrete observations of the process. The asymptotic distribution of the estimator depends explicitly on the order and the length of the filter. We compare the numerical values of the asymptotic variances for various choices of filters, including finite-difference and wavelet-based filters. This is joint work with Ciprian Tudor (Sorbonne I) and Frederi Viens (Purdue University).

Tuesday, September 29, 2009, 10:30 AM in HAAS 111
Michael Röckner, Fakultät für Mathematik, Universität Bielefeld, Germany
Fokker--Planck equations on Hilbert spaces

We consider a stochastic differential equation in Hilbert space with time dependent coefficients for which no general existence and uniqueness results are known. We prove, under suitable assumptions, existence and uniqueness of a measure valued solution, for the corresponding Fokker-Planck equation. In particular, we verify the Chapman-Kolmogorov equations and get an evolution system of transition probabilities for the stochastic dynamics informally given by the stochastic differential equation.

Tuesday, September 29, 2009, 09:30 AM in HAAS 111
Francesco Russo, Institut Galilée,Mathématiques, Université Paris 13, and Projet MATHFI, INRIA Rocquencourt & Cermics Ecole des Ponts
Probabilistic representation of a partial differential equation with monotone discontinuous coefficients and related fields

We consider a partial differential equation over the the real line with monotone discontinuous coefficients and prove a probabilistic representation of its solution in terms of an associated microscopic diffusion. We will distinguish between two different situations: the so-called non-degenerate and degenerate cases. In the first case we show existence and uniqueness, however in the second one for which we only show existence. Some comments about an associated stochastic PDE with multiplicative noise will be provided. This talk is based on two joint papers: the first with Ph. Blanchard and M. Röckner, the second one with V. Barbu and M. Röckner.

Wednesday, September 30, 2009, 09:30 AM in LWSN 1142
David Nualart, Department of Mathematics, University of Kansas
Central limit theorem for the modulus of continuity of the Brownian local time

In this talk we present a proof of the central limit theorem for the modulus of continuity of the Brownian local time based on the Clark-Ocone stochastic integral representation formula and an asymptotic version of Knight's theorem. We plan also to discuss the application of the techniques of Malliavin calculus to derive central limit theorems for Skorohod stochastic integrals.

Wednesday, September 30, 2009, 10:30 AM in LWSN 1142
Giovanni Peccati, Université Paris Ouest Nanterre and Université Paris 6, France
Stein's method meets Malliavin calculus: from Berry-Esseen to universality

We discuss some applications of Malliavin calculus and Stein's method to the derivation of explicit bounds in limit theorems involving functionals of a general Gaussian field. Some applications to isotropic spherical fields are described. A universality result is also presented: this is a crucial tool in the CLT for spectral measures of non-Hermitian random matrix ensembles discussed in Nourdin's talk. Based on joint works with I. Nourdin (Paris 6) and G. Reinert (Oxford).

Wednesday, September 30, 2009, 11:30 AM in LWSN 1142
Ivan Nourdin, Laboratoire de Probabilités, Université Paris 6, France
Universal Gaussian fluctuations of non-Hermitian matrix ensembles

I will explain how to prove multi-dimensional central limit theorems for the spectral moments (of arbitrary degrees) associated with random matrices with real-valued i.i.d. entries, satisfying some appropriate moment conditions. The used techniques rely on a universality principle for the Gaussian Wiener chaos as well as some combinatorial estimates. Unlike other related results in the probabilistic literature, the fact that the law of the entries has a density with respect to the Lebesgue measure is not required. This talk is based on a joint work with Giovanni Peccati (Paris Ouest), and use an invariance principle obtained in a joint work with G. P. and Gesine Reinert (Oxford).

Wednesday, September 30, 2009,
Stochastic Analysis at Purdue 2009 Workshop

Thursday, October 1, 2009,
Stochastic Analysis at Purdue 2009 Workshop

Thursday, October 1, 2009, 10:30 AM in LWSN 1142
Ionut Florescu, Department of Mathematical Sciences, Stevens Institute of Technology
A study of an integro-differential parabolic problem arising in Mathematics of Finance

In Finance one of the most studied problems is pricing options when the underlying equity follows a stochastic process. If the underlying process is a regular diffusion the problem is reduced to solving a Partial Differential Equation. However, if the underlying process possesses jumps (or more general a Lévy component) an integral term arises in the defining equation due to the associated Levy measure. This produces the so called Partial Integro-Differential Equations. Problems of existence, uniqueness and determination of solutions for such equations are still open. In this talk I will present a proof of existence on general domains under suitable conditions on the integral operator. The proof is based on the method of upper and lower solutions and also provides an algorithm to approximate the solution. The work is based on the collaboration with Prof. Maria C. Mariani from University of Texas at El Paso

Thursday, October 1, 2009, 11:30 AM in LWSN 1142
Fabrice Baudoin, Department of Mathematics, Purdue University
Generalized Bochner formulas and subelliptic heat kernels estimates

We will prove generalized Bochner formulas for some subelliptic Hormander's type operators. As a consequence, we shall derive Li-Yau type estimates for the corresponding semigroup and heat kernels Gaussian bounds.

Thursday, October 1, 2009, 09:30 AM in LWSN 1142
José E. Figueroa-Lopez, Department of Statistics, Purdue University
Optimal portfolios and admissible strategies in a Lévy market

In this talk, we give characterizations for the dual solution of Merton's portfolio optimization problem in a non-Markovian market driven by a Lévy process. Our approach is based on a multiplicative optional decomposition for nonnegative supermartingales due to F"ollmer and Kramkov as well as a closure property for integrals with respect to a fixed Poisson random measure. Under certain constraints on the jumps of the price process, we characterize explicitly the admissible trading strategies and show that the dual solution is a risk-neutral local martingale.

Wednesday, October 28, 2009, 04:30 PM in LWSN 1142
Douglas Nychka, Director of the Institute for Mathematics Applied to Geosciences (IMAGe) and Senior Scientist at National Center for Atmospheric Research (NCAR)
Joint Department of Statistics and Earth and Atmospheric Sciences Special Colloquium
Climate Past, Present and Future

A grand scientific challenge for this century is to understand the complex interrelationships among the physical processes, and human activities that define the Earth's climate. One specific concern is the warming of our climate brought about the increase of greenhouse gases, such as carbon dioxide, being released into the atmosphere. What do we know about the Earth's past climate and is the global warming over the last century real? What is a climate model and how is it used to understand changes to climate for the future? For each of these questions statistical science can play a role in quantifying the uncertainty in scientific conclusions, for combining different kinds of information and summarizing complex data sets.

Biography

Dr. Doug Nychka is currently the Director of the Institute for Mathematics Applied to Geosciences (IMAGe) and also a Senior Scientist at National Center for Atmospheric Research (NCAR). He received his PhD from the University of Wisconsin in 1983 and then spent 14 years as a professor in the Statistics Department at North Carolina State University before he moved to NCAR in 1997. Dr. Nychka's research areas cover spline, inverse problems, spatial statistics, Bayesian methods, nonparametric regression, statistical computing, and spatial designs. He made outstanding contributions to the statistical sciences, both theory and practice, atmospheric science, climatology, environmetrics and the geosciences. Dr. Nychka is a leading expert on interdisciplinary research. That work has covered many aspects of atmospheric sciences from ocean winds to dispersion of pollutants, from design of monitoring networks to data assimilation in precipitation models, and from assessment of model fits to global warming. As director of IMAGe, Dr. Nychka is also heavily invested in mentoring younger scientists, including those who come to NCAR as postdoctoral researchers.