Stochastic Analysis at Purdue '09 Workshop

Schedule of Lectures

Tuesday September 29

09:30 am to 10:15 am in HAAS 111: Francesco Russo
10:30 am to 11:15 am in HAAS 111: Michael Röckner
11:30 am to 12:15 pm in HAAS 111: Carlo Marinelli
12:30 pm to 02:30 pm Lunch Break
02:30 pm to 03:15 pm in HAAS 111: Jason Swanson
03:30 pm to 04:15 pm in HAAS 111: Alexandra Chronopoulou

Wednesday September 30

09:30 am to 10:15 am in room LWSN 1142: David Nualart
10:30 am to 11:15 am in room LWSN 1142: Giovanni Peccati
11:30 am to 12:15 pm in room LWSN 1142: Ivan Nourdin

Thursday October 1

09:30 am to 10:15 am in room LWSN 1142: José E. Figueroa-lopez
10:30 am to 11:15 am in room LWSN 1142: Ionuţ Florescu
11:30 am to 12:15 pm in room LWSN 1142: Fabrice Baudoin

Location

All talks on the campus of Purdue University, W. Lafayette, IN.

HAAS is the Department of Statistics building, at the Southeast corner of University Street and 3rd Street.

LWSN (Lawson) is the Computer Science building, at the Northwest corner of University Street and 3rd Street.

See the Purdue Campus map, with directions and parking info.

Abstracts

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.

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 $q>2$ , $Z^{(q,H)}$ 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).

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.

Ionuţ 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.

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 $L_{1}$ spaces with respect to an invariant measure.

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).

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.

Giovanni Peccati, Centre de Recherche Modal'x, Université Paris Ouest Nanterre and Laboratoire de Statistique Théorique et Appliquée, 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).

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.

Francesco Russo, Institut Galilée, Mathématiques, Université Paris 13, and Projet MATHFI, INRIA Rocquencourt & Cermics Ecole des Ponts, France,
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.

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, $B_{j}(t)$ , where $B_{j}(0)$ has a rapidly decreasing, smooth density function $f$ . The empirical quantiles, or pointwise order statistics, are denoted by $B_{j:n}(t)$ , and we are interested in a sequence of quantiles Q_n(t) = B_j(n):n(t), where j(n)/n — α ε (0,1). This sequence converges in probability in $C[0,\infty )$ to $q(t)$ , the $\alpha $ -quantile of the law of $B_{j}(t)$ . Our main result establishes the convergence in law in $C[0,\infty )$ of the fluctuation processes F_n = n^1/2(Q_n — 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 $B^{1/4}$ , 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 $\gamma <1/4$, and (at least locally) it has increments whose correlation is negative and of the same order of magnitude as those of $B^{1/4}$ .

Lectures open to the public free of charge, no advanced registration required.

Contact Professor Frederi Viens for more information: viens@purdue.edu.