Probability, Fall 2009
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Schedule
Tuesday, September 1, 2009,
No Probability Seminar
Tuesday, September 8, 2009, 10:30 AM in REC 307
Professor Mateusz Kwasnicki, Wroclaw University of Technology, Poland
Half-Laplacian on the half-line
The eigenvalues of Dirichlet square root of the Laplacian on a half-line are single. The eigenfunctions and the corresponding semigroup can be represented by fairly explicit formulas obtained by involved calculations on the complex-plane.
Tuesday, September 15, 2009, 03:30 PM in REC 113
Lucian Beznea, Institute of Mathematics of the Romanian Academy, Bucharest, Romania
Tightness of capacities, compact superharmonic functions, and path regularity
We discuss the relations between the existence of the L-superharmonic functions that have compact level sets (L being the generator of a right Markov process), the path regularity of the process, and the tightness of the induced capacities. We present several examples, mainly in infinite dimensional situations, like the case when L is the Gross-Laplace operator on an abstract Wiener space. We deduce the còadlòag property of the paths of a class of measure-valued branching process associated with nonlinear operators of the form Lu+ ö(u), where ö is "branching mechanism".
The talk includes results from joint works with Nicu Boboc and Michael Röckner.
Tuesday, September 15, 2009, 10:30 AM in REC 307
Professor Bartek Siudeja, University of Illinois at Urbana Champaign
Sharp bounds for Neumann eigenvalues of triangles
Among all triangles with fixed diameter, we prove the degenerate acute isosceles triangle minimizes the Neumann fundamental tone. In the other direction, if we fix perimeter (or area) then the equilateral triangle maximizes the Neumann fundamental tone. Our approach involves variational principles and geometric transformations of the domain. We also prove symmetry/antisymmetry for eigenfunctions of isosceles triangles.
Tuesday, September 22, 2009, 10:30 AM in REC 307
Cheng Ouyang, Golomb Assistant Professor of Mathematics, Purdue University
Near expiry asymptotics of the implied volatility in local volatility models and stochastic volatility models
Using the heat kernel expansion technique, we give the first term in the asymptotics of European call option prices with respect to the time to the expiry T. We use this formula to calculate both the leading value of the implied volatility σI and the first order deviation of σI from its leading value. Some geometric interpretations will be discussed for these two terms. In particular, the leading value of the implied volatility could be interpreted as the Riemannian distance under the metric determined by the equation satisfied by the stock price S.
A quick survey of the background of the problem including some back-ground knowledge in mathematical finance will be given at the beginning of the talk.
Tuesday, September 29, 2009, 10:30 AM in UNUSUAL PLACE: HAAS 111
Professor Dr. Michael Röckner, Professor, Fakultät für Mathematik, Universität Bielefeld, Germany and Adjunct Professor, Department of Statistics, Purdue University
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, October 6, 2009,
No Probability Seminar
Tuesday, October 13, 2009,
No Probability Seminar
October Break
Tuesday, October 20, 2009, 10:30 AM in REC 307
Frederi Viens, Professor of Statistics and Mathematics, Purdue University
Extensions of the Nourdin-Peccati Analysis on Wiener space
Ivan Nourdin and Giovanni Peccati have recently established results in stochastic analysis by which, for a scalar random variable X which is differentiable in the sense of Malliavin calculus (|DX| is square-integrable), an important quantity to study is the random variable G=<DX,-DMX>, where M is the pseudo-inverse of the so-called Ornstein-Uhlenbeck semigroup generator on Wiener space. For instance G is a random way of measuring the dispersion of X since Var[X]=E[G]; plus, G is constant if and only if X is Gaussian; and comparing G to a constant can yield comparisons of X to the Gaussian law, as in limit theorems or density formulas. We will review such results, their extensions to vector-valued X's, and to iterations of the map taking X to G. These are collected in papers by Airault, Malliavin, and the speaker; Nourdin and Peccati; Nourdin, Peccati, and Reveillac; Nourdin and the speaker; and a single-authored paper by the speaker.
Tuesday, October 27, 2009, 10:30 AM in REC307
Professor Frederi Viens, Department of Statistics, Purdue University
Extensions of the Nourdin-Peccati Analysis on Wiener space, Part II
Continuation of last week's talk. Ivan Nourdin and Giovanni Peccati have recently established results in stochastic analysis by which, for a scalar random variable X which is differentiable in the sense of Malliavin calculus (|DX| is square-integrable), an important quantity to study is the random variable G=<DX,-DMX>, where M is the pseudo-inverse of the so-called Ornstein-Uhlenbeck semigroup generator on Wiener space. For instance G is a random way of measuring the dispersion of X since Var[X]=E[G]; plus, G is constant if and only if X is Gaussian; and comparing G to a constant can yield comparisons of X to the Gaussian law, as in limit theorems or density formulas. We will review such results, their extensions to vector-valued X's, and to iterations of the map taking X to G. These are collected in papers by Airault, Malliavin, and the speaker; Nourdin and Peccati; Nourdin, Peccati, and Reveillac; Nourdin and the speaker; and a single-authored paper by the speaker.
Tuesday, November 3, 2009, 10:30 AM in REC 307
Panki Kim, Department of Mathematical Sciences, Seoul National University (Korea) Visiting the Department of Mathematics, University of Illinois at Urbana-Champaign
The boundary Harnack principle of the independent sum of a Brownian motion and a symmetric stable process
In this talk, we consider the family of pseudo differential operators {Δ + b Δα/2; b∈ [0, 1]}$ that evolves continuously from Δ to Δ+ Δα2. We establish a uniform boundary Harnack principle with explicit boundary decay rate for nonnegative functions which are harmonic with respect to Δ +b = Δα/2 (or equivalently, the sum of a Brownian motion and an independent symmetric α-stable process with constant multiple b1/α) in C1, 1 open sets.
For a pdf version, please click here.
Tuesday, November 10, 2009, 10:30 AM in REC 307
Krzysztof Bogdan, Visiting Professor of Mathematics, Purdue University, Permanent Position at Wroclaw University of Technology, Poland
Heat Kernel Estimates for the Fractional Laplacian
Tuesday, November 17, 2009,
No Probability Seminar
Tuesday, November 24, 2009,
No Probability Seminar
Tuesday, December 1, 2009, 10:30 AM in REC 307
Aaron Nung Kwan Yip, Professor of Mathematics, Purdue University
TBA
Tuesday, December 8, 2009, 10:30 AM in REC 307
Professor Parthanil Roy, Department of Statistics and Probability, Michigan State University
Ergodic Properties of Stable random Fields
We establish characterization results for the ergodicity of symmetric α-stable stationary random fields. We first show that the result of Samorodnitsky (2005) remains valid in the multiparameter setting, i.e., a stationary SαS (0<α<2) random field is ergodic (or equivalently, weakly mixing) if and only if it is generated by a null group action. By establishing multiparameter versions of Stochastic and Birkhoff Ergodic Theorems, we give a criterion for ergodicity of these random fields which is valid for all dimensions and new even in the one-dimensional case. The similarity of the spectral representations for sum- and max-stable random fields yields parallel characterization results in the max-stable setting.
This talk is based on a joint work with Yizao Wang and Stilian A. Stoev.
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