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# Stochastic Analysis

DMS Stochastics Seminar: Tianxiao Wang
Sep 27, 2017 01:00 PM
Parker Hall 246

Speaker: Tianxiao Wang, Sichuan University

Title: Optimal control theories of stochastic Volterra equations

Abstract: In this talk we will show some recent progress on the optimal control theories of stochastic Volterra integral equations (SVIEs). Unlike the differential systems, Volterra integral systems inherently lose the semi-group property, which makes some basic tools, say, change-of-variable formula, become no longer powerful here. Here we develop new tricks and techniques to treat the stochastic linear quadratic problems, the maximum principles of optimal controls, and other related problems.

Host: Yongsheng Han

DMS Stochastics Seminar
Aug 30, 2017 01:00 PM
Parker Hall 246

Speaker: Ms. Xiangqian Meng

Title:  Space-time fractional stochastic partial differential equations with Lévy Noise

Abstract:  We consider non-linear time-fractional  stochastic heat type equation with Poisson random measure or  compensated Poisson random measure. We prove existence and uniqueness of mild solutions to this equation. Our results  extend the results in the case of parabolic stochastic partial differential equations obtained before. Under some mild assumptions we show that solution of the time fractional stochastic partial differential equation grows  exponentially with respect to the time. We also establish the non-existence of the random field solution of both  of these stochastic partial differential equations under faster than linear growth of the drift coefficient.

Stochastics Seminar
Apr 27, 2017 01:00 PM
Parker Hall 224

Speaker: Dr. Ming Liao

Title:  Convolution semigroups of probability measures on compact Lie groups (continued)

Abstract: We present an asymptotic condition on the Levy measure of a convolution semigroup to ensure it has a smooth density.

This will be the last seminar of the semester.

Stochastics Seminar
Apr 20, 2017 01:00 PM
Parker Hall 224

Speaker: Dr. Jerzy Szulga​

Title: Poisson chaos in Wiener's famous 1938 paper

Abstract: Using the intuition derived from Gibbs' statistical mechanics, Norbert Wiener introduced what we today call an abstract Poisson process'' or a Poisson random measure.'' In fact, his analytic approach was quite concrete, down-to-earth. The construction will be presented in the seminar. In a sense, the topic complements Olav's talk a month ago.

Stochastics Seminar
Apr 13, 2017 01:00 PM
Parker Hall 224

Speaker: Olav Kallenberg

Title: Point process entropy and local energy

Abstract: My aim is to highlight some connections between stochastic processes, information theory, and statistical mechanics. After reviewing the classical Shannon--McMillan theorem, I will discuss the difficulties encountered in continuous time and/or state space, and indicate how they can be overcome. Then I will introduce the entropy of a point process, and explain in what sense the Poisson process may be regarded as a process with maximum entropy. If time permits, I may finally introduce the notion of Gibbs kernel, formalizing the notion of external conditioning, and give a precise meaning to the notion of local energy. (My discussion will mostly be elementary, and no special knowledge of physics is required.)

POSTPONED

Speaker: Dr. Jerzy Szulga

Title and abstract: TBA

Stochastics Seminar
Apr 06, 2017 01:00 PM
Parker Hall 224

Speaker: Dr. Ming Liao

Title:  Convolution semigroups of probability measures on compact Lie groups

Abstract: As mentioned in the previous talk, if a convolution semigroup has an $$L^2$$ density at time $$t > 0$$, then the density is continuous and converges to 1 uniformly as $$t$$ goes to infinity. In this talk, some useful conditions ensuring the existence of an $$L^2$$ density are provided.

Stochastics Seminar
Mar 30, 2017 01:00 PM
Parker Hall 224

Speaker: Yinan Ni (student of Dr. Erkan Nane)

Title:  Path stability of the solution of a stochastic differential equation driven by a time-changed Lévy noise

Abstract: This talk is based on a paper where I am studying the path stabilities of solutions to stochastic differential equations (SDE) driven by time-changed L´evy noise. Conditions are given for the solution of a time-changed SDE to be path stable and exponentially path stable. Moreover, we reveal the important role of the time drift in determining the path stability properties of the solution. Related examples are provided.

Stochastics Seminar
Mar 23, 2017 01:00 PM
Parker Hall 224

Speaker: Olav Kallenberg

Title: Some early Poisson discoveries---the untold story.

Abstract: Though the Poisson process is arguably as basic as Brownian motion, its theory and early history are virtually unknown by the general probabilistic community, beyond the most basic facts. The story of its discovery goes back to none other than Sir Isaac Newton (and in a way even to Darwin), just as Einstein godfathered the birth of Brownian motion. I will comment on its early use in the contexts of telecommunication, insurance mathematics, and particle physics, and then highlight the roles of Lévy, Itô, and Wiener, before moving on to some more recent discoveries.

Stochastics Seminar
Mar 09, 2017 01:00 PM
Parker Hall 224

Speaker: Dr. Erkan Nane

Title: Some non-existence results for a class of stochastic partial differential equations

Abstract: I will consider the parabolic type stochastic partial differential equations with fractional Laplacian, which is the generator of a symmetric stable L'evy process. Under some conditions, namely faster than linear growth condition of a function and the initial condition, I will show non-existence of global random field solutions. These results are new and complement those of P. Chow, among others:

P.-L. Chow. Unbounded positive solutions of nonlinear parabolic Ito equations. Commun.  Stoch. Anal., 3(2) (2009), 211—222.

P.-L. Chow. Explosive solutions of stochastic reaction-diffusion equations in mean  $$l_{p}$$-norm.  J. Differential Equations, 250(5) (2011), 2567—2580.

Stochastics Seminar
Mar 02, 2017 01:00 PM
Parker Hall 224

Speaker: Dr. Jerzy Szulga

Title: Reversal of classical inequalities and topologies in some functional Banach spaces

Abstract: A little 1960s book Some Random Series of Functions by Jean-Pierre Kahane ignited a new mathematical discipline in the intersection of probability and functional analysis - probability theory on Banach spaces. Of course, the factual origins can be traced many decades back.

A classical problem: a Fourier series is transformed by inserting arbitrary signs at the coefficients. Does this Hilbertian isometry extend to Lp-spaces for p distinct from 2? Can the signs be arbitrary, or are they subject to some constraints? Yes, they are: almost all but not all signs are admissible. This is one of many ways random series in a Banach space arise. Random signs can be further extended to random multipliers, and so probability on Banach spaces is born.

In the talk I will address a quite interesting category of random spaces where the classical rules are reversed. For example, Jensen's or Chebyshev's inequalities will hold in reverse, entailing equivalences of vector topologies that normally are strictly distinct.

Last Updated: 09/11/2015