STOCHASTIC ANALYSIS

Meets Wednesdays, 2 PM in Parker 236 (PLEASE NOTE NEW ROOM)

 

Regular speakers and their current areas of interest:


Olav Kallenberg: stochastic analysis and dynamical systems, with applications to statistical mechanics and biological evolution

Ming Liao: stochastic processes in Lie groups

Erkan Nane: fractional diffusions and iterated processes

Jurek Szulga


2014 - 2015



LAST seminar of semester 

April 29

Speaker: Dr. Ming Liao

Title:  Mathematical finance, part 3

Abstract:  I will just finish the unfinished business from last talk (pricing of stock options and Black-Scholes formula) after a brief review.

 
 
April 22
 

Speaker: Dr. Erkan Nane 

Title: Stochastic Representation of subdiffusion processes with space-time dependent drift.

Abstract: 

In Physics, subdiffusion processes are characterized by certain power-law deviations from the classical Brownian linear time dependence of the mean square displacement. For the mathematical description of subdiffusion, one uses fractional Fokker-Planck equations. In this talk I will introduce a stochastic process, whose probability density function is the solution of the fractional Fokker-Planck equation with space-time dependent drift. This process is obtained by subordinating two-dimensional Langevin equations driven by appropriate Brownian and Levy noises. 

--- I will also talk about some current problems we are working on currently.

 
April 15
 
Speaker: Olav Kallenberg
 
Title: Some martingale results useful in mathematical finance
 
Abstract: Mathematical (or stochastic) finance is an area of applied stochastic processes, based on martingale theory and stochastic calculus. For example, the assumption of no ''arbitrage'' (riskfree gain) is equivalent to the existence of an equivalent probability measure turning the basic risk process into a martingale, whereas ''completeness of the market'' is equivalent to the representation of certain processes as stochastic integrals with respect to the mentioned martingale. In this talk, I shall attempt to survey some of the underlying martingale theory, which is very classical and well-known to any serious student of advanced stochastic processes.
 
 
April 8

Speaker: Dr. Jerzy Szulga

Title: (no title given)

Abstract: I gave a short talk on "operators on toy Fock spaces" at the AMS conference in Huntsville. The topic involves a melange of random chaos (probability and functional analysis), Walsh-Fourier series (dyadic harmonic analysis), and tensor products (multilinear algebra), wrapped around so called ``quantum probability.''

The seminar talk, a sort of a follow-up, might clarify the dry formalism and underline the directions of the research.

 

April 1

Speaker: Dr. Ming Liao

Title:  Stochastic finance, part 2.

Abstract:  After a brief review of part 1 (some basic notions and martingale measure), I will describe the completeness of a market in connection with martingale representation, and the pricing of a stock option, including the Black and Scholes formula.

 

March 18

Speaker: Dr. Erkan Nane

Title: Moment bounds for a class of fractional stochastic heat equations

Abstract: I will give a survey of recent results on the moments of the solution of  a class of stochastic partial differential equations (SPDEs) with noise that is white in time and colored in space. Under suitable assumptions, we show that the second moments of the solution grow exponentially with time. Then I will state some results for time fractional SPDEs. I will  also mention a collection of open problems for  SPDEs. The results are our recent work with Mohammud Foondun.

 

March 11

Speaker: Olav Kallenberg
 
Title: Line processes and particle systems
 
Abstract: Particles moving with constant velocities through space generate straight lines in a space-time diagram. In this way, any random collection of non-interacting particles in $R^d$ is equivalent to a line process in $R^{d+1}$, and either description illuminates the other. In this talk, I will focus on the long-term behavior of stationary particle systems, and related criteria for a stationary line process in space to be mixed Poisson.

 

March 4

Speaker: Dr. Jerzy Szulga

Abstract: Integrals of functions with respect to a stochastic process (a financial instrument) can be viewed as net financial gains under corresponding investments. A reasonable stochastic process (e.g., a Levy process) entails a natural (to be explained) metrizable topological space of integrands. For Brownian motion, it is just a Hilbert space and non-Gaussian stable processes yield L^p-spaces, p<2. In general, Levy process carry associated Orlicz spaces but they are not necessarily Banach (e.g., when the mean is infinite). The lack of local convexity restricts the use of functionals yet there are ways around this obstacle.I will show how the Levy measures determine such spaces, and how their properties (usually the type of growth) determine the properties of the underlying metric vector space, in particular, the local boundedness.

February 25

Speaker: Dr. Erkan Nane

Title: Moment bounds for a class of fractional stochastic heat equations

Abstract: I will give a survey of recent results on the moments of the solution of  a class of stochastic partial differential equations (SPDEs). Under suitable assumptions, we show study that the  exponential growth of second moment of the solution with time. Then I will state some results for time fractional SPDEs. I will  also mention a collection of open problems for  SPDEs. 

February 18 

Speaker: Dr. Ming Liao

Title: Stochastic finance

Abstract: We will describe a stock market model based on semimartingales (as stock prices).  This includes the more traditional model based on stochastic differential equations driven by Brownian motion, but is much more general.  There are two basic theorems.  The first theorem says that the assumption of no arbitrage (that is, no opportunity for riskless profit after discounting interest), is essentially equivalent to the existence of a new probability measure under which the discounted stock price process is a martingale.  This allows the powerful tool of martingales to be applied.  The second theorem deals with the completeness of the stock market, which guarantees that any stock option can be exactly fulfilled by a suitably chosen investment portfolio.  This is a consequence of the martingale representation theorem, and provides a precise way to price stock options, and leads to a rigorous treatment of the famous Black-Scholes formula.

February 11 

Speaker: Olav Kallenberg

Title: Stationary line and flat processes

Abstract: The line and flat processes form a major area of stochastic geometry, where I have been working for years. Here one of the main problems is to give conditions for a stationary process of lines or flats in a Euclidean space to be a Cox process. In higher dimensions this leads to some extremely intricate geometrical problems, intertwined with some equally subtle problems of measure and probability. In this talk I will give a totally elementary introduction to the area, focusing on some more accessible aspects of line processes. 

February 4 

Speaker: Dr. Jerzy Szulga

Title: On capacities

Abstract: The notion was introduced in 1950s by Gustave Choquet as an extension of probability that relaxed its additivity property. It attracted quite a great attention for about 20 years but then the topic per se went dormant. However, in the last few years it came back, has enjoyed a significant revival, documented by numerous new publications with new findings.

 

January 28 

Speaker: Dr. Ming Liao

Title: Limiting properties of Brownian motions in semisimple Lie groups and symmetric spaces

Abstract: We will describe Dynkin's result on limiting properties of Brownian motion in certain symmetric spaces, and extensions to more general processes.

January 21

Speaker: Olav Kallenberg

Title: On the legacy of E.B. Dynkin, 1924-2014

Abstract: The famous probabilist E.B. Dynkin died last November at the age of 90. At the time of his death, he was arguably the greatest probabilist still alive and one of the last representatives of the great pioneering generation. A student of Kolmogorov, he became himself a professor at Moscow University, where he conducted his own seminar and developed systematically the modern theory of Markov processes, as documented by a series of pathbreaking monographs. Many participants in Dynkin's seminar became famous probabilists in their own right. Dynkin was also world famous for his work on Lie algebras. --- Dynkin's father "disappeared" in the Gulag, and his own career was constantly hampered by his Jewish ancestry, until he managed to emigrate to the US in 1977. I was privileged to know Dynkin personally. In this talk I shall comment, in all modesty, on some aspects of Dynkin's life and work.

 

December 3

Speaker: Dr. Ming Liao

Title:  Population genetics and coalescent process, part 3

Abstract: In the previous discussion, we have mainly considered a haploid population.  Roughly speaking, an individual in a haploid population has only one copy of a gene, inherited from its single parent.  It is then relatively easy to trace the ancestors of a sample of genes taken at the present time, and it has been shown that, as the population size tends to infinity, the ancestral process converges in distribution to a continuous time Markov chain, called Kingman's coalescent process.  For a diploid population, each individual has two copies of a gene, one from each of its two parents, so the ancestral process of a gene sample has a quite complicated structure, but by a simple convergence result of Markov chains, it can be shown that the ancestral process still converges in distribution to Kingman's coalescent.

November 19

Speaker: Olav Kallenberg

Title: Inner and outer conditioning in point processes and particle systems

Abstract: A point process $\xi$ on a suitable space $S$ may be regarded as the model of a particle system in space. A basic problem is to study the conditional distribution of $\xi$ in a bounded set $B\subset S$, given the configuration of $\xi$ outside of $B$. In statistical mechanics, such studies go back to the fundamental work of J.W.\ Gibbs (1902), leading up to the theory of Gibbs measures. The merger of the physical and mathematical theories was initiated in the 1970's and 80's. In this talk I shall outline the elements of a general theory of conditioning, involving the notions of Palm and Gibbs kernels, obtained by dual disintegrations of a compound Campbell measure.

November 12

Speaker: Professor Jerzy Szulga

Title: On the semicentennial of Bell’s Inequality

Abstract: In November 1964 John Stewart Bell published a short but influential article “On the Einstein-Podolsky-Rosen Paradox.” It contained a simple probabilistic inequality, showing that the classical probability approach to quantum mechanics fails. I will talk about this and related issues in the context of quantum – or noncommutative probability, into which the classical probability can be embedded in a suitable sense.

 

November 5

Speaker: Dr. Erkan Nane

Title: A Strong Law of Large Numbers with Applications to Self-Similar Stable Processes:

Abstract: In this talk I will prove a strong law of large numbers (SLLN) under general moment condition for sequence of random variables. We  extend the maximal inequalities of Chobanyan, Levental and Salehi (2005), and obtain a general version of SLLN. This SLLN then can be applied conveniently to a wide class of self-similar processes with stationary increments including stable processes. This is a recent  joint work with A. Zeleke and Y. Xiao , Michigan State University

 

October 29

Speaker: Dr. Ming Liao

Title: Population genetics and the coalescent process, part 2

Abstract: After a review of Wright-Fisher model and Kingman's coalescent process, two extensions will be considered.  The first is the inclusion of neutral mutations, and the second is a structured coalescent process taking care of migration between subpopulations.


October 22

Speaker: Olav Kallenberg

Title: Random fractals in stochastic evolution processes

Abstract: We show how random fractals arise naturally in the study of stochastic evolution processes. Informally, a fractal is characterized by the properties of fractional dimension and scaling invariance (even known as self-similarity). A biological population consists of individuals that move around in space, multiply, and eventually die, all inherently random phenomena. In the diffusion limit one gets a measure-valued process, whose associated support process exhibits some basic fractal properties. Though the process itself may be thought of as a randomly evolving diffuse cloud, its associated ancestral process forms a discrete skeleton of coalescing branches.

October 15

Speaker: Dr. Erkan Nane

Title: Correlation structure of time-changed Pearson diffusions

Abstract: The stochastic solution to  diffusion equations with polynomial coefficients is called a Pearson diffusion. If the  time derivative is replaced by a distributed fractional derivative, the stochastic solution is called a fractional Pearson diffusion. We  develop a formula  for the covariance function of a fractional Pearson diffusion in steady state, in terms of generalized Mittag-Leffler functions. That formula shows that fractional Pearson diffusions are long-range dependent, with a correlation that falls off like a power law, whose exponent equals the smallest order of the distributed fractional derivative. This is a recent Joint work with Jebessa Mijena.

 

October 8

Speaker: Dr. Ming Liao

Title:  Markov chains, population genetic models and coalescent process

Abstract:  I will start with a brief review of Markov chains, discrete and continuous times, as covered in our Applied Stochastic Processes course (math7820), and then will mention the Wright-Fisher model both in forward and backward times. The backward model traces the ancestry of a sample in present, and in the limit leads to Kingman's coalescent process. 

October 1

Speaker: Dr. Olav Kallenberg

Title: Some highlights of random measure theory

Abstract: I am currently writing a book on Random Measures, which is due to appear within the next couple of years. In this talk and its possible sequels, I am planning to give an informal introduction to some of the main ideas and results. Though this is one of the fundamental theories underlying the modern theory of stochastic processes, much of this material is relatively unknown, and many results are new. 

 

September 24

Speaker: Dr. Jerzy Szulga

Title: “Quantum probability and Lorentz transformation from an elementary point of view. Part II”

Abstract: Previously, a background was presented (quantum probability, Maxwell equations). Three topics will be addressed (almost no higher mathematics, i.e., beyond undergraduate algebra and calculus):

1.Discontinuity of the eigensystem of a Maxwell matrix.

2. Surjectivity of the exponential map, Maxwell -> Lorentz.

3. Pauli coding, sliders and jaw operators.

 

September 17

Speaker: Dr. Jerzy Szulga

Title: Quantum probability and Lorentz transformation from an elementary point of view

Abstract: I will refresh the framework of quantum probability, which essentially adapts the theory of subspaces of the space of linear continuous operators on a Hilbert space.  Then I will discuss some aspects of the Lorentz group and show how to derive several of its properties. These properties are well known but usually they  are presented within a quite advanced setup of differential geometry and tensor theory. In fact, at least some of these results can be obtained just by using undergraduate  linear algebra and calculus, and not even tediously.

 

 

September 10 

Speaker: Dr. Erkan Nane

Topic:  Intermittence and  time fractional stochastic partial differential equations.

Abstract: In this talk, I consider time fractional stochastic heat type equations. The time fractional stochastic heat type equations might be used to model phenomenon with random effects with thermal memory. In this talk I discuss: (i) existence and uniqueness of solutions and existence of a continuous version of the solution; (ii) absolute moments of the solutions of this equation grow exponentially; and (iii) the distances to the origin of the farthest high peaks of those moments grow exactly linearly with time.  These results extend the results of Mohammud Foondun and Davar Khoshnevisan, (Intermittence and nonlinear parabolic stochastic partial differential equations, Electron. J. Probab. 14 (2009), no. 21, 548--568) and Daniel Conus and Khoshnevisan (On the existence and position of the farthest peaks of a family of stochastic heat and wave equations, Probab. Theory Related Fields 152 (2012), no. 3-4, 681--701)  on the parabolic stochastic heat equations. --- This is a recent joint work with Jebessa B Mijena.


September 3 

Speaker: Dr. Ming Liao

Title:  Independent increments and semimartingale property on Lie groups

Abstract:  On a Euclidean space, it is well known that a process with independent increments is a semimartingale if and only if its (deterministic) drift has a finite variation.  The same holds on a Lie group.  Various representations of a process with independent increments in a Euclidean space are not affected by whether it is a semimartingale, but the proofs of these formulae may be easier in the semimartingale case.  However, such a process in a Lie group has a simpler and more direct martingale representation when it is a semimartingale.


August 27

Speaker: Olav Kallenberg

Topic: Tangential existence and comparison

Abstract: By a fundamental theorem of Jacod, a semimartingale X has independent increments iff its local characteristics are a.s. nonrandom. This seems to suggest that we may reduce the study of X to the independence case by a simple conditioning. Though this doesn't work in general, we can always construct an approximating tangential process with conditionallly independent increments. In this talk, I shall try to explain and elaborate on these ideas.

 

Last updated: 04/27/2015