Probability and Statistics Seminar at Boston University
Seminar on Stable Processes

Day:	Thursdays and/or Tuesdays	Click here for directions to the Boston University Department of Mathematics. Campus text map, gif map , and a general area map.
Time:	10am-noon
Place:	Room 135, Department of Mathematics, 111 Cummington St., Boston University

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Schedule for 2001-2002 (updated weekly)

Go to the most recent scheduled talk.

This is a research oriented seminar, coordinated by Professor Murad Taqqu. The main theme, this fall semester will be

In a series of talks, we will explore connections between stable non-Gaussian processes and non-singular flows. In the first talks, we will introduce stable processes and we will focus, in particular, on various properties of their integral representations. The following talks will be on non-singular flows and will cover their representations and decompositions. With the necessary background in hand, we will then make preliminary connections between stable processes and non-singular flows. In particular, we will go over the fundamental work of Rosinski where stationary stable processes are shown to be determined by non-singular flows and then are decomposed according to the nature of the underlying flow. We will then focus on stable processes which are self-similar and also have stationary increments. We will show how connections to non-singular flows can be used to decompose an important subclass of such processes called self-similar mixed moving averages.

Other topics, however, may also sometimes be covered particularly when there is an outside speaker.

Last year we talked about Risk analysis and Heavy tails. Click here to see last year's lisk of talks.

This seminar has now become a regular feature of Boston University and is also attended by mathematicians, scientists and postdoctoral fellows in the greater Boston area. Announcements will be done by email and through my Web Page:

This semester the talks will usually be on Thursdays (10-12). The first talk will be on September 20, 2001. That day we will have an outside speaker, Professor Youri Davydov, from the University of Lille, France.

Thursday, September 20, 2001:

On the estimation of the parameters of multivariate stable distributions

Youri Davydov (Universite of Lille, France)

The dependence structure of a multivariate stable distribution is characterized by a spectral measure. We present a new estimator for the spectral measure and show that it is asymptotically normal. We also introduce an estimator for the density of a multivariate stable distribution and study its properties. The stability exponent $\alpha$ plays an important role. We shall discuss the dependence of the stable density on the exponent \alpha and the spectral measure.

If there is time, we will also consider the role played by extreme summands when a sum of i.i.d. random vectors is asymptotically \alpha-stable, focusing on what happens these extreme summands are removed.

Thursday, September 27, 2001:

No talk because of Yom Kippur

Thursday, October 4, 2001:

Properties of Levy Flights on an Interval with Absorbing Boundaries

Sergey Buldyrev (Boston University)

We consider a L\'evy flyer of order $\alpha$ that starts from a point $x_0$ on an interval $[O,L]$ with absorbing boundaries. We find a closed-form expression for an arbitrary average quantity, characterizing the trajectory of the flyer, such as mean first passage time, average total path length, probability to be absorbed by one of the boundaries. Using fractional differential equations with a Riesz kernel, we find exact analytical expressions for these quantities in the continuous limit. We find numerically the eigenfunctions and the eigenvalues of these equations. We study how the results of Monte-Carlo simulations of the L\'evy flights with different flight length distributions converge to the continuous approximations. We show that if $x_0$ is placed in the vicinity of absorbing boundaries, the average total path length has a minimum near $\alpha=1$, corresponding to the Cauchy distribution. We discuss the relevance of these results to the problem of biological foraging and transmission of light through cloudy atmosphere.

Thursday, October 11, 2001:

Stable non-Gaussian processes and their connections to non-singular flows I

Vladas Pipiras (Boston University)

In this first talk, we will introduce stable processes and their integral representations. If there is time, we will start introducing minimal integral representations.

Thursday, October 18, 2001:

Stable non-Gaussian processes and their connections to non-singular flows II

Vladas Pipiras (Boston University)

In this second talk, we will further discuss minimal integral representations of stable processes. If there is time, we will start speaking about rigidity property of integral representations of stable processes.

Thursday, October 25, 2001:

Stable non-Gaussian processes and their connections to non-singular flows III

Vladas Pipiras (Boston University)

We will continue to discuss rigidity properties of integral representations of stable processes. Then, we will start with a short introduction to flows and some of their functionals.

No talk on Thursday, November 1, 2001

Thursday, November 8, 2001:

Stable non-Gaussian processes and their connections to non-singular flows IV

Vladas Pipiras (Boston University)

We will go over the fundamental work of Jan Rosinski where stationary stable processes are shown to be determined by non-singular flows and then are decomposed according to the nature of the underlying flow.

Thursday, November 15, 2001:

Stable non-Gaussian processes and their connections to non-singular flows V

Vladas Pipiras (Boston University)

We will further explore connections between stationary stable processes and flows. Then, we will discuss extensions to self-similar stable processes and to stationary increments stable processes.

Friday , November 16, 2001:

Dependence of Extreme Events in Financial Markets I

Brendan Bradley (Boston University)

We give an overview of the problem.

Tuesday , November 20, 2001:

Dependence of Extreme Events in Financial Markets II

Brendan Bradley (Boston University)

We review Extreme values in 1 dimension.

Thursday, November 29, 2001:

Stable non-Gaussian processes and their connections to non-singular flows VI

Vladas Pipiras (Boston University)

We will talk about stationary stable processes which are harmonizable. We will draw their connections to flows and show how to identify them in stationary stable processes.

Thursday, December 6, 2001:

Stable non-Gaussian processes and their connections to non-singular flows VII

Vladas Pipiras (Boston University)

We will focus on ergodic properties of stationary stable processes and explore their relationship with properties of the underlying flow.

Friday, December 7, 2001:

Dependence of Extreme Events in Financial Markets III

Brendan Bradley (Boston University)

We introduce Poisson point processes and draw their relantionship to extremes.

Tuesday, December 11, 2001:

Dependence of Extreme Events in Financial Markets IV

Brendan Bradley (Boston University)

We consider multivariate extremes.

Wednesday, December 12, 2001:

Statistical Learning Theory and Model Selection for Bagging

Massimiliano Ponti (University of Siena, Italy)

We begin with an overview of statistical learning theory. This field consists of a number of approaches to the classical learning problem of neural network theory, in which examples of an input-output function are given, and the entire input-output function must be constructed (learned). The technique of "bagging" has been a popular meta-approach to a number of learning problems. We give an introduction to this approach. We then present a simple extension of bagging which consists of averaging a number of learning machines, such as so-called support vector machines, each trained on a possibly small bootsrapped set of examples. This method has two main advantages:

(a) It provides a feasible way for training a learning algorithm on large training sets, and

(b) it can stabilize the underlying learning algorithm.

We present a theoretical analysis of bagging based on the framework of Stability and Generalization of Bousquet and Elisseeff. Within this framework, we can derive new bounds on the distance between empirical and expected error for bagging. This leads to the implication that for bagging using small sub-samples, the empirical error of a bagging combination of a machine can be a good indicator of its predictive performance. It is therefore possible to use the empirical error of these combinations of machines for model selection. We report experiments on a number of datasets using support vector machines. The results validate our theoretical framework and confirm that bagging by random sub-samples is an appropriate technique for improving stability of learning machines.

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SPRING SEMESTER 2002

Thursday, January 31, 2002:

Stable non-Gaussian processes and their connections to non-singular flows

Vladas Pipiras (Boston University)

The probability seminars are resuming.

We will continue exploring connections between stable non-Gaussian processes and non-singular flows started in the fall semester. This semester, the focus will be on stable self-similar processes with stationary increments. We will show how connections to flows can be used to decompose an important subclass of such processes called self-similar mixed moving averages. We start with a review.

In this first talk of the semester, we will recall the basic ideas behind connections of stable stationary processes and flows. We will do so by applying these ideas to stable self-similar processes and to stable stationary increments processes.

TUESDAY, February 5, 2002:

The Weierstrass function and Fractional Brownian motion

Murad Taqqu (Boston University)

Benoit Mandelbrot suggested that Weierstrass' nowhere differentiable function can be modified and randomized so as to approximate fractional Brownian motion, which is a Gaussian self-similar process whose paths are almost surely non-differentiable. The randomization involves introducing independent and identically distributed random variables with finite variance in the definition of the Weierstrass function. We will show how one then obtains fractional Brownian motion in the limit.

We will also describe various modifications and indicate what happens if, for example, one introduces in the above randomization strongly dependent random variables instead of independent ones or if one uses infinite variance random variables instead of finite variance ones.

Joint work with Vladas Pipiras.

TUESDAY, February 12, 2002:

Szego theorem on Toeplitz determinants and asymptotic behavior of prediction error for stationary processes

Mamikon Ginovyan (Northeastern University, Institute of Mathematics of Armenian Academy of Sciences)

We discuss G. Szego's well-known "weak" theorem on asymptotic behavior of Toeplitz determinants D_n(f) generated by a nonnegative integrable function f(t). We describe classes of generating functions f(t) having singularities, for which the remainder term is O(n^a) or o(n^a), where a is in (0,1).

We also consider the problem of asymptotic behavior of the linear prediction error for discrete-time stationary process with a spectral density function possessing singularities. We describe the rate of decrease to zero of the prediction error, and show that for b b in (0,1/4), the estimates O(n^{-b}) or o(n^{-b}) are possible for sufficiently broad classes of spectral densities under certain restrictions on the types of their zeros.

TUESDAY, February 26, 2002:

Linear Stochastic Equations with Anticipative Initial Data and Asset Prices with Correlated Gains and Heavy Tails

Andrew Lyasoff (Boston University)

The now famous Black and Scholes option pricing formula derives from a model in which the logarithm of the stock price follows a Brownian motion process plus a deterministic linear function of time. In such a model the current stock price and the future gains from investing in the stock are independent random variables and, consequently, no reason to buy or sell the stock could be found in observing present and past values of the stock. In practice, however, stocks are bought and sold almost exclusively as a result of a careful or, sometimes, not so careful examination of their current prices and price histories. This talk is about a work a considerable part of which was done jointly with D. W. Stroock from M.I.T. that grew out of the desire to develop Financial Calculus similar to Black and Scholes, but in which stock prices and future gains are not independent, i.e. stock prices do reflect some knowledge about future gains. This investigation naturally lead to a model in which the stock price solves a linear stochastic equation of Skorohod type, i.e. an equation in which the Ito stochastic integral is replaced by an integral whose integrand anticipates the integrator (= Brownian Motion). Equivalently, this could be treated as a standard Ito equation in which the initial data depends on the entire future path of the driving Brownian motion. Such objects have been studied well before Mathematical Finance was en vogue in connection with the so called anticipative stochastic calculus (often referred to as Malliavin's calculus), but it is the financial interpretation (our main object of interest) that leads to closed form solutions which seem impossible to guess by other means.

It has been recognized for quite some time that the empirical distributions of stock prices exhibit much heavier tails than those prescribed by the geometric Brownian motion (the B&S model). Somewhat surprisingly, the model involving stochastic equations of the Skorohod type does exhibit heavy tails. Phenomena like the so called "smile" are commonly accepted as further evidence of inconsistencies in the Black-Scholes model. Roughly speaking, the "smile" phenomenon suggests that the actual distribution of the stock price is considerably thinner around 0 than the distribution prescribed by the standard (Ito type) geometric Brownian motion. It appears that the geometric Brownian motion of Skorohod type leads to a probability distribution which is also "thin" around 0.

The talk will present some new results in the mathematical theory of Skorohod stochastic equations and will discuss its implications to finance in issues like arbitrage and control over interest rates.

TUESDAY, March 19, 2002:

Asymptotically efficient estimation of spectral density functionals

Mamikon Ginovyan (Northeastern University, Institute of Mathematics of Armenian Academy of Sciences)

We consider the problem of construction of asymptotically efficient estimators for linear and some nonlinear smooth spectral functionals. We describe classes of spectral densities having singularities for which the statistic L(I_T), where I_T is the periodogram of the underlying stationary Gaussian process X(t) with an unknown spectral density f(u), is IK-asymptotically efficient (or locally asymptotically minimax) estimator for a linear functional L(f).

We show that for a nonlinear smooth functional \Phi(f) a IK-asymptotically efficient estimator is the statistic \Phi(f_T), where f_T is a suitable sequence of T^{1/2}-consistent estimators of unknown spectral density f(u). We also present exact asymptotic bounds for the minimax mean square risks of the estimators.

Tuesday, April 2, 2002:

Stable non-Gaussian processes and their connections to non-singular flows IX

Vladas Pipiras (Boston University)

We start a classification of an important subclass of stable self-similar processes with stationary increments, called self-similar mixed moving averages. We relate the stable self-similar mixed moving averages to flows and show that they can be decomposed into processes related to dissipative flows and processes related to conservative flows.

Tuesday, April 9, 2002:

A central limit theorem for wavelet-based estimators of multifractal spectra: the fractional Brownian motion case I

Carlos Morales (Boston University)

Multifractal processes have become a successful, widely used modeling tool with applications in areas such as turbulence, Internet traffic and biomedical engineering. Their appeal as a modeling tool stems from the fact that these processes exhibit intricate patterns of locally varying scaling behavior similar to that encountered in real data-sets. A multifractal spectrum (MFS) of a process quantifies the presence of the possibly multiple local scaling behavior from sample paths of these processes. The wavelet transform (WT) has been successfully employed in the estimation of the MFS. In this talk we explore statistical properties of estimators of MF spectra base on the discrete WT for the case of fractional Brownian motion (fBm). By developing appropriate tools to study the across-scale covariance structure of functions of wavelet coefficients of fBm, we show the asymptotic normality of new and standard wavelet-based estimators.

Tuesday, April 16, 2002:

A central limit theorem for wavelet-based estimators of multifractal spectra: the fractional Brownian motion case II

Carlos Morales (Boston University)

Continuation of the previous week's talk.

Tuesday, April 23, 2002:

Stable non-Gaussian processes and their connections to non-singular flows X

Vladas Pipiras (Boston University)

We study stable self-similar mixed moving averages related to dissipative flows. We show that these processes have a canonical representation, provide examples, discuss related spaces of integrands and also examine uniqueness properties.

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