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 2000-2001 (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

Since the tremendous growth of the derivatives market in the early 1970s financial institutions have dedicated themselves to the study and quantification of risk. It is now, however, well known that the assumption of independent normally distributed (log) returns, consistent with the Black-Scholes model, is problematic for quantifying risk. Even for a well diversified portfolio such as the Standard & Poors 500, it can be seen that returns are not independent (volatility clustering) and that rare events happen more often than they should under the assumption of normality, i.e. returns are heavy tailed. For example any Value at Risk calculation under the assumption of normality will underrate the maximum loss (over a given horizon) at a given confidence level leaving the institution open to more frequent large unexpected losses. In an effort to better model financial risk the mathematical, actuarial and financial communities have joined forces. The goal of this series of talks is to present more adequate ways to analyse risk.

Click here for the outline in postscript format. and here for the outline in acrobat pdf format. Other topics, however, may also sometimes be covered particularly when there is an outside speaker.

Last year we talked about self-similarity and applications to telecommunications. 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 Tuesdays (10-12). The first talk will be on September 12, 2000. That day we will have an outside speaker, Professor Herold Dehling from the Netherlands. His talk is not about finance, but about dynamical systems and correlation dimensions. The first talk on Financial Risk Analysis will be on September 19.

Tuesday, September 12, 2000:

From dimension estimation to asymptotics of dependent U-statistics

Herold Dehling (University of Groningen, The Netherlands)

Motivated by statistical issues concerning the estimation of correlation dimensions in dynamical systems, we study the asymptotic distribution of U-statistics with dependent observations. Special attention will be given to functionals of absolutely regular processes as these arise naturally in dynamical systems.

Tuesday, September 19, 2000:

Financial risk analysis I

Brendan Bradley (Boston University)

To gain a better understanding of financial risk analysis, we start with a brief historical review starting with the work of Markowitz, factor models, the Capital Asset Pricing Model (CAPM), and the Arbitrage Pricing Theory (APT).

Friday, September 22, 2000 at 2pm:

Spectral estimation of local power law processes

George Papanicolaou (Stanford University)

I will describe a method for estimating the spectrum of a process when it is approximately a variable power law. I will apply the method to atmospheric temperature data and describe a Matlab-based collection of software tools for implementing the estimation method.

To reach Papanicolaou's home page and on-line publications click here.

Papanicolaou will also be speaking at the BU Mathematical Finance Day 2000 here in Boston on September 24.

The talk will be held on September 22, 2000 at 2:00 P.M., at Boston University, Center for Adaptive Systems and Department of Cognitive and Neural Systems, 677 Beacon Street, Room B03. This talk is jointly sponsored with the Center for Adaptive Systems

Saturday , September 23 and Sunday September 24, 2000 :

BU Mathematical Finance Day 2000

Program in acrobat format . For more information BU Mathematical Finance Day 2000

Tuesday, September 26, 2000:

Financial risk analysis II

Brendan Bradley (Boston University)

We will continue developing Mean Variance portfolio theory,and discuss the benefits of diversification.

Tuesday, October 3, 2000:

Financial risk analysis III

Brendan Bradley (Boston University)

We discuss the CAPM, APT and factor models and their implications for the decomposition of risk.

Tuesday, October 10, 2000:

Financial risk analysis IV

Brendan Bradley (Boston University)

We will start our investigation of tail related risk measures. Value at Risk (VaR), and its different implementations will be discussed.

Tuesday, October 24, 2000:

Financial risk analysis V

Brendan Bradley (Boston University)

In an effort to calculate Value at Risk we will discuss modeling of the return distribution. Several models including that of RiskMetrics, the ARCH/GARCH type and jump-diffusion models will be discussed.

Tuesday, October 31, 2000:

Financial risk analysis VI

Brendan Bradley (Boston University)

After discussing several measures of risk and problems surrounding their calculation we backtrack and discuss the validity of these measures. Specifically we define what is meant by a coherent measure of risk and examine its implications on commonly used measures of risk.

Tuesday, November 7, 2000:

Financial risk analysis VII

Brendan Bradley (Boston University)

In order to quantify the risk associated with a financial portfolio, the full joint distribution of the assets in the portfolio must be specified. For example, Markowitz assumed the the joint distribution of assets was multivariate normal and the dependence structure was therefore given by the covariance matrix. We will examine the modelling of multivariate risks beyond the multivariate normality assumption.

Tuesday, November 14, 2000:

Financial risk analysis VIII

Brendan Bradley (Boston University)

We will continue to investigate alternative measures of dependence. In addition we will look at some of the properties of spherical and elliptical distributions and show that within this framework linear correlation is a natural measure of dependence.

Tuesday, November 28, 2000:

Financial risk analysis IX

Brendan Bradley (Boston University)

We will discuss some of the implications of elliptical distributions on risk management. Time permitting we will also discuss some of the issues involved with picking a copula.

Tuesday, December 5, 2000:

Financial risk analysis X

Brendan Bradley (Boston University)

We will examine the estimation of a copula for real financial data. Implications for risk management will be discussed.

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SPRING SEMESTER 2001

Tuesday, January 23, 2001

A Multiresolution Analysis for Likelihoods

Eric Kolaczyk (Boston University)

Much of the recent work in adaptive, nonlinear statistical methods employs a paradigm that begins with the traditional ``signal plus noise'' model and then seeks an alternative representation with respect to an orthonormal basis or redundant expansion, appropriate for the assumed underlying structure -- Fourier and wavelet-based methods, of course, being canonical examples. However, in many nonparametric statistical problems the ``signal plus noise'' model can be less applicable or even inappropriate, yet a more general likelihood-based model may suggest itself in a natural manner. In such cases, the concept of an orthogonal basis decomposition may be replaced by analogy with a factorization of the data likelihood.

I will present a theory for a certain class of multiscale likelihood factorizations i.e., with structure capturing similarly localized position/scale information as do wavelet expansions. The results are formulated in analogy to the standard multiresolution analysis (MRA) associated with orthonormal wavelet bases, and hence may be viewed as an MRA for likelihoods. The conditions underlying this MRA can be used to characterize families of distributions whose members permit a multiscale likelihood factorization, as I will illustrate. Additionally, I will show how certain fundamental connections between these factorizations, recursive partitioning and a class of generalized Haar bases, can be exploited to enable a general class of nonparametric, complexity penalized likelihood problems to be solved using analogues of the best-subset and best-basis methods in the literature on wavelets. If time permits, I will illustrate the performance of such methods with a problem of segmentation for Poisson time series satellite data in high-energy astrophysics.

Tuesday, January 30, 2001

A Multiresolution Analysis for Likelihoods, part 2

Eric Kolaczyk (Boston University)

I will continue our discussion of multiresolution analysis for likelihoods, first reviewing the multiscale factorizations derived in part I, and then showing how certain fundamental connections between these factorizations, recursive partitioning and a class of generalized Haar bases, can be exploited to enable a general class of nonparametric, complexity penalized likelihood problems to be solved using analogues of the best-subset and best-basis methods in the literature on wavelets. I'll then finish up with illustration of the resulting tools and techniques to applications in astronomy and geography, as time permits.

Tuesday, February 6, 2001

Decomposition of Distributions characterized by Slow Variation

Takaaki SHIMURA (Boston University)

The usual central limit theorem provides conditions for normalized sums of independent random variables with finite variance to converge to a normal distribution. If the variance is infinite, the limit is not necessarily a normal distribution, it may be a stable distribution. However, one can also get a normal limit if the summands have infinite variance, provided that the truncated variance is slowly varying, e.g. as log x. In the first talk, we will discuss properties of the domain of attraction of a normal distribution, that is, the class of distributions with slowly varying truncated variances, as related to convolution.

We will focus on conditions under which, a random variables with slowly varying truncated moments can be expressed:

(1) as a sum of two independent random variables with the same property.

(2) as a product of two independent random variables with the same property.

Tuesday, February 13, 2001

Decomposition of Distributions characterized by Slow Variation II

Takaaki SHIMURA (Boston University)

We will review the results stated last time, namely when a random variables with slowly varying truncated moments can be expressed as a sum of two independent random variables with the same property. This talk will focus on the ides behind the proofs.

Tuesday, February 27, 2001

Decomposition of Distributions characterized by Regular Variation III

Takaaki SHIMURA (Boston University)

In the previous talks, we discussed properties of the domain of attraction of a normal distribution, that is, the class of distributions with slowly varying truncated variances, as related to convolution. We now consider distributions with regularly varying tails. These distributions appear not only in the domain of attraction of non- Gaussian stable distributions but also in other limit theorems. The presentation will parallel the previous ones. Namely, after giving several general results on decomposition of monotone regularly varying functions, we shall apply them to the decomposition problem of distributions with regularly varying tails. The results, however, will be quite different, due to important differences between slowly varying functions and functions that are regularly varying.

Tuesday, March 13, 2001

An Insider view of Bond Trading

Michael Wilberton (Citizens Bank)

We will present an overview of the life-cycle of a bond issue from issuance through maturity. We will focus on the role of the trading desk and describe the typical day of a trader in either a large or small firm. We will also discuss the different types of bond investors and their portfolio management styles.

Tuesday, March 20, 2001

Shortfall, an Alternative to Variance in Portfolio Selection

Geoffrey J. Lauprete (MIT)

When asset returns have a multivariate elliptical distribution, mean-variance portfolio selection can be shown to be optimal for all investors. Without the ellipticity assumption, mean-variance looses this optimality property.

We propose an alternative to variance in risk-reward portfolio selection, which we call shortfall. Shortfall is a lower partial moment indexed by a probability level, like VaR (Value-at-Risk). Shortfall asymmetrically penalizes upside and downside risk, and can therefore capture investor sentiment towards return skewness, other things being equal.

Tuesday, March 27, 2001

Shortfall, an Alternative to Variance in Portfolio Selection II

Geoffrey J. Lauprete (MIT)

We continue our discussion of shortfall and focus on sampling properties. We develop an efficient algorithm to solve the shortfall optimization problem on historical data. We obtain the rates of convergence of shortfall estimators towards their population counterparts. We illustrate shortfall portfolio selection on artificial and real data.

We next focus on the special but important case when returns have a multivariate ellliptical distribution. We argue that shortfall is to variance in portfolio selection, what least-absolute-deviation (LAD) is to least-squares in regression. In particular shortfall may offer a more robust alternative to variance even when one is after the optimal mean-variance portfolio weights. We illustrate this idea using sample data which is multivariate elliptical, but has fatter tails, and stronger tail-dependence, than the multivariate Gaussian.

We conclude by overviewing our ongoing research on the application of shortfall to portfolio selection.

Tuesday, April 3, 2001

Using Extreme Values in Financial Risk Analysis I

Brendan Bradley (Boston University)

Financial firms and regulators are concerned with extreme market movements. For example, certain regulators require banks to keep a minimum capital reserve equal to (a multiple of) their Value at Risk with a 99% confidence level for a ten day horizon. This means that it is most important to model the tail of the probability distribution of returns (profits and losses) correctly. Extreme Value Theory approaches the modelling of the tails of the distribution in a statistically sound way. For historical reasons, we will first introduce the modelling of the maximum of a collection of random variables and show its possible applications in finance and risk management.

Tuesday, April 10, 2001

Using Extreme Values in Financial Risk Analysis II

Brendan Bradley (Boston University)

We will continue our discussion on the application of extreme value theory (EVT) to finance. The more modern approach to modelling extreme events is to attempt to focus not only on the largest (maximum) events, but on all events greater than some large threshold. This is referred to as peaks over threshold (POT) modelling. We will discuss two approaches currently found in the literature. The first is a semiparametric approach based on a Hill type estimator of the tail index. The second is a fully parametric approach based on the generalized Pareto distribution.

Tuesday, April 17, 2001

The structure of stable self-similar processes with stationary increments

Vladas Pipiras (Boston University)

In this talk, I shall use some connections between stable processes and the theory of non-singular flows to classify a large subclass of stable self-similar processes with stationary increments

Tuesday, April 24, 2001

Compatible sequences and a slow Winkler percolation

Peter Gacs (Boston University)

Two infinite 0-1 sequences are called compatible when it is possible to cast out 0's from both in such a way that they become complementary to each other. Answering a question of Peter Winkler, we show that if the two 0-1-sequences are random i.i.d. and independent from each other, with probability p of 1's, then if p is sufficiently small they are compatible with positive probability. The question is equivalent to a certain dependent percolation with a power-law behavior: the probability that the origin is blocked at distance n but not closer decreases only polynomially fast and not, as usual, exponentially.

Go to the top of the list for time and place.