UW Math Probability Seminar Fall 2004

Thursdays in 901 Van Vleck Hall at 2:25 PM, unless otherwise noted.

Organized by Timo Seppäläinen

[Past Schedules]

Schedule and Abstracts

The monopolist game is a multi-player ruin process that has been used to model the behavior of certain learning algorithms. We prove that when the players begin with equal stakes, the expected duration of the game is proportional to the square of their collective initial wealth. This proves a conjecture of Amano, Tromp, Vitányi, and Watanabe. More generally, we find that the expected duration is proportional to a quadratic function that reflects the uniformity of the initial stakes, and calculate the expected duration exactly for 3 players.

The optimal operation of a hydro-electric reservoir in a deregulated electricity market can be treated as a stochastic dynamic programming problem. However, the market trading period (usually 1 hour or less) may be much shorter than the inherent time scale of the reservoir (often many months). This talk describes an attempt to devise a simple stochastic model which can represent both time scales.

Abstract

For certain R^d-valued processes X we determinine Chung-type functional LIL's for the maximal process given by M(t) := sup_{0 \leq s \leq t}|X(s)|. Under mild conditions, we are able to determine such an LIL for Levy processes X with X(1) residing in the domain of attraction of a strictly alpha-stable random vector, as well as for the partial sum process constructed from iid summands in such a domain of attraction. In particular, our results capture an LIL for any alpha-stable process in R^1 which is not a subordinator, extending and complimenting previously known results. We will discuss some methods entailed in the proof, the conditions placed on the result, and applications if time permits.

A previously unoccupied site on a integer lattice becomes occupied (with a certain probability, or at a certain rate) once it is in contact with at least a threshold number of already occupied sites in its neighborhood. As a finite occupied set grows, it assumes a deterministic asymptotic shape, which is, however, in most cases not known explicitly. As the range of the neighborhood increases, such shape starts to resemble an explicitly computable object, obtained by combining equilibrium solutions of one-dimensional integro-differential equations.

Shocks developed by the hydrodynamic limit of particle systems are well understood at the macroscopic level of the hydrodynamic equation. Finding the corresponding microscopic structure in the model itself is much more difficult. It first appeared in Derrida et al. that under special conditions, some simple product measures of simple exclusion are stationary as seen by the so-called second class particle, and show all properties of a shock. Belitsky and Schutz found later that the very same measures, only under the very same conditions, evolve to linear combinations of themselves as seen from a non-moving, fixed position. This phenomena can be given an interpretation in terms of "random walking shocks". I will try to explain these results and will also show both of them for a special deposition model called the exponential bricklayers' process. Once again, the two type of results occur only under the very same conditions.

We describe a martingale approach to quenched invariance principles (functional central limit theorems) for random walk in random environment. Quenched means that the result is valid for fixed realizations of the environment. At this point we have applied the strategy to certain walks with a strong drift in some spatial direction. (Joint work with Firas Rassoul-Agha, Ohio State University.)

The motion of a distinguished particle interacting with others in a zero-range particle system is considered. The "zero-range" system consists of a collection of dependent random walks on the integer lattice whose interaction is of "speed change" type. The problem considered is that of the asymptotic fluctuations of the distinguished particle in equilibrium. We show that the fluctuations are diffusive in dimensions d=1 and d>2 under some conditions. Also, we show for a specific case in d=1 that the centered diffusively rescaled distinguished particle position converges to a Brownian motion. Some background and open problems will also be discussed.

I will discuss some (very) recent results showing how techniques from the theory of Gaussian Hilbert spaces can be used to solve a number of open problems in discrete Fourier analysis. Joint work Ryan O'Donnell and Krzysztof Oleszkiewicz.

In financial mathematics, markets can be modeled using Brownian motion and stochastic differential equations. The Black-Scholes method can then be used to price and hedge derivative securities. In this model, a standard assumption is that the market is "frictionless" in the sense that there are no transaction costs. If we try to introduce these costs into the model, we are led to quantities related to the total variation of the processes involved. Unfortunately, like Brownian motion, these processes are of unbounded variation and the transaction costs are thus found to be infinite.

One way to try to study these costs in a nontrivial way is to use scaling. I will present a general result regarding the scaled p-th variation of a certain class of continuous martingales. I will then show how the special case p = 1 can be used to give an alternate proof of certain results in finance related to the limit of scaled transaction costs.

Abstract

Abstract

In 1973, Thorp introduced the following card shuffling procedure. Cut the deck into two equal piles. Drop the first card from the left pile or the right pile according to the outcome of a fair coin flip; then drop from the other pile. Continue this way, flipping an independent coin each time, until both piles are empty. Despite its simple description, the Thorp shuffle has been hard to analyze. It has long been believed that the mixing time is polynomial in log of the number of cards. We prove the first such bound.