Probability Seminars

Thursday, June 25, 2009

2:00 PM

901 Van Vleck Hall

Henry Lam, Harvard University, will speak on

“Corrections to the Central Limit Theorem for heavy-tailed random variables”

The classical Central Limit Theorem for scaled i.i.d. sums can be refined by adding subsequent terms through the Edgeworth expansion, up to an order that depends on the number of moments available. In the talk I will provide correction terms beyond this expansion, in the case of regularly varying distributions. The extra terms that we obtain depend interestingly on the parity and integrity of the tail power and the symmetry of the distribution. Furthermore, the terms blend smoothly with known heavy-tailed large deviations asymptotics even outside the central region when applied formally to the spatial scales of the Central Limit Theorem.

Friday, June 26, 2009

2:00 PM

901 Van Vleck Hall

Sergio Pulido, Cornell University, will speak on

“Futures contracts in markets with short-selling constraints.”

The current financial crisis, product of the burst of the alleged real estate bubble, has brought back the attention of the financial and academic community to the study of the causes and implications of asset price bubbles.  In recent works Jarrow, Protter and Shimbo (2006, 2008) and Cox and Hobson (2005) developed an arbitrage-free pricing theory for bubbles in complete and incomplete markets. These papers approach the subject by using the insights and tools of mathematical finance, rather than equilibrium arguments where substantial structure, such as investor optimality and market clearing mechanisms, has to be imposed. In their framework, bubbles occur because the market's valuation measure is a local martingale measure which is not a martingale measure and hence the discounted asset's price is above the expectation of its future cash-flows. The existence of bubbles does not contradict the condition of no free lunch with vanishing risk (NFLVR), because short-selling constraints, given by an admissibility condition on the set of trading strategies, do not allow investors to make a riskless profit from the overpriced securities. The aim of this talk is to explain how the previous work extends to models where some assets cannot be sold short whatsoever and explore how financial instruments such as futures behave in such models.

Monday, June 29, 2009

2:00 PM

901 Van Vleck Hall

Ankit Gupta, University of Wisconsin-Madison, will speak on

“Stochastic model for cell polarity.”

We study the model proposed by Altschuler, Angenent and Wu for the localization of particles in a yeast cell. The cell contains particles in the cytoplasm and on the membrane. The membrane particles can pull particles from the cytoplasm and this causes cluster formation on the membrane. We are mainly interested in these clusters. For finite population size N, we can model this phenomenon as a Markov Process over the space of measures. As N approaches infinity, we show that this process converges to a measure-valued diffusion process called a Fleming-Viot process. Using particle representation for the Fleming-Viot process, we will answer a lot of interesting questions about the structure of this limiting random measure.

Tuesday, June 30, 2009

2:00 PM

901 Van Vleck Hall

Rohini Kumar, University of Wisconsin-Madison, will speak on

“Current fluctuations for independent random walks.”

In a system of independent, identical random walks, the hydrodynamic limit of particle density satisfies the transport pde. The characteristics of this transport pde are straight lines with slope v, where v is the mean velocity of the random walks. We will look at fluctuations of the particle current across characteristics. These fluctuations are subdiffusive. In the one-dimensional random walk model we construct a two-parameter current process indexed by time and spatial shifts in the characteristic line. The limiting scaled current process is found to be a mean-zero, two-parameter Gaussian process with given covariance. We define a distribution-valued current process for the random walk model in multiple dimensions and show that it's scaled limit is a distribution-valued Gaussian process with given covariance. Some large deviation results for the current process in the one-dimensional case will be presented.

Wednesday, July 1, 2009

2:30 PM [NOTE later start time]

901 Van Vleck

Xu Sun, Illinois Institute of Technology, will speak on
“An Impact of noise on invariant manifolds in dynamical systems.”

Invariant manifolds provide geometric structures for understanding dynamical behavior of nonlinear systems. However, these nonlinear systems are often subject to random fluctuations or noises. It is thus desirable to quantify the impact of noises on the invariant manifolds. When the noise intensity is small, we estimate the impact via asymptotic analysis in the context of Liapunov-Perron formulation. Namely, the random invariant manifold is represented as a perturbation of the deterministic invariant manifold, with a well-defined bound for the deviation.