Stochastics Seminar
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Fall 2022 Friday 3:004:00 PM
Room for inperson: TBA
Zoom information:
Meeting ID: 910 5244 8043
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Date  Speaker  Title (click for abstract, if available) 

Friday, September 9th 
Yu Gu
University of Maryland 
We discuss three different proofs of the central limit theorem for the KPZ equation on a torus. All proofs rely on the geometric ergodicity of the socalled projective process, which, in this case, is the endpoint distribution of the corresponding directed polymer. I will try to draw connections to the subjects of homogenization, Malliavin calculus and the product of random matrices. I will also discuss what happens if we enlarge the torus as time increases. 
Friday, September 16th 
Andrey Sarantsev
University of Nevada, Reno 
A system of stochastic differential equations models the stock market, with coefficients depending on size of a stock relative to a benchmark. We fit the model using realworld financial data, and study longterm stability. This article is published in 2021 in Annals of Finance. We continue this research with current PhD students. 
Friday, September 23rd 
Alex Dunlap
NYU 
I will discuss a twodimensional stochastic heat equation with a nonlinear noise strength, and consider a limit in which the correlation length of the noise is taken to 0 but the noise is attenuated by a logarithmic factor. I will discuss how pointwise statistics of this equation can be related to a forwardbackward stochastic differential equation (FBSDE) depending on the nonlinearity, as long as the nonlinearity is such that the FBSDE can be solved for long enough with Lipschitz decoupling function. I will also discuss conditions on the nonlinearity for this property to hold, and explain several cases in which the FBSDE can be solved explicitly. No previous knowledge of forwardbackward SDEs will be assumed. This talk will be based on current joint work with Cole Graham and older joint work with Yu Gu. 
Friday, September 30th 
Kevin Yang
UC Berkeley 
This talk has two goals. The first is the derivation of a timedependent KPZ equation (TDKPZ) from a timeinhomogeneous GinzburgLandau model, which describes, among many other things, stochastic dynamics of ferromagnets whose physics have been known to be timedependent (due to Eddy currents) since the 1800s. The TDKPZ has a nonlinear twist that is not seen in the usual KPZ equation, making it a more interesting SPDE. The second goal is universality of the method for deriving (TD)KPZ, which should work beyond GinzburgLandau. In particular, we answer a question of deriving (TD)KPZ from asymmetric particle systems under natural fluctuationversions of the assumptions in Yau's relative entropy method; the only additional assumption we need is a logSobolev inequality. Time permitting, future directions (of pure and applied flavors) will be discussed. 
Friday, October 7th 
Hao Shen
UWMadison 
We will discuss stochastic quantization of the YangMills model on two and three dimensional torus. In stochastic quantization we consider the Langevin dynamic for the YangMills model which is described by a stochastic PDE. We construct local solution to this SPDE and prove that the solution has a gauge invariant property in law, which then defines a Markov process on the space of gauge orbits. We will also describe the construction of this orbit space, on which we have welldefined holonomies and Wilson loop observables. Based on joint work with Ajay Chandra, Ilya Chevyrev, and Martin Hairer. 
Friday, October 21st 
Fabian Mies
RWTH Aachen University 
The linear fractional stable motion generalizes two prominent classes of stochastic processes, namely stable Lévy processes, and fractional Brownian motion. For this reason it may be regarded as a basic building block for continuous time models. We study a stylized model consisting of a superposition of independent linear fractional stable motions, which nests various processes encountered in highfrequency financial economentrics as special cases. Applying an estimating equations approach, we construct estimators for the whole set of parameters of the mixed fractional stable motion, and we derive asymptotic normality in a highfrequency regime. The conditions for consistency turn out to be sharp for two prominent special cases: (i) for Lévy processes, i.e. for the estimation of the successive BlumenthalGetoor indices, and (ii) for the mixed fractional Brownian motion introduced by Cheridito. In the remaining cases, our results reveal a delicate interplay between the Hurst parameters and the indices of stability. This talk is based on joint work with Mark Podolskij. 
Friday, October 28th 
Carsten Chong
Columbia University 
In recent years, there has been substantive empirical evidence that stochastic volatility is rough. In other words, the local behavior of stochastic volatility is much more irregular than semimartingales and resembles that of a fractional Brownian motion with Hurst parameter H < 0.5. In this paper, we derive a consistent and asymptotically mixed normal estimator of H based on highfrequency price observations. In contrast to previous works, we work in a semiparametric setting and do not assume any a priori relationship between volatility estimators and true volatility. Furthermore, our estimator attains a rate of convergence that is known to be optimal in a minimax sense in parametric rough volatility models.

Wednesday, November 2nd 
Jingyu Huang
University of Birmingham 

Friday, November 4th 
Bei Wang
University of Utah 
Hypergraphs capture multiway relationships in data, and they have consequently seen a number of applications in higherorder network analysis, computer vision, geometry processing, and machine learning. We develop theoretical foundations for studying the space of hypergraphs using ingredients from optimal transport. By enriching a hypergraph with probability measures on its nodes and hyperedges, as well as relational information capturing local and global structures, we obtain a general and robust framework for studying the collection of all hypergraphs. First, we introduce a hypergraph distance based on the cooptimal transport framework of Redko et al. and study its theoretical properties. Second, we formalize common methods for transforming a hypergraph into a graph as maps between the space of hypergraphs and the space of graphs, and study their functorial properties and Lipschitz bounds. Finally, we demonstrate the versatility of our Hypergraph CoOptimal Transport (HyperCOT) framework through various examples. If time permits, I will discuss the application of optimal transport in scientific data visualization 
Friday, November 11th 
Ofer Busani
University of Bonn 
Canceled 
Friday, November 18th 
Jack Hanson
City College of NY 
Several wellknown problems in firstpassage percolation relate to the behavior of infinite geodesics: whether they coalesce and how rapidly, and whether doubly infinite ``bigeodesics'' exist. In the plane, a version of coalescence of ``parallel'' geodesics has previously been shown; we will discuss new results that show infinite geodesics from the origin have zero density in the plane. We will describe related forthcoming work showing that geodesics coalesce in dimensions three and higher, under unproven assumptions believed to hold below the model's upper critical dimension. If time permits, we will also discuss results on the bigeodesic question in dimension three and higher. 
Friday, December 2nd 
David Harper
Georgia Tech 
In firstpassage percolation (FPP), we let $\tau_v$ be i.i.d. nonnegative weights on the vertices of a graph and study the weight of the minimal path between distant vertices. If $F$ is the distribution function of $\tau_v$, there are different regimes: if $F(0)$ is small, this weight typically grows like a linear function of the distance, and when $F(0)$ is large, the weight is typically of order one. In between these is the critical regime in which the weight can diverge but does so sublinearly. This talk will consider a dynamical version of critical FPP on the triangular lattice where vertices resample their weights according to independent rateone Poisson processes. We will discuss results that show that if the sum of $F^{1}(1/2+1/2^k)$ diverges, then a.s. there are exceptional times at which the weight grows atypically, but if sum of $k^{7/8} F^{1}(1/2+1/2^k)$ converges, then a.s. there are no such times. Furthermore, in the former case, we compute the Hausdorff and Minkowski dimensions of the exceptional set and show that they can be but need not be equal. Then we will consider what the model looks like when passage times are unusually small by considering an analogous construction to Kesten's incipient infinite cluster in the FPP setting. This is joint work with M. Damron, J. Hanson, W.K. Lam. 
Friday, December 9th 
Lekha Patel
Sandia National Labs 
The Hawkes Processes is a popular type of selfexciting point process that has found application in the modeling of financial stock markets, earthquakes, and social media cascades. Their continuous time framework, however, necessitates that data collected for inference be accurate. However, for realtime monitors of data, for example in remote sensing or cybersecurity, accurate detection of events is challenging. In the first part of this talk, we develop a novel frequentist inference mechanism for binned data, i.e., continuous data that is aggregated within discrete bins upon recording. The results of this will be highlighted on real computer network data, which exhibit high levels of selfexcitation. We then highlight that understanding the underlying Hawkes’ source and pattern of excitation is important for many realworld applications, such as criminal behavior. In the second part, we develop a novel Bayesian nonparametric model for a Hawkes process whose excitation kernel is be flexibly modeled, to allow for different levels of selfexcitation. The utility of this model is presented for modeling and predicting extreme terror attacks in Afghanistan. 
Stochastics Seminar for Spring 2022 is organized at the University of Utah by B. Cooper Boniece and Xiao Shen.
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