# CUNY Probability Seminar, Spring 2016

The CUNY Probability Seminar is typically held on Tuesdays at 4pm by the CUNY Graduate Center Math Department. The exact dates, times and locations are mentioned below.

Feb 9, 2016 4:15pm, Room 6494
Speaker: Maxim Bichuch, Johns Hopkins
Title: Optimal Investment with Transaction Costs and Stochastic Volatility
Abstract: Two major financial market complexities are transaction
costs and uncertain volatility, and we analyze their joint impact on
the problem of portfolio optimization. When volatility is constant,
the transaction costs optimal investment problem has a long history,
especially in the use of asymptotic approximations when the cost is
small. Under stochastic volatility, but with no transaction costs, the
Merton problem under general utility functions can also be analyzed
with asymptotic methods. Here, we look at the final time optimal
investment and consumption problem, when both complexities are
present, using separation of time scales approximations. We find the
first term in the asymptotic expansion in the time scale parameter, of
the optimal value function, consumption, and of the optimal strategy,
for fixed small transaction costs. We give a proof of accuracy in the
case of fast mean-reverting stochastic volatility. Additionally, we
deduce the optimal long-term growth rate. This is a joint work with
Ronnie Sircar.

Feb 16, 2016 4:15pm, Room 6494
Speaker: Jack Hanson, City College CUNY
Title: Chemical distances in critical 2D percolation
Abstract: In critical two-dimensional Bernoulli percolation, fraction 1/2 of the edges of the graph Z^2 are erased independently. The resulting graph has connected components and “holes” appearing on all scales. As a result, the chemical (graph) distance inside large connected components is conjectured to grow superlinearly in the Euclidean distance, and some results in this direction are known. For instance, the shortest crossing of the box [-n, n]^2 has length S_n >  n^{1 + \epsilon} with high probability, and is no longer than the unique lowest crossing, whose length L_n is known to scale as n^{4/3}. Kesten and Zhang asked whether S_n = o(L_n); we will discuss recent work which gives an affirmative answer to this question, as well as some results on point-to-point and point-to-box distances.

Feb 23, 2016 4:15pm, Room 6494
Speaker: Jay Rosen, CUNY
Title:Tightness for cover times in two dimensions: the right tail
Abstract: For general smooth two dimensional manifolds M we consider the \ep cover time, that is, the time need for Brownian motion on M to come within \ep of every point. We present a conjecture that, when appropriately normalized, the \ep cover time is tight. We discuss a proof of this conjecture for the right tail of the \ep cover time for the sphere.

March 1, 2016 4:15pm, Room 6494
Speaker: Hao Shen, Columbia
Title: Regularity structure theory and its applications
Abstract: I will review the basic ideas of the regularity structure theory recently developed by Martin Hairer, as well as its applications to stochastic PDE problems. I will then discuss some joint works with Hairer on well-posedness of the sine-Gordon equation, and central limit theorems for KPZ equation.

March 8, 2016 4:15pm, Room 6494
Speaker: David Kelly, NYU
Title:  Fast-slow systems with chaotic noise
Abstract:  It has long been observed that multi-scale systems, particularly those in climatology, exhibit behavior typical of stochastic models, most notably in the unpredictability and statistical variability of events. This is often in spite of the fact that the underlying physical model is completely deterministic. One possible explanation for this stochastic behavior is deterministic chaotic effects. In fact, it has been well established that the statistical properties of chaotic systems can be well approximated by stochastic differential equations. In this talk, we focus on fast-slow ODEs, where the fast, chaotic variables are fed into the slow variables to yield a diffusion approximation. In particular we focus on the case where the fast process is multi-dimensional and enters the slow equation as multiplicative “noise”. The tools from rough path theory prove useful in this difficult setting. From a stochastic modeling perspective, the limiting slow variables are somewhat surprising. Even though the noise is approximated by a smooth (chaotic) signal, one does not obtain a Stratonovich integral in the limiting equation.

March 15, 2016 4:15pm, Room 6494
Speaker: Wei-Kuo Chen, Minnesota
Title: Parisi formula, disorder chaos and fluctuation for the ground state energy in the spherical mixed p-spin models
Abstract:
Spin glasses are disordered spin systems originated from the desire of understanding the strange magnetic behaviors of certain alloys in physics. As mathematical objects, they are often cited as examples of complex systems and have provided several fascinating structures and conjectures. In this talk, we will focus on the spherical mixed p-spin mean-field spin glass model. We will present the Parisi formula and some fluctuation properties for the maximum energy. In addition, we will discuss results concerning the chaotic nature of the location of the maximum energy under small perturbations to the disorder. This talk is based on a joint work with Arnab Sen.

March 22, 2016 4:15pm, Room 6494
Speaker: Hana Kogan, CUNY
Title: Coarsening with Frozen Vertices.
Abstract: Coarsening occurs in the Stochastic Ising model with zero temperature Glauber dynamics. When the initial sign of spins is assigned according to i.i.d. Bernoulli (1/2) the model describes the dynamics of a ferromagnet with zero external field quenched from infinite to 0 temperature. On the Z^d lattice the behavior of the model is an open problem for d=3 and higher; for d=1&2, every spin changes sign infinitely many times. It is known that when the initial condition is Bernoulli(\beta) with \beta sufficiently large, the model converges to consensus. It is conjectured that the same is true for any \beta>1/2. A number of related models has been developed in an attempt to better understand the behavior of this model on Z^d in higher dimensions. We will present results about the symmetrical (\beta=1/2) model for any d with some vertices frozen to +1 at time 0. We consider both the random and deterministic sets of frozen vertices.

April 5, 2016 4:15pm, Room 6494
Speaker: Murad S. Taqqu, Boston University
Title: Behavior of the generalized Rosenblatt process at extreme critical exponent values
Abstract: The Rosenblatt process is one of the simplest non-Gaussian version of a self-similar process. Its marginal distributions are mixtures of chi-squares. The Rosenblatt process, however, involves a single critical exponent. The generalized Rosenblatt process is obtained by replacing that exponent by two different exponents living in the interior of a triangular region. What happens to the generalized Rosenblatt process as these critical exponents approach the boundaries of the triangle? We show that on each of the two symmetric boundaries, the limit is non-Gaussian. On the third boundary, the limit is Brownian motion. The rates of convergence to some of these boundaries are also given. The situation is particularly delicate as one approaches the corners of the triangle, because the limit process will depend on how these corners are approached. This is joint work with Shuyang Bai.

April 12, 2016 4:15 pm. Room 6494                                                                                               Speaker: Dmitry Dolgopyat, University of Maryland                                                                                  Title:  Local Limit Theorem for Markov chains                                                                                            Abstract:  We prove a local limit theorem for inhomogeneous Markov chains with finite
state space under conditions similar to the conditions of Dobrushin’s Central
Limit Theorem. This is a joint work with Omri Sarig.

April 19, 2016 4:15pm, Room 6494
Speaker: Ramon van Handel, Princeton University
Title: Chaining, interpolation, and convexity
Abstract: A significant achievement of modern probability theory is the development of sharp connections between the boundedness of random processes and the geometry of the underlying index set. In particular, the generic chaining method of Talagrand provides in principle a sharp understanding of the suprema of Gaussian processes. The multiscale geometric structure that arises in this method is however notoriously difficult to control in any given situation. In this talk, I will exhibit a surprisingly simple but very general geometric construction, inspired by real interpolation of Banach spaces, that is readily amenable to explicit computations and that explains the behavior of Gaussian processes in various interesting situations where classical entropy methods are known to fail. No prior knowledge of this topic will be assumed in the talk.

May 3, 2016 4:15pm, Room 6494 — CANCELED
Speaker: Andrey Sarantsev, University of California, Santa Barbara
Title: Competing Brownian particles
Abstract: Consider a system (finite or infinite) of rank-based Brownian particles on the real line. Each particle moves as Brownian motion with drift and diffusion coefficients based on its current rank relative to other particles. We study properties of such systems: existence, uniqueness, stability, ergodicity, collisions, propagation of chaos and applications in finance.

May 10, 2016 4:15pm, Room 6494
Speaker: Omri Sarig, Weizmann Institute
Title: The deterministic random walk
Abstract: The simple random walk can be generated dynamically by picking x in the unit interval randomly uniformly, iterating the map T(x)=2x mod 1 on the unit interval, and telling the walker to take a step to the left when T^k(x) is in [0,1/2) and a step to the right when T^k(x) is in [1/2,1) (k=1,2,3,…)
The map T(x)=2x mod 1 is very chaotic: it is mixing, has positive entropy, and has exponential sensitivity to initial conditions at every point.
The question arises what happens when T(x) is replaced by a very “non-chaotic” map like the irrational rotation S(x)=x+a mod 1 (not weak mixing, zero entropy, no exponential sensitivity to initial conditions at any point).
I will discuss some properties of this “deterministic random walk”, and in particular its local time. This is joint work with Avila, Dolgopyat, and Duriev.

May 17, 2016 4:15 PM, Room 6494
Speaker: Clement Hongler, EPFL
Title: Ising interfaces and CLE(3)
Abstract: In the last six years, a series of results about the critical 2D Ising model has allowed one to study the interfaces of the model with great precision. I will explain the main ideas behind these results, that bring us from discrete holomorphic observables to the description of the full scaling limit of the interfaces by CLE(3).
Based off joint works with S. Benoist, D. Chelkak, H. Duminil-Copin, A. Kemppainen, K. Kytola, P. Nolin and S. Smirnov