# CUNY Probability Seminar, Fall 2015

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.

Oct 6, 2015 4:15pm, Room 6494
Speaker: Agostino Capponi, Columbia
Title: Optimal Investment under Information Driven Default Contagion
Abstract:
We introduce a novel dynamic optimization framework to analyze optimal portfolio allocations within an information driven default contagion model. The investor allocates his wealth across several defaultable stocks whose growth rates and default intensities are driven by a hidden Markov chain. We provide a rigorous analysis of default contagion arising through recursive dependence of the optimal strategies on the gradient of value functions. We establish uniform bounds for solutions to a sequence of approximation problems, show their convergence to the unique Sobolev solution of a recursive system of Hamilton-Jacobi-Bellman partial differential equations (HJB PDEs), and prove the corresponding verification theorem. We provide a numerical study to illustrate the sensitivity of the strategies to default contagion, stock volatility, and risk aversion.

Oct 13, 2015 4:15pm, Room 6494
Title: Random multiplicative functions
Abstract: A function f is said to be multiplicative if f(m n) = f(m) f(n) for all integers m,n.
Such an f is determined by its behavior on primes and prime powers. By letting the value on the primes be random and independent of each other one obtains a random multiplicative function. On the integers the random variables f(m) are highly dependent and their study presents numerous challenges. For example the limiting distribution of the partial sums is
not known to exist, but following work of Harper, we know that it cannot be Gaussian.

I will describe some of the main results and questions pertaining to random multiplicative functions and their connections to number theory and harmonic analysis. In particular I’ll discuss in more details Helson’s conjecture on the first absolute moment of the partial sums and some partial progress towards disproving this conjecture (joint work with Adam Harper and Ashkan Nikeghbali).

Oct 27, 2015 4:15pm, Room 6494
Speaker: Partha Dey, UIUC
Title: Longest increasing path within the critical strip
Abstract: Consider a Poisson Point Process of intensity one in the two-dimensional square $[0,n]^2$. In Baik-Deift-Johansson (1999), it was shown that the length $L_n$ of a longest increasing path (an increasing path that contains the most number of points) when properly centered and scaled converges to the Tracy-Widom distribution. Later Johansson (2000) showed that all maximal paths lie within the strip of width $n^{2/3+\epsilon}$ around the diagonal with probability tending to 1 as $n\to \infty$. We consider the length $L_n^{(\gamma)}$ of maximal increasing paths restricted to lie within a strip of width $n^{\gamma}, \gamma<2/3$ around the diagonal and show that when properly centered and scaled it converges to a Gaussian distribution. We also obtain tight bounds on the expectation and variance of $L_n^{(\gamma)}$. Joint work with Matthew Joseph and Ron Peled.

Nov 3, 2015 4:15pm, Room 6494
Speaker:    Christian Benes, Brooklyn College
Title:    The scaling limit of the loop-erased random walk Green’s function
Abstract: We show that the probability that a planar loop-erased random walk passes through a given edge in the interior of a lattice approximation of a simply connected domain converges, as the lattice spacing goes to zero, to a multiple of the SLE(2) Green’s function. This is joint work with Greg Lawler and Fredrik Viklund.

Tuesday, November 10, 4:15 PM, Rm. 6494
Speaker: Shirshendu Chatterjee, City College
Title: Phase transition for the threshold contact process, an “annealed approximation” of heterogeneous random Boolean networks
Abstract: We consider a model for heterogeneous gene regulatory networks that is an “annealed approximation” of Kauffmann’s (1969) original random Boolean networks. In this model, genes are represented by the nodes of a random directed graph Gn on n vertices with specified degree distribution, and the interactions among the genes are approximated by an appropriate threshold contact process (in which a vertex with at least one occupied in-neighbor at time t will be occupied at time t+1 with probability q, and vacant otherwise) on Gn. We characterize the order-chaos phase transition curve for the threshold-contact process on Gn segregating the chaotic and ordered random Boolean networks.

Tuesday, November 24, 4:15 PM, Rm. 6494
Speaker: Shelemyahu Zacks, SUNY-Binghamton
Title: Compound Poisson Process with a Poisson subordinator
Abstract: A compound Poisson process whose randomized time is an independent Poisson process is called a compound Poisson process with Poisson subordinator. We provide its probability distribution, which is expressed in terms of the Bell polynomials, and investigate in detail both the special cases in which the compound Poisson process has exponential jumps and normal jumps. Then for the iterated Poisson process we discuss some properties and provide convergence results to a Poisson process. The first crossing time problem for the iterated Poisson process is finally tackled in the cases of (i) a decreasing and constant boundary, where we provide some closed-form results, and (ii) a linearly increasing boundary, where we propose an iterative procedure to compute the first crossing time density and survival functions. (This is joint work with Antonio di Crescenzo and Barbara Martinucci from Universita degli studi di Salerno.)
Reference: http://projecteuclid.org/euclid.jap/1437658603

Tuesday, December 1, 4:15 PM, Rm. 6494
Speaker: Krishnamurthi Ravishankar, NYU-Shanghai
Title: Hydrodynamics and quenched strong equilibrium for asymmetric zero range process with sitewise disorder
Abstract: We consider a one dimensional zero range process on $\mathbb{Z}$. The state space for this process is {0, 1, 2, …}^Z, with z a particle at site z is moved to one of its two nearest neighbors at a rate α(z)g(η(z))p to the right and α(z)g(η(z))(1 − p) to the left where p > q. g is an increasing function with 0 = g(0) < g(1) and $\lim_{n \to \infty} g(n) = 1$. The sitewise disorder 0 < c < α ≤ 1 is assumed to be ergodic with respect to space shifts. This process can be used to model traffic in a one lane road where the sites are cars and η(z) is the gap between cars at z and z + 1. Since the maximum flux at some sites are bounded by c one would expect that this process has a critical site density above which invariant measures do not exist. This was proved in earlier works by Benjamini, Ferrari and Landim [1] and Andjel, Ferrari, Guiol, Landim [2]. We prove the existence of a hydrodynamic limit given by a scalar conservation law starting from general initial states including regions with supercritical empirical density for quenched disorder. We also prove a quenched strong local equilibrium result.

References:
[1] Benjamini, I.; Ferrari, P. A.; Landim, C. Asymmetric conservative processes with random rates. Stochastic Process. Appl. 61 (1996), no. 2, 181–204.

[2] Andjel, E.; Ferrari, P.A.; Guiol, H.; Landim, C. Convergence to the maximal invariant measure for a zero-range process with random rates. Stoch. Proces. Appl. 90 (2000) 67–81.

Tuesday, December 8, 4:15 PM, Rm. 6494
Speaker: Jeffrey Kuan, Columbia
Title: Stochastic duality of two-component ASEP via symmetry of quantum groups of rank two
Abstract: We study two generalizations of the asymmetric simple exclusion process (ASEP) with two types of particles. Particles of type 1 can jump over particles of type 2, while particles of type 2 can only influence the jump rates of particles of type 1. We prove that the processes are self-dual and explicitly write the duality function. The construction and proofs of duality are accomplished using symmetry of the quantum groups $$Uq(gl_3)$$ and $$Uq(sp_4)$$, generalizing the $$U_q(sl_2)$$ symmetry of ASEP.