Fall 2016

The modified Euler scheme for SDEs driven by fractional Brownian motions is a natural generalization of the classical Euler scheme in the Brownian motion case. In the first part of the talk we focus on the rate of convergence and the asymptotic error distribution of this numerical scheme, and compare these results to the Brownian motion case. In the second part we consider some higher-order Euler schemes (or Taylor schemes) and their variations. Almost sure rate of convergence and ${\scriptsize L_p}$-rate of convergence are obtained for these numerical schemes, which allow us to design the best Euler-type numerical scheme for the almost sure or the ${\scriptsize L_p}$-convergence.

We study the existence and (Hölder) regularity of local time for SDEs driven by fractional Brownian motions, in particular, the regularity in the time variable. The difficulty comes from the fact that the underlying process is non-Gaussian and non-Markovian. Hence many known technics can not be directly applied. We obtain our result by a sharp estimate for the joint density of finite dimensional distributions of the underlying process. The talk is based on a joint work with Shuwen Lou.

We study a market where are traders with different levels of information. Insiders observe exclusive, non-public information which affects the volatility of the price process, and the information levels are different even among insiders. We extend Lee and Song (2007), Kang and Lee (2014) and Park and Lee (2016) to the case where there are multiple information processes, both discrete and continuous. Also, we study local risk minimization strategies of insiders of various levels under stochastic volatility models.

Resistance forms are crucial in the study of diffusions and analysis on post-critically finite fractals, from the Sierpinski triangle to continuous random trees. These forms can be understood from the limit of a discrete electrical networks. We shall discuss Sub-Gaussian heat kernel estimates on these resistance spaces. We are interested the geometry associated with quantum graphs -- continuous spaces made from a network of wires. Of particular focus is gradient estimates for these spaces and the functional inequalities which are implied. Finally, we shall talk about fractals which are best understood as the limits of quantum graphs.

We establish a functional central limit theorem for long memory stationary infinitely divisible processes with heavy tailed marginals. The class of central limit theorems we consider involves a significant interaction of probabilistic and ergodic theoretical ideas and tools. The limiting process constitutes a new class of stable processes and is expressed in terms of an integral representation involving a stable random measure, due to the heaviness of marginal tails, and the Mittag-Leffler process, due to long memory. If, in particular, the original sequence has negative dependence, the Brownian motion appears as well in the limiting process, due to the second-order cancellation property. This is joint work with Gennady Samorodnitsky (Cornell) and Paul Jung (University of Alabama).

This talk will introduce the log-potential of a random matrix. In the large-matrix limit, these log-potentials converge to a random Gaussian harmonic function in the upper half plane. This limiting object appears frequently in modern probability, for example in branching random walk, the Gaussian free field, and even (largely conjecturally) in the Riemann zeta function. We will introduce these objects and some conjectures concerning the maximum of these recentered log-potentials, including a method to prove the law of large numbers for the maximum of these log-potentials.

We study a stochastic differential equation driven by a fractional Brownian motion with a discontinuous coefficient. We also give an approximation to the solution of the equation.

We discuss the stability problem for the Gaussian logarithmic Sobolev inequality from two perspectives: the first is in the context of the Brenier map arising in mass transport theory; the second is based on Fourier analysis and involves the Beckner-Hirschmann entropic uncertainty principle.

We consider a space- and time- inhomogeneous Markov chain on the integer lattice whose transition probability (environment) is random. The Markov chain is assumed to be a martingale (balanced). When the transition probability is uniformly elliptic, ergodic and independent of time, a central limit theorem was proved by Lawler in the 80s. In this talk, I will present a local limit theorem for the more general space- and time- inhomogeneous case, which states that the rescaled hitting probability of the walk converges to the density of the limiting Brownian motion. This is a joint work with Jean-Dominique Deuschel.

We study Brinkman's equations with microscale properties that are highly heterogeneous in space and time. The time variations are controlled by a stochastic particle dynamics described by an SDE. Our main results include the derivation of macroscale equations and showing that the macroscale equations are deterministic. We use the asymptotic properties of the SDE and the periodicity of the Brinkman coefficient in the space variable to prove the convergence result. The SDE has a unique invariant measure that is ergodic and strongly mixing. The macro scale equations are derived through an averaging principle of the slow motion (fluid velocity) with respect to the fast motion (particle dynamics) and also by averaging Brinkman's coefficient with respect to the space variable. Our results can be extended to more general nonlinear diffusion equations with heterogeneous coefficients. This is a joint work with Yalchin Efendiev and Florian Maris.

The exponential transform of a vector-valued path, also known as the signature of a path, is the formal sequence of associated iterated path integrals. While the description of a path involves its local behaviors and their interactions, the signature is a global algebraic quantity encoding the total increment, geometric signed area and all higher order areas of the underlying path. It is widely believed (and surprisingly) that the signature contains essentially all information about the underlying path. In this talk, we will prove that every (rough) path is uniquely determined by its signature up to certain tree-like equivalence. Moreover, looking into its probabilistic counterpart, we will obtain stronger uniqueness results for sample paths of Gaussian processes by applying the Malliavin calculus.

We consider an exactly solvable variant of the corner growth model in which the weights are geometrically distributed with site dependent parameters randomly drawn from an ergodic distribution. Then the shape function is strictly concave in a certain region that can be a proper subset of the first quadrant. In this talk, we show that the limit fluctuations are given by the Tracy-Widom GUE distribution in the strictly concave region. We also present a right tail bound for the last-passage times. For these results, we utilize an available Fredholm determinant representation for the last-passage time distributions.

We consider the stochastic wave equation on the real line driven by a linear multiplicative Gaussian noise, which is white in time and whose spatial correlation corresponds to that of a fractional Brownian motion with Hurst index ${\scriptsize H\in (\frac14,\frac12)}$. First, we prove that this equation has a unique solution (in the Skorohod sense) and obtain an exponential upper bound for the ${\scriptsize p}$-th moment of the solution, for any ${\scriptsize p\geq 2}$. The condition ${\scriptsize H>\frac14}$ turns out to be necessary for the existence of solution. Secondly, we show that this solution coincides with the one interpreted in the Itô sense. Finally, we prove that the solution of the equation in the Skorohod sense is weakly intermittent. The talk is based on a joint work with Raluca Balan (Univ. of Ottawa) and Maria Jolis (Autonomous Univ. of Barcelona).