PCOC (pronounced as peacock) is an acronym for French words "Processus Croissant pour l'Ordre Convexe", words for an integrable process which is increasing in the convex order. The class is important because it characterizes the property that a martingale can "mimic" the one-dimensional distribution of the process in this class. Baker and Yor presented a simple but remarkable application of this equivalence to mathematical finance; they showed that the time-average of a geometric Brownian motion is a PCOC. This fact implies that the price of an Asian option is increasing in the volatility and maturity. Recently, we extend the result by Baker-Yor to the cases of fractional Brownian motion. We prove that the time-average of the exponential of a fractional Brownian motion is a PCOC, and use it to analyze option prices written on fractional volatility model.

Large networks have become increasingly popular over the last decades, and their modeling and investigation have led to interesting and new ways to apply analytical and statistical methods. The introduction of exponential random graphs has aided in this pursuit, as they are able to capture a wide variety of common network tendencies by representing a complex global configuration through a set of tractable local features. This talk will look into the asymptotic structure and dynamics of weighted exponential random graphs and formulate a quantitative theory of phase transitions. The main techniques that we use are variants of statistical physics (both equilibrium and non-equilibrium). Based on joint work with multiple collaborators.

The bell-shaped curve (or more technically the normal law) arises in many statistical applications ranging from coin tossing, scores on calculus exams, heights of individuals, to name but three. Mathematically, the normal law arises when independent outcomes of an experiment are averaged; for example, the number of times heads appears in n tosses of a coin.

In the last two decades, new universal laws have been discovered when the events are strongly dependent. The applications of these new universal laws range from stochastic growth of an interface, analysis of large data sets, to sorting algorithms in computer science.

This lecture will give a nontechnical introduction to these developments.

Random walk on the integer lattice $\mathbb Z$ is arguably one of the most important (and simple) Markov processes. When there is more than one walker, for the problem to be interesting, the walkers must in some way interact. The asymmetric simple exclusion process (ASEP) is one such generalization where infinitely many particles execute random walk but with an exclusion rule.

ASEP is an example of integrable probability which more or less means there are analytical techniques to derive explicit formulas that can be analyzed asymptotically. Furthermore, ASEP is important because it leads to solutions to the KPZ equation which describe fluctuations of the height of a stochastically growing interface.

This lecture will give an introduction to ASEP and KPZ Universality.

This presentation will review two methods for proving the Central Limit Theorems (CLT): the Lindeberg’s method, introduced by Jarl Waldemar Lindeberg in 1922 and the principle of conditioning, introduce in 1986 by Adam Jakubowski. We will discuss both methods and their applications concerning random fields, martingales, and ARCH and GARCH processes in finance introduced respectively by Robert Engel in 1982 and by Bollerslev in 1986.

The rough paths theory can be seen as a technique which allows to define very general noisy differential systems with a minimum amount of probability structure. I will first give an introduction and some motivation for this area of research, and also highlight some of the main applications to stochastic differential equations and stochastic partial differential equations. Then ll try to explain the main mechanisms behind the rough paths method. I will eventually give some results about noisy differential systems which can be achieved from the rough paths perspective.

In this talk, we give combinatorial formulas for the joint moments of the Brownian motions taken at different times. We expect that these computations can be used to recover some identity in law, by using methods of moments and successive integrations by parts.

A Motzkin path is a lattice path from $(0,0)$ to $(n,0)$ which does not fall below the horizontal axis, and uses only the ascents $(1,1)$, the descents $(1,-1)$ or the level steps $(1,0)$. It is known that after scaling a random Motzkin path converges to a Brownian excursion. It turns out that asymptotic fluctuations of the components of such a path that count separately the ascent steps, the descent steps, and the level steps are linear combinations of two independent processes: a Brownian motion and a Brownian excursion. The proofs rely on the Laplace transforms.

The talk is based on a joint paper with Dalibor Volny on the convergence to the Brownian motion of random fields, which are not started from their equilibrium, but rather known a fixed past trajectory. The initial class considered is that of orthomartingales and then the result is extended to a more general class of random fields by approximating them, in some sense, with an orthomartingale. We construct an example which shows that there are orthomartingales which satisfy the CLT but not in its quenched form.

The Eggenberger-Polya urn model had been studied for a long time and there are several generalizations and applications in the literature. In this talk, we first review some generalized Eggenberger-Polya urn models. Then we will give some asymptotic study.

In this talk, we review an infinite urn scheme investigated by Karlin (1967). We shall sketch the proof for the central limit theorem for the so-called odd-occupancy process, and explain how the Poissonization technique helps in this case.

By providing instances of approximation of linear diffusions by birth-death processes, Feller, in the 50s, has offered an original path from the discrete world to the continuous one. In this talk, by identifying an intertwining relationship between squared Bessel processes and linear birth-death processes, we show that this connection is in fact more intimate and goes in the two directions. This procedure can be interpreted as a (de)-quantization between the two worlds. A notable consequence of this relationship, is the identification of the discrete self-similarity property enjoyed by the linear birth-death process. It is expressed in terms of the so-called binomial thinning operator, introduced by Steutel and van Harn. We proceed by explaining that the same quantization procedure applied between positive-valued self-similar Markov processes and a class of discrete self-similar skip-free Markov chains that we introduce. We end the talk by discussing several consequences of this connection. This is a joint work with L. Miclo.

Statistical network analysis typically deals with inference concerning various parameters of an observed network. In several applications, especially those from social sciences, behavioral information concerning groups of subjects are observed. In such data sets, even though a network structure is present it is not typically observed. These are referred to as implicit networks. In this presentation, we describe a model-based framework to uncover the implicit network structure and address related inferential questions. We also describe extensions of the methodology to time series of grouped observations.