Organizer: Mogens Bladt (IIMAS, UNAM)
Some multivariate distributions with exponential marginal
Bo Friis Nielsen (Technical University of Denmark)
We present examples of distributions with exponential marginals that can be classified as multivariate phase-type distributions. Particularly we put some emphasis on a bivariate distribution recently proposed by Bladt and Nielsen that allows for arbitrary values of the correlation coefficient. This distribution can be interpreted as a mixture of order statistics, which allows for the generalisation to higher dimensions.
Estimation of paleo-temperatures with ice-core data via integrated diffusion processes
Fernando Baltzar Larios (UNAM)
We present a method for obtaining maximum likelihood estimates of parameters in diffusion models when the data is a discrete time sample of the integral of the process, while no direct observations of the process itself are available. The data are, moreover, assumed to be contaminated by measurement errors. The data can be viewed as incomplete observations of a model with a tractable likelihood function. Therefore we propose a simulated EM-algorithm to obtain maximum likelihood estimates of the parameters in the diffusion model. The method is applied to a set of ice-core data on oxygen isotopes used to investigate paleo-temperatures.
Non-stabilizability of queueing networks with infinite supplies
Leonardo Rojas Nandayapa (The University of Queensland)
Queueing network models are very useful for describing congestion and resource scarcity phenomena occurring in manufacturing, service and telecommunication applications. A typical queueing network model consists of servers, buffers and routes as well as stochastic assumptions on the rates and durations of processing. For a stochastic queueing network model and an associated control policy, stability implies that the number of items in the system remains stochastically bounded over time. Finding sensible control laws that stabilize the network is crucial for applications yet often not trivial.
We consider a family of queueing networks that generate their own arrivals. Such networks are interesting because they allow servers to be fully utilized yet can remain stable. In this talk we present results of a negative flavour: non-stabilizability. We show that in some cases queueing networks that generate their own arrivals cannot be stabilized while remaining fully utilized. The proof method of this last result is based on a novel linear martingale argument.
This is joint work with Yoni Nazarathy and Thomas Salisbury.
Lévy processes and applications
Organizer: Löic Chaumont
A Lamperti type representation of real-valued self-similar of Markov Processes
Henry Panti Trejo (Universidad Autónoma de Yucatán)
Lamperti in 1972 and several subsequent papers, establish that all positive self-similar Markov processes can be expressed as the exponential of Lévy processes time changed by the inverse of their exponential functional. This relation between positive self-similar Markov processes and Lévy processes is called the Lamperti representation. In this work we obtain a Lamperti type representation for real-valued self-similar Markov processes killed at their first hitting time of zero. One of the main result in this work states that all real-valued self-similar Markov processes killed at their first hitting time of zero can be expressed as time changed multiplicative invariant processes. We provide two examples where the characteristics of the multiplicative invariant process in the Lamperti representation can be computed explicitly: the alpha stable process and the alpha stable process conditioned to avoid zero.
This is a joint work with Loïc Chaumont and Victor Rivero.
Random stable looptrees
Igor Kortchemski (Ecole Normal Superièure)
We introduce a class of random compact metric spaces $L_\alpha$ indexed by $\alpha \in (1,2)$ and which we call stable looptrees. They are made of a collection of random loops glued together along a tree structure, can be informally be viewed as dual graphs of $\alpha$ -stable Lévy trees and are coded by a spectrally positive $\alpha$-stable Lévy process. We study their properties and see in particular that the Hausdorff dimension of $L_\alpha$ is almost surely equal to $\alpha$. We also show that stable looptrees are universal scaling limits, for the Gromov–Hausdorff topology, of various combinatorial models. We finally see that the stable looptree of parameter $3/2$ is closely related to the scaling limits of cluster boundaries in critical site-percolation on large random triangulations.
Based on joint works with Nicolas Curien.
Sharp two-sided Dirichlet heat kernel estimates for subordinate Brownian motions.
Renming Song (University of Illinois)
A subordinate Brownian motion is a Levy process which can be obtained from a Brownian motion by replacing the time parameter o Brownian motion by an independent subordinator. Subordinate Brownian motions form a large subclass of Levy processes. Recently, a lot of progress has been made in establishing sharp two-sided heat kernel estimates for subordinate Brownian motions in smooth open sets. In this talk, I will give a survey of some of these recent results.
Stochastic control and applications
Organizer: Daniel Hernández-Hernández (CIMAT)
Controlling a n-Dimensional Lévy Process
Harold Moreno (CIMAT)
Integral equations for the ruin problem with discounted factor
Carlos Pacheco (CINVESTAV)
We will discuss the renewal property behind a discounted insurance model. As in the classical case (non-discounted), such property gives rise to some characterization of the ruin probability using integral equations. We will also discuss some equations arising from characterizing the time of ruin.
Probability and analysis
Organizer: José Alfredo López Mimbela
Global and nonglobal solutions for a nonautonomous semilinear system with Dirichlet boundary condition
Aroldo Pérez Pérez (UJAT)
Reversibility and Deviation from Equilibrium in Quantum Markov Semigroups
Roberto Quezada (UAM-I)
Quantum Markov Semigroups (QMS) where introduced in physics to model the non-unitary evolution of a quantum system interacting with its environment. The concept of equilibrium steady state for this class of semigroups is relatively well understood and there exists several characterizations, quantum detailed balance among them. However, there are few mathematically rigorous approaches to the notion of non-equilibrium steady state, even when these states play an important role in the study of complex systems. In this talk we shall discuss this notion through a definition of time reversal QMS and entropy production.
Fractional processes as models in mathematical finance
Marco Dozzi (Universidad de Nancy)
Classical models in mathematical finance, such as pricing and interest rate models are often formulated in terms of Lévy processes (i.e. processes with stationary and independent increments). We propose in this talk possible extensions of such models to processes with memory, which are not necessarily semimartingales and show how classical problems in finance can be treated in this context. More precisely, we present (i) a class of filtered compound Poisson processes whose jump intensities are stochastic and with memory, and give elements of the stochastic calculus for such processes, (ii) a new way to solve pricing problems in non-arbitrage conditions in the presence of fractional processes.
This is joint work with S. El Rahouli (University of Luxemburg).
Statistical Inference for Stochastic Processes
Organizer: J. Enrique Figueroa-López
Bayesian Inference via Filtering Equations for Ultra-High Frequency Data
We propose a general partially-observed framework of Markov processes with marked point process observations for ultra-high frequency (UHF) data, allowing other observable economic or market factors. We develop the corresponding Bayesian inference via filtering equations to quantify parameter and model uncertainty. Specifically, we derive filtering equations to characterize the evolution of the statistical foundation such as likelihoods, posteriors, Bayes factors and posterior model probabilities. Given the computational challenge, we provide a powerful convergence theorem, enabling us to employ the Markov chain approximation method to construct consistent, easily-parallelizable, recursive algorithms. The algorithms calculate the fundamental statistical characteristics and are capable of implementing the Bayesian inference in real-time for streaming UHF data. The general theory is illustrated by specific models built for U.S. Treasury Notes transactions data from GovPX and by Heston stochastic volatility model.
Maximum Likelihood Estimation of Misspecified Diffusion Models
We study the consequences of model misspecification for diffusion processes with a new asymptotic framework when maximum likelihood estimators (MLEs) are used for estimation and inference. The new asymptotic theory leads to the following results; 1) the probability limits of MLEs (pseudo-true values) for diffusion function parameters do not depend on drift function specifications and are different from those with data observed at fixed time intervals, 2) pseudo-true values of drift function parameters depend on diffusion function specifications and approximation methods of transition densities. We derive new two-tier information criteria, namely the primary and secondary criteria, which measure the divergence from a true diffusion to another diffusion based on the Kullback-Leibler information criterion. Pseudo-true values of diffusion and drift function parameters minimize the primary and secondary information criteria sequentially respectively. Based on our main results, we also study estimation under partial misspecification.
Optimal threshold estimators for Levy Jump Diffusion Models
Jeffrey A. Nisen
When estimating the parameters of a jump diffusion process it is natural to use threshold type estimators. While the asymptotic properties for the class of threshold estimators has been well document in the literature, the problem of how to objectively select the threshold parameter has previously not received much attention. In noticing that every threshold estimator gives rise to three sources of estimation error; false positive jump detection error, false negative jump detection error, and intrinsic natural variability, we introduce a novel and well-posed optimization problem designed to select estimators which minimize the first two sources. We analyze this problem within the class of Levy Jump Diffusion processes and demonstrate the existence and uniqueness of an "optimal'' threshold estimator. Moreover, we provide an explicit infill asymptotic characterization of the optimal threshold parameter and propose novel estimation algorithms which allow for feasible implementation of these estimators. Numerical experiments highlight the improved finite sample performance of the new estimators relative to some commonly used alternatives. These results open new and interesting directions for statistical estimation of stochastic processes with jumps. This is based on join work with Dr. Figueroa-Lopez.
Long-time convergence rates for Markov processes: probabilistic and functional methods
Organizer: Joaquín Fontbona (Center for Mathematical Modelling-Universidad de Chile)
Bakry-Emery meet Villani
Fabrice Baudoin (Purdue University)
We study gradient bounds for solutions of degenerate Fokker-Planck equations on manifolds, including as a special case, the celebrated kinetic Fokker-Planck equation. Our method generalizes to hypoelliptic operators the Bakry-Emery's approach and allows to recover and strengthen hypocoercive estimates obtained by Villani.
Quantitative estimates for the long time behavior of an ergodic variant of the telegraph process
Hélène Guérin (Centre de Recherches Mathématiques, Université de Montréal)
Motivated by stability questions on piecewise deterministic Markov models of bacterial chemotaxis, we study the long time behavior of a variant of the classic telegraph process having a non-constant jump rate that induces a drift towards the origin. We compute its invariant law and show exponential ergodicity, obtaining a quantitative control of the total variation distance to equilibrium at each instant of time. This result rely on an explicit construction of an original coalescent coupling for both velocity and position. This is a joint work with F. Malrieu and J. Fontbona.
A tractorial interpretation of entropy dissipation
Joaquín Fontbona (Center for Mathematical Modelling-Universidad de Chile)
We introduce and develop a pathwise description of the dissipation of general convex entropies for continuous time Markov processes based on time reversal. The entropy is in this setting the expected value of a backward submartingale. By expliciting its Doob Meyer decomposition in the case of (non necessarily reversible) Markov diffusion processes, we provide a stochastic analogue of the well known entropy dissipation formula, for general convex entropies (including total variation). As a first application of this approach, using stochastic flow techniques we obtain a new Bakry-Emery criterion for exponential convergence of the entropy to 0, which depends on the square root of the diffusion matrix. Secondly, we derive some conditions for exponential convergence to equilibriuim for kinetic Ornstein-Uhlenbeck processes. Some open questions are then discussed.
Based on joint works with Benjamin Jourdain and with Leonardo Sepulveda.
Organizer: Arno Siri-Jegousse
E. coli in a forest of Yule trees: A weak mutation - strong selection model for experimental
Adrián González-Casanova, TU Berlin
Inspired by the Lenski experiment for the evolution of E. coli, we discuss a model with random reproduction that under a suitable rescaling leads to a stochastic differential equation. We quantify assumptions which lead to a separation of timescales for the effects of mutation and selection. This makes the model tractable and gives some explanation of the form of the fitness curve observed in the long term experiment.
Joint work with A. Wakolbinger and L. Yuan
Superprocesses and travelling waves
Antonio Murillo, Universidad de Guanajuato
We shall discuss the existence of travelling waves solutions to some FKPP-type equations related tosome class of superprocesses. This is done by means of the so-called spine decomposition.
The backbone decomposition for spatially dependent supercritical superprocess
José Luis Pérez Garmendia, ITAM
Consider any supercritical Galton-Watson process which may become extinct with positive probability. It is a well-understood and intuitively obvious phenomenon that, on the survival set, the process may be pathwise decomposed into a stochastically "thinner" Galton-Watson process, which almost surely survives and which is decorated with immigrants, at every time step, initiating independent copies of the original Galton-Watson process conditioned to become extinct. The thinner process is known as the backbone and characterizes the genealogical lines of descent of proli?c individuals in the original process. Here, proli?c means individuals who have at least one descendant in every subsequent generation to their own. Starting with Evans and O'Connell, there exists a cluster of literature, describing the analogue of this decomposition (the so-called backbone=decomposition) for a variety of different classes of superprocesses and continuous-state branching processes. Note that the latter family of stochastic processes may be seen as the total mass process of superprocesses with non-spatially dependent branching mechanism.
In this talk we consolidate the aforementioned collection of results concerning backbone decompositions and describe a result for a general class of supercritical superprocesses with spatially dependent branching mechanisms. Our approach exposes the commonality and robustness of many of the existing arguments in the literature.
Stochastic processes for frequency models
Organizer: Matteo Ruggiero
The arrangement problem for Poisson-Dirichlet random measures and partitions
Alexander Gnedin, Queen Mary University of London
There are three natural orders on the collection of masses of the two-parameter PD measure, respectively on the set of blocks of the associated integer partition: by decrease of sizes, the size biased order associated with a stick-breaking construction, and the regenerative order associated with a subordinator construction. For Ewens' subfamily the size-biased and the regenerative orders coincide, but for other measures/partitions the relation between the orders is more delicate. The talk is devoted to a construction of the regenerative order in the general PD case.
Based on a joint work with C. Haulk and J. Pitman.
Large Deviations for Frequency Counts
Shui Feng, McMaster University
The frequency counts are studied for the random sample from the two-parameter Poisson-Dirichlet process. More specifically the talk will focus on large deviations for the frequency counts under the conditional and unconditional laws. Unlike the fluctuation results, large deviation result under the conditional law turns out to be the same as under the unconditional laws. The meaning of these results and their potential applications will be discussed.
This is a joint work with Stefano Favaro.
Polynomial Spectrum and moment dualities for time-dependent Gamma and Dirichlet random
Dario Spanò, University of Warwick
A wide class of continuous time-dependent random measures with gamma or Dirichlet stationary law will be introduced by means of kernels which map polynomials to polynomials of the same degree. All such kernels allow for a computable and transparent probabilistic structure in connection with special function theory and Population Genetics. The connection between gamma and Dirichlet polynomial kernels is fully understood only in two very well known cases of Dawson Watanabe measure-valued diffusions and neutral, parent independent Fleming-Viot processes. One way to undertand their connection is via their moment duality with two classes of discrete stochastic processes intimately related to Kingman’s coalescent process. We will use such duality results, well-known in Population Genetics, to shed some light on the interplay between gamma and dirichlet polynomial kernels beyond the known cases.