Une formule pour calculer la distribution du maximum d'un processus stochastique

We give a formula for the distribution of the supremum on an interval of a random process with a real parameter. Our method is based on the number of crossings of a level. Two procedures are described: 1) for processes having C^∞ paths; 2) for processes having only continuous paths. The example of stationary Gaussian processes is considered.     

Piecewise deterministic Markov processes

In this paper we construct and study a class of Markov processes whose sample paths are piecewise deterministic and whose random jumps can have limiting points in finite time intervals. A characterization of the infinitesimal generators of such processes will be given     

2-microlocal analysis of martingales and stochastic integrals

Recently, a new approach in the fine analysis of sample paths of stochastic processes has been developed to predict the evolution of the local regularity under (pseudo-)differential operators. In this paper, we study the sample paths of continuous martingales and stochastic integrals. We proved that the almost sure 2-microlocal frontier of a martingale can be obtained through the local regularity of its quadratic variation. It allows to link the Hölder regularity of a stochastic integral to the regularity of the integrand and integrator processes. These results provide a methodology to predict the local regularity of diffusions from the fine analysis of its coefficients. We illustrate our work with examples of martingales with unusual complex regularity behaviour and square of Bessel processes.     

Correlation Inequalities and Applications to Vector-Valued Gaussian Random Variables and Fractional Brownian Motion

In this paper we extend certain correlation inequalities for vector-valued Gaussian random variables due to Kolmogorov and Rozanov. The inequalities are applied to sequences of Gaussian random variables and Gaussian processes. For sequences of Gaussian random variables satisfying a correlation assumption, we prove a Borel-Cantelli lemma, maximal inequalities and several laws of large numbers. This extends results of Be?ka and Ciesielski and of Hytönen and the author. In the second part of the paper we consider a certain class of vector-valued Gaussian processes which are α-Hölder continuous in p-th moment. For these processes we obtain Besov regularity of the paths of order α. We also obtain estimates for the moments in the Besov norm. In particular, the results are applied to vector-valued fractional Brownian motions. These results extend earlier work of Ciesielski, Kerkyacharian and Roynette and of Hytönen and the author.     

Asymptotic properties of autoregressive integrated moving average processes

In this paper we study the asymptotic behavior of so-called autoregressive integrated moving average processes. These processes constitute a large class of stochastic difference equations which includes among many other well-known processes the simple one-dimensional random walk. They were dubbed by G.E.P. Box and G.M. Jenkins who found them to provide useful models for studying and controlling the behavior of certain economic variables and various chemical processes. We show that autoregressive integrated moving average processes are asymptotically normally distributed, and that the sample paths of such processes satisfy a law of the iterated logarithm. We also establish a law which determines the time spent by a sample path on one or the other side of the “trend line” of the process.     

Weak convergence of approximate solutions of random equations

Approximations of random operator equations are considered where the stochastic inputs and the underlying deterministic equation are approximated simultaneously. The main convergence result asserts that, under reasonable and verifiable assumptions, a sequence of weak solutions of approximate random equations converges weakly to a weak solution of the original equation. It is shown that this theorem extends and unifies results already known. We apply our theory to approximations of random differential equations involving stochastic processes with discontinuous paths and to projection methods for nonlinear random Hammerstein integral equations in spaces of integrable functions.     

Skorohod–Loynes Characterizations of Queueing,Fluid, and Inventory Processes

We consider queueing, fluid and inventory processes whose dynamics are determined by general point processes or random measures that represent inputs and outputs. The state of such a process (the queue length or inventory level) is regulated to stay in a finite or infinite interval – inputs or outputs are disregarded when they would lead to a state outside the interval. The sample paths of the process satisfy an integral equation; the paths have finite local variation and may have discontinuities. We establish the existence and uniqueness of the process based on a Skorohod equation. This leads to an explicit expression for the process on the doubly-infinite time axis. The expression is especially tractable when the process is stationary with stationary input–output measures. This representation is an extension of the classical Loynes representation of stationary waiting times in single-server queues with stationary inputs and services. We also describe several properties of stationary processes: Palm probabilities of the processes at jump times, Little laws for waiting times in the system, finiteness of moments and extensions to tandem and treelike networks.     

Sample paths of generalized random linear mappings<Superscript>*</Superscript>

In this paper, generalized random linear mappings are viewed as random field indexed by random parameters. This point of view leads to questions about sample paths properties. We obtain various necessary and sufficient conditions ensuring the existence of nice sample paths of a generalized random linear mapping. We also give illustrative examples and applications.     

Singularity sets of Lévy processes

We completely describe the size and large intersection properties of the Hölder singularity sets of Lévy processes. We also study the set of times at which a given function cannot be a modulus of continuity of a Lévy process. The Hölder singularity sets of the sample paths of certain random wavelet series are investigated as well.     

The Median of the Number of Simple Paths on Three Vertices in the Random Graph

We study the asymptotic behavior of the random variable equal to the number of simple paths on three vertices in the binomial random graph in which the edge probability equals the threshold probability of the appearance of such paths. We prove that, for any fixed nonnegative integer b and a sufficiently large number n of vertices of the graph, the probability that the number of simple paths on three vertices in the given random graph is b decreases with n. As a consequence of this result, we obtain the median of the number of simple paths on three vertices for sufficiently large n.

     

Extending the Wong-Zakai theorem to reversible Markov processes

We show how to construct a canonical choice of stochastic area for paths of reversible Markov processes satisfying a weak H?lder condition, and hence demonstrate that the sample paths of such processes are rough paths in the sense of Lyons. We further prove that certain polygonal approximations to these paths and their areas converge in p-variation norm. As a corollary of this result and standard properties of rough paths, we are able to provide a significant generalization of the classical result of Wong-Zakai on the approximation of solutions to stochastic differential equations. Our results allow us to construct solutions to differential equations driven by reversible Markov processes of finite p-variation with p<4. Received May 18, 2001 / final version received April 3, 2001?Published online April 8, 2002     

Random self-similar trees and a hierarchical branching process

We study self-similarity in random binary rooted trees. In a well-understood case of Galton–Watson trees, a distribution on a space of trees is said to be self-similar if it is invariant with respect to the operation of pruning, which cuts the tree leaves. This only happens for the critical Galton–Watson tree (a constant process progeny), which also exhibits other special symmetries. We extend the prune-invariance setup to arbitrary binary trees with edge lengths. In this general case the class of self-similar processes becomes much richer and covers a variety of practically important situations. The main result is construction of the hierarchical branching processes that satisfy various self-similarity definitions (including mean self-similarity and self-similarity in edge-lengths) depending on the process parameters. Taking the limit of averaged stochastic dynamics, as the number of trajectories increases, we obtain a deterministic system of differential equations that describes the process evolution. This system is used to establish a phase transition that separates fading and explosive behavior of the average process progeny. We describe a class of critical Tokunaga processes that happen at the phase transition boundary. They enjoy multiple additional symmetries and include the celebrated critical binary Galton–Watson tree with independent exponential edge length as a special case. Finally, we discuss a duality between trees and continuous functions, and introduce a class of extreme-invariant processes, constructed as the Harris paths of a self-similar hierarchical branching process, whose local minima has the same (linearly scaled) distribution as the original process.     

Modeling of an Asymptotically Central Markov Process on 3D Young Graph

The paper discusses the asymptotics of path probabilities in Markov processes close to a central one on 3D Young graph. We consider a one-parameter family of such processes and compute the parameter value in such a way that the centrality condition is satisfied with the greatest possible accuracy. We defined a normalized dimension of paths in the 3D Young graph. We study the growth and oscillations of these normalized dimensions along greedy and random trajectories of the processes using large computer experiments involving 3D Young diagrams with millions of boxes.     

A new approach to stochastic evolution equations with adapted drift

In this paper we develop a new approach to stochastic evolution equations with an unbounded drift A which is dependent on time and the underlying probability space in an adapted way. It is well-known that the semigroup approach to equations with random drift leads to adaptedness problems for the stochastic convolution term. In this paper we give a new representation formula for the stochastic convolution which avoids integration of non-adapted processes. Here we mainly consider the parabolic setting. We establish connections with other solution concepts such as weak solutions. The usual parabolic regularity properties are derived and we show that the new approach can be applied in the study of semilinear problems with random drift. At the end of the paper the results are illustrated with two examples of stochastic heat equations with random drift.     

Diffusions of multiplicative cascades

A multiplicative cascade can be thought of as a randomization of a measure on the boundary of a tree, constructed from an iid collection of random variables attached to the tree vertices. Given an initial measure with certain regularity properties, we construct a continuous time, measure-valued process whose value at each time is a cascade of the initial one. We do this by replacing the random variables on the vertices with independent increment processes satisfying certain moment assumptions. Our process has a Markov property: at any given time it is a cascade of the process at any earlier time by random variables that are independent of the past. It has the further advantage of being a martingale and, under certain extra conditions, it is also continuous. For Gaussian independent increment processes we develop the infinite-dimensional stochastic calculus that describes the evolution of the measure process, and use it to compute the optimal Hölder exponent in the Wasserstein distance on measures. We also discuss applications of this process to the model of tree polymers.     

On increasing continuous processes

It is shown that increasing continuous semimarkov processes are functional inverses of increasing processes with independent (non-stationary) increments. In particular, every increasing continuous strong Markov process has deterministic paths, up to random killing.     

Analysis of Bidirectional Ballot Sequences and Random Walks Ending in Their Maximum

Consider non-negative lattice paths ending at their maximum height, which will be called admissible paths. We show that the probability for a lattice path to be admissible is related to the Chebyshev polynomials of the first or second kind, depending on whether the lattice path is defined with a reflective barrier or not. Parameters like the number of admissible paths with given length or the expected height are analyzed asymptotically. Additionally, we use a bijection between admissible random walks and special binary sequences to prove a recent conjecture by Zhao on ballot sequences.     

Self-similar fragmentations derived from the stable tree II: splitting at nodes

We study a natural fragmentation process of the so-called stable tree introduced by Duquesne and Le Gall, which consists in removing the nodes of the tree according to a certain procedure that makes the fragmentation self-similar with positive index. Explicit formulas for the semigroup are given, and we provide asymptotic results. We also give an alternative construction of this fragmentation, using paths of Lévy processes, hence echoing the two alternative constructions of the standard additive coalescent by fragmenting the Brownian continuum random tree or using Brownian paths, respectively due to Aldous-Pitman and Bertoin.Mathematics Subject Classification (2000): 60J25, 60G52Acknowledgement Many thanks to Jean Bertoin for many precious comments on this work, and to Jean-François Le Gall for discussions related to the stable tree. Thanks also to an anonymous referee for a careful reading and very helpful comments that helped to consequently improve the presentation of this work.     

Forward equations for reflected diffusions with jumps

In this paper we obtain the forward equations associated with the evolution of the density, if it exists, of reflected diffusions on the positive orthant with jumps which form a marked point process whose random jump measure possesses a stochastic intensity. These results generalize the so-called generalized Dynkin equations for piecewise deterministic jump processes due to Davis. We then consider the stationary case where the existence of a stochastic intensity is not needed. The techniques are based on local times and the use of random jump measures. We discuss the application of these results to problems arising in queuing and storage processes as well as stationary distributions of diffusions with delayed and jump reflections at the origin.This research was supported in part by the Quebec-France Cooperative Research Program and by the Natural Sciences and Engineering Research Council of Canada under Grant OGP 0042024.     

A Markov chain Monte Carlo Expectation Maximization Algorithm for Statistical Analysis of DNA Sequence Evolution with Neighbor-Dependent Substitution Rates

The evolution of DNA sequences can be described by discrete state continuous time Markov processes on a phylogenetic tree. We consider neighbor-dependent evolutionary models where the instantaneous rate of substitution at a site depends on the states of the neighboring sites. Neighbor-dependent substitution models are analytically intractable and must be analyzed using either approximate or simulation-based methods. We describe statistical inference of neighbor-dependent models using a Markov chain Monte Carlo expectation maximization (MCMC-EM) algorithm. In the MCMC-EM algorithm, the high-dimensional integrals required in the EM algorithm are estimated using MCMC sampling. The MCMC sampler requires simulation of sample paths from a continuous time Markov process, conditional on the beginning and ending states and the paths of the neighboring sites. An exact path sampling algorithm is developed for this purpose.