Pierre Del Moral

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• Ph D 1994, HRD 2002
• Research Director at National Institute for Research in Computer Science and Control

The article presents new entropic continuity bounds for conditional expectations and conditional covariance matrices. These bounds are expressed in terms of the relative entropy between different coupling distributions. Our approach combines Wasserstein coupling with quadratic transportation cost inequalities. We illustrate the impact of these resu...

In this article we obtain several new results and developments in the study of entropic optimal transport problems (a.k.a. Schr\"odinger problems) with general reference distributions and log-concave target marginal measures. Our approach combines transportation cost inequalities with the theory of Riccati matrix difference equations arising in fil...

We develop a novel semigroup contraction analysis based on Lyapunov techniques to prove the exponential convergence of Sinkhorn equations on weighted Banach spaces. This operator-theoretic framework yields exponential decays of Sinkhorn iterates towards Schr\"odinger bridges with respect to general classes of $\phi$-divergences as well as in weight...

The Diffusion Monte Carlo method with constant number of walkers, also called Stochastic Reconfiguration as well as Sequential Monte Carlo, is a widely used Monte Carlo methodology for computing the ground-state energy and wave function of quantum systems. In this study, we present the first mathematically rigorous analysis of this class of stochas...

Entropic optimal transport problems are regularized versions of optimal transport problems. These models play an increasingly important role in machine learning and generative modelling. For finite spaces, these problems are commonly solved using Sinkhorn algorithm (a.k.a. iterative proportional fitting procedure). However, in more general settings...

In this paper we examine the numerical approximation of the limiting invariant measure associated with Feynman-Kac formulae. These are expressed in a discrete time formulation and are associated with a Markov chain and a potential function. The typical application considered here is the computation of eigenvalues associated with non-negative operat...

In this paper, we consider a genetic evolution model associated to a given Feynman–Kac flow (called also the simple genetic algorithm). We first obtain an estimate of the contraction coefficient of this interacting particle system in some suitable metric, independent of the number of particles in the system. Second, by transport-entropy inequality...

We consider the problem of Bayesian estimation of static parameters associated to a partially and discretely observed diffusion process. We assume that the exact transition dynamics of the diffusion process are unavailable, even up to an unbiased estimator and that one must time-discretize the diffusion process. In such scenarios it has been shown...

We study a class of time-inhomogeneous diffusion: the self-interacting one. We show a convergence result with a rate of convergence that does not depend on the diffusion coefficient. Finally, we establish a so-called Kramers' type law for the first exit-time of the process from domain of attractions when the landscapes are uniformly convex.

We consider the long time behavior of Wong-Zakai approximations of stochastic differential equations. These piecewise smooth diffusion approximations are of great importance in many areas, such as those with ordinary differential equations associated to random smooth fluctuations ; e.g. robust filtering problems. In many examples, the mean error es...

We propose a new solvable class of multidimensional quantum harmonic oscillators for a linear diffusive particle and a quadratic energy absorbing well associated with a semi-definite positive matrix force. Under natural and easily checked controllability conditions, for non necessarily reversible models with possibly transient free particle diffusi...

The purpose of this review is to present a comprehensive overview of the theory of ensemble Kalman–Bucy filtering for continuous-time, linear-Gaussian signal and observation models. We present a system of equations that describe the flow of individual particles and the flow of the sample covariance and the sample mean in continuous-time ensemble fi...

The stability analysis of possibly time varying positive semigroups on non-necessarily compact state spaces, including Neumann and Dirichlet boundary conditions is a notoriously difficult subject. These crucial questions arise in a variety of areas of applied mathematics, including nonlinear filtering, rare event analysis, branching processes, phys...

We study a class of time-inhomogeneous diffusion: the self-interacting one. We show a convergence result with a rate of convergence that does not depend on the diffusion coefficient. Finally, we establish a so-called Kramers' type law for the first exit-time of the process from domain of attractions when the landscapes are uniformly convex.

The stability analysis of possibly time varying positive semigroups on non necessarily compact state spaces, including Neumann and Dirichlet boundary conditions is a notoriously difficult subject. These crucial questions arise in a variety of areas of applied mathematics, including nonlinear filtering, rare event analysis, branching processes, phys...

In this article we consider the estimation of the log-normalization constant associated to a class of continuous-time filtering models. In particular, we consider ensemble Kalman–Bucy filter estimates based upon several nonlinear Kalman–Bucy diffusions. Using new conditional bias results for the mean of the aforementioned methods, we analyze the em...

We present a novel backward Itô-Ventzell formula and an extension of the Alekseev-Gröbner interpolating formula to stochastic flows. We also present some natural spectral conditions that yield direct and simple proofs of time uniform estimates of the difference between the two stochastic flows when their drift and diffusion functions are not the sa...

We study the exit-time from a domain of a self-interacting diffusion, where the Brownian motion is replaced by $\sigma B_t$ for a constant $\sigma$. The first part of this work consists in showing that the rate of convergence (of the occupation measure of the self-interacting process toward some explicit Gibbs measure) previously obtained in \cite{...

The stability and contraction properties of positive integral semigroups on locally compact Polish spaces are investigated. We provide a novel analysis based on an extension of V-norm, Dobrushin-type, contraction techniques on functionally weighted Banach spaces for Markov operators. These are applied to a general class of positive and possibly tim...

Discrete algebraic Riccati equations and their fixed points are well understood and arise in a variety of applications, however, the time-varying equations have not yet been fully explored in the literature. In this article we provide a self-contained study of discrete time Riccati matrix difference equations. In particular, we provide a novel Ricc...

Despite the widespread usage of discrete generation Ensemble Kalman particle filtering methodology to solve nonlinear and high dimensional filtering and inverse problems, little is known about their mathematical foundations. As genetic-type particle filters (a.k.a. sequential Monte Carlo), this ensemble-type methodology can also be interpreted as m...

We propose a new solvable class of multidimensional quantum harmonic oscillators for a linear diffusive particle and a quadratic energy absorbing well associated with a semi-definite positive matrix force. Under natural and easily checked controllability conditions, the ground state and the zero-point energy are explicitly computed in terms of a po...

Мы представляем обратный диффузионный поток (т.е. обратное по времени стохастическое дифференциальное уравнение), маргинальное распределение которого в любой (более ранний) момент времени равно сглаживающему распределению, когда конечное состояние (в заключительный момент) распределено согласно распределению фильтра. Это новая интерпретация сглажив...

We present a new Bayesian inference method for compartmental models that takes into account the intrinsic stochasticity of the process. We show how to formulate a SIR-type Markov jump process as the solution of a stochastic differential equation with respect to a Poisson Random Measure (PRM), and how to simulate the process trajectory deterministic...

In this article we consider the estimation of the log-normalization constant associated to a class of continuous-time filtering models. In particular, we consider ensemble Kalman-Bucy filter based estimates based upon several nonlinear Kalman-Bucy diffusions. Based upon new conditional bias results for the mean of the afore-mentioned methods, we an...

This Note and its extended version [7] present a novel backward Itô–Ventzell formula and an extension of the Aleeksev–Gröbner interpolating formula to stochastic flows. We also present some natural spectral conditions that yield direct and simple proofs of time uniform estimates of the difference between the two stochastic flows when their drift an...

The purpose of this review is to present a comprehensive overview of the theory of ensemble Kalman-Bucy filtering for linear-Gaussian signal models. We present a system of equations that describe the flow of individual particles and the flow of the sample covariance and the sample mean in continuous-time ensemble filtering. We consider these equati...

We present a new Bayesian inference method for compartmental models that takes into account the intrinsic stochasticity of the process. We show how to formulate a SIR-type Markov jump process as the solution of a stochastic differential equation with respect to a Poisson Random Measure (PRM), and how to simulate the process trajectory deterministic...

We present a backward diffusion flow (i.e. a backward-in-time stochastic differential equation) whose marginal distribution at any (earlier) time is equal to the smoothing distribution when the terminal state (at a latter time) is distributed according to the filtering distribution. This is a novel interpretation of the smoothing solution in terms...

We present forward-backward stochastic semigroup formulae to analyse the difference of diffusion flows driven by different drift and diffusion functions. These formulae are expressed in terms of tangent and Hessian processes and can be interpreted as an extension of the Aleeksev-Gr{\"o}bner lemma to diffusion flows. We present some natural spectral...

We propose a second order differential calculus to analyze the regularity and the stability properties of the distribution semigroup associated with McKean-Vlasov diffusions. This methodology provides second order Taylor type expansions with remainder for both the evolution semigroup as well as the stochastic flow associated with this class of nonl...

The article presents a novel variational calculus to analyze the stability and the propagation of chaos properties of nonlinear and interacting diffusions. This differential methodology combines gradient flow estimates with backward stochastic interpolations, Lyapunov linearization techniques as well as spectral theory. This framework applies to a...

This work is concerned with the stability properties of linear stochastic differential equations with random (drift and diffusion) coefficient matrices and the stability of a corresponding random transition matrix (or exponential semigroup). We consider a class of random matrix drift coefficients that involves random perturbations of an exponential...

The article presents a rather surprising Floquet-type representation of time-varying transition matrices associated with a class of nonlinear matrix differential Riccati equations. The main difference with conventional Floquet theory comes from the fact that the underlying flow of the solution matrix is aperiodic. The monodromy matrix associated wi...

The article presents a novel variational calculus to analyze the stability and the propagation of chaos properties of nonlinear and interacting diffusions. This differential methodology combines gradient flow estimates with backward stochastic interpolations, Lyapunov linearization techniques as well as spectral theory. This framework applies to a...

The stability properties of matrix-valued Riccati diffusions are investigated. The matrix-valued Riccati diffusion processes considered in this work are of interest in their own right, as a rather prototypical model of a matrix-valued quadratic stochastic process. In addition, this class of stochastic models arise in signal processing and data assi...

Consider the continuous-time matrix Riccati operator Ricc(Q)=AQ+QA′−QSQ+R. In this work, we consider the robustness of this operator to direct perturbations of the matrices (A, R, S) and, in particular, the flow robustness of the corresponding Riccati differential equation. For a given class of perturbation, we show that the corresponding different...

Continuous time Feynman-Kac measures on path spaces are central in applied probability, partial differential equation theory, as well as in quantum physics. This article presents a new duality formula between normalized Feynman-Kac distribution and their mean field particle interpretations. Among others, this formula allows us to design a reversibl...

The article presents a rather surprising Floquet-type representation of time-varying transition matrices associated with a class of nonlinear matrix Riccati differential equations. The main difference with conventional Floquet theory comes from the fact that the underlying flow of the solution matrix is aperiodic. The monodromy matrix associated wi...

This work is concerned with the stability properties of linear stochastic differential equations with random (drift and diffusion) coefficient matrices, and the stability of a corresponding random exponential semigroup. We consider a class of random matrix drift coefficients that involves random perturbations of an exponentially stable flow of dete...

Random matrix theory plays a central role in statistical physics, computational mathematics and engineering sciences, including data assimilation, signal processing, combinatorial optimization, compressed sensing, econometrics and mathematical finance, among numerous others. The mathematical foundations of the theory of random matrices are technica...

This article is concerned with the fluctuation analysis and the stability properties of a class of one-dimensional Riccati diffusions. This class of Riccati diffusion is quite general, and arises, for example, in data assimilation applications, and more particularly in ensemble (Kalman-type) filtering theory. These one-dimensional stochastic differ...

This article is concerned with the fluctuation analysis and the stability properties of a class of one-dimensional Riccati diffusions. These one-dimensional stochastic differential equations exhibit a quadratic drift function and a non-Lipschitz continuous diffusion function. We present a novel approach, combining tangent process techniques, Feynma...

We analyse various perturbations and projections of Kalman-Bucy semigroups and Riccati equations. For example, covariance inflation-type perturbations and localisation methods (projections) are common in the ensemble Kalman filtering literature. In the limit of these ensemble methods, the regularised sample covariance tends toward a solution of a p...

This article provides a comprehensive, rigorous, and self-contained introduction to the analysis of Wishart matrix moments. This article may act as an introduction to some aspects of random matrix theory, or as a self-contained exposition of Wishart matrix moments. Random matrix theory plays a central role in nuclear and statistical physics, comput...

These lecture notes provide a comprehensive, self-contained introduction to the analysis of Wishart matrix moments. This study may act as an introduction to some particular aspects of random matrix theory, or as a self-contained exposition of Wishart matrix moments. Random matrix theory plays a central role in statistical physics, computational mat...

Matrix differential Riccati equations are central in filtering and optimal control theory. The purpose of this article is to develop a perturbation theory for a class of stochastic matrix Riccati diffusions. Diffusions of this type arise, for example, in the analysis of ensemble Kalman-Bucy filters since they describe the flow of certain sample cov...

Matrix differential Riccati equations are central in filtering and optimal control theory. The purpose of this article is to develop a perturbation theory for a class of stochastic matrix Riccati diffusions. Diffusions of this type arise, for example, in the analysis of ensemble Kalman-Bucy filters since they describe the flow of certain sample cov...

This short note provides an explicit description of the the Fr\'echet derivatives of the principal square root matrix functional. A Taylor expansion with an integral remainder term is also provided.

This short note provides an explicit description of the Fr\'echet derivatives of the principal square root matrix functional at any order. We present an original formulation that allows to compute sequentially the Fr\'echet derivatives of the matrix square root at any order starting from the first order derivative. A Taylor expansion at any order w...

This article provides a new theory for the analysis of the particle Gibbs (PG) sampler Andrieu et al. (2010). Following the work of Del Moral et al. (2017) we provide some analysis of the particle Gibbs sampler, giving first order expansions of the kernel and minorization estimates. In addition, first order propagation of chaos estimates are derive...

This article provides a new theory for the analysis of forward and backward particle approximations of Feynman-Kac models. Such formulae are found in a wide variety of applications and their numerical (particle) approximation is required due to their intractability. Under mild assumptions, we provide sharp and non-asymptotic first order expansions...

Let $\Xa=XX^{\prime}$ be a random matrix associated with a centered $r$-column centered Gaussian vector $X$ with a covariance matrix $P$. In this article we compute expectations of matrix-products of the form $\prod_{1\leq i\leq n}(\Xa P^{v_i})$ for any $n\geq 1$ and any multi-index parameters $v_i\in\NN$. We derive closed form formulae and a simpl...

We introduce a new class of Monte Carlo based approximations of expectations of random variables such that their laws are only available via certain discretizations. Sampling from the discretized versions of these laws can typically introduce a bias. In this paper, we show how to remove that bias, by introducing a new version of multi-index Monte C...

The purpose of this work is to analyse the effect of various perturbations and projections of Kalman-Bucy semigroups and Riccati equations. The original motivation was to understand the behaviour of various regulation methods used in ensemble Kalman filtering (EnKF). For example, covariance inflation-type methods (perturbations) and covariance loca...

In this article, we consider the multilevel sequential Monte Carlo (MLSMC) method of Beskos et al. (Stoch. Proc. Appl. [to appear]). This is a technique designed to approximate expectations w.r.t. probability laws associated to a discretization. For instance, in the context of inverse problems, where one discretizes the solution of a partial differ...

Particle island models [31] provide a means of parallelization of sequential Monte Carlo methods, and in this paper we present novel convergence results for algorithms of this sort. In particular we establish a central limit theorem—as the number of islands and the common size of the islands tend jointly to infinity—of the double bootstrap algorith...

In this article we consider importance sampling (IS) and sequential Monte Carlo (SMC) methods in the context of 1-dimensional random walks with absorbing barriers. In particular, we develop a very precise variance analysis for several IS and SMC procedures. We take advantage of some explicit spectral formulae available for these models to derive sh...

In this article we consider importance sampling (IS) and sequential Monte Carlo (SMC) methods in the context of 1-dimensional random walks with absorbing barriers. In particular, we develop a very precise variance analysis for several IS and SMC procedures. We take advantage of some explicit spectral formulae available for these models to derive sh...

This article analyses a new class of advanced particle Markov chain Monte Carloalgorithms recently introduced by Andrieu, Doucet, and Holenstein (2010). We presenta natural interpretation of these methods in terms of well known unbiasedness propertiesof Feynman-Kac particle measures, and a new duality with Feynman-Kac models.This perspective sheds...

The Kalman-Bucy filter is the optimal state estimator for an Ornstein-Ulhenbeck diffusion given that the system is partially observed via a linear diffusion-type (noisy) sensor. Under Gaussian assumptions, it provides a finite-dimensional exact implementation of the optimal Bayes filter. It is generally the only such finite-dimensional exact instan...

The Kalman-Bucy filter is the optimal state estimator for an Ornstein-Uhlenbeck diffusion given that the system is partially observed via a linear diffusion-type (noisy) sensor. Under Gaussian assumptions, it provides a finite-dimensional exact implementation of the optimal Bayes filter. It is generally the only such finite-dimensional exact instan...

Collision between satellites and space debris seldom happens, but the loss of a satellite by collision may have catastrophic consequences both for the satellite mission and for the space environment. To support the decision to trigger off a collision avoidance manoeuver, an adapted tool is the determination of the collision probability between debr...

Sequential Monte Carlo (SMC) methods have been used successfully in many applications in engineering, statistics and physics. However, they are seldom used in financial option pricing literature and its practice. We present an SMC method for pricing barrier options with continuous and discrete monitoring of the barrier condition. With our method, s...

The exponential stability and the concentration properties of a class of extended Kalman-Bucy filters are analyzed. New estimation concentration inequalities around partially observed signals are derived in terms of the stability properties of the filters. These non asymptotic exponential inequalities allow to design confidence interval type estima...

This article is concerned with the exponential stability and the uniform propagation of chaos properties of a class of Extended Ensemble Kalman-Bucy filters with respect to the time horizon. This class of nonlinear filters can be interpreted as the conditional expectations of nonlinear McKean Vlasov type diffusions with respect to the observation p...

The exponential stability and the concentration properties of a class of extended Kalman-Bucy filters are analyzed. New estimation concentration inequalities around partially observed signals are derived in terms of the stability properties of the filters. These non asymptotic exponential inequalities allow to design confidence interval type estima...

This article is concerned with the exponential stability and the uniform propagation of chaos properties of a class of Extended Ensemble Kalman-Bucy filters with respect to the time horizon. This class of nonlinear filters can be interpreted as the conditional expectations of nonlinear McKean Vlasov type diffusions with respect to the observation p...

The Ensemble Kalman filter is a sophisticated and powerful data assimilation method for filtering high dimensional problems arising in fluid mechanics and geophysical sciences. This Monte Carlo method can be interpreted as a mean-field McKean-Vlasov type particle interpretation of the Kalman-Bucy diffusions. In contrast to more conventional particl...

The Ensemble Kalman filter is a sophisticated and powerful data assimilation method for filtering high dimensional problems arising in fluid mechanics and geophysical sciences. This Monte Carlo method can be interpreted as a mean-field McKean-Vlasov type particle interpretation of the Kalman-Bucy diffusions. In contrast to more conventional particl...

Estimating rare event probability with accuracy is of great interest for safety and reliability applications. In this paper, we focus on rare events which can be modeled by a threshold exceedance of a deterministic input–output function with random inputs. Some parameters of this function or density parameters of input random variables may be fixed...

This article considers the sequential Monte Carlo (SMC) approximation of ratios of normalizing constants associated to posterior distributions which in principle rely on continuum models. Therefore, the Monte Carlo estimation error and the discrete approximation error must be balanced. A multilevel strategy is utilized to substantially reduce the c...

This article considers the sequential Monte Carlo (SMC) approximation of ratios of normalizing constants associated to posterior distributions which in principle rely on continuum models. Therefore, the Monte Carlo estimation error and the discrete approximation error must be balanced. A multilevel strategy is utilized to substantially reduce the c...

In the last decade, the area of multiple target tracking has witnessed the introduction of important concepts and methods, aiming at establishing principled approaches for dealing with the estimation of multiple objects in an efficient way. One of the most successful classes of multi-object filters that have been derived out of these new grounds in...

In the following article, we investigate a particle filter for approximating Feynman–Kac models with indicator potentials and we use this algorithm within Markov chain Monte Carlo (MCMC) to learn static parameters of the model. Examples of such models include approximate Bayesian computation (ABC) posteriors associated with hidden Markov models (HM...

A formulation of the hypothesised filter for independent stochastic populations (hisp) is proposed, based on the concept of association measure, which is a measure on the set of observation histories. Using this formulation, a particle approximation is introduced at the level of the association measure for handling the exponential growth in the num...

This article is concerned with moderate deviation principles of a class of interacting empirical processes. We derive an explicit description of the rate function, and we illustrate these results with Feynman-Kac particle models arising in nonlinear filtering, statistical machine learning, rare event analysis, and computational physics. We discuss...

This Note and its extended version [10] present a new duality formula between genetic type genealogical tree based particle models and Feynman–Kac measures on path spaces. Among others, this formula allows us to design reversible Gibbs–Glauber Markov chains for Feynman–Kac integration on path spaces. Our approach yields new Taylor series expansions...

Biips is a software platform for automatic Bayesian inference with interacting particle systems. Biips allows users to define their statistical model in the probabilistic programming BUGS language, as well as to add custom functions or samplers within this language. Then it runs sequential Monte Carlo based algorithms (particle filters, particle in...

We consider an elliptic and time-inhomogeneous diffusion process with time-periodic coefficients evolving in a bounded domain of R d with a smooth boundary. The process is killed when it hits the boundary of the domain (hard killing) or after an exponential time (soft killing) associated with some bounded rate function. The branching particle inter...

This article provides a new theory for the analysis of forward and backward particle approximations of Feynman-Kac models. Such formulae are found in a wide variety of applications and their numerical (particle) approximation are required due to their intractability. Under mild assumptions, we provide sharp and non-asymptotic first order expansions...

Particle island models (Verg\'e et al., 2013) provide a means of parallelization of sequential Monte Carlo methods, and in this paper we present novel convergence results for algorithms of this sort. In particular we establish a central limit theorem - as the number of islands and the common size of the islands tend jointly to infinity - of the dou...

The article deals with the propagation of chaos for a system of interacting particles. Under suitable assumptions, if the system at time t=0 is chaotic (that is to say the particles are independent), this chaos propagates as the number of particles goes to infinity. Here, we deal with a case in which the system at time t=0 is not chaotic and we sho...

The purpose of this paper is to present a law of large numbers and a central limit theorem for Adaptive Multilevel Splitting algorithms. In rare event estimation, Multilevel Splitting is a sequential Monte Carlo method to estimate the probability of a rare event as well as to simulate realisations of this event. Contrarily to the fixed-levels versi...