Laboratoire de Mathématiques d'Orsay

We are concerned with different properties of backward stochastic differential equations and their applications to finance. These equations, first introduced by Pardoux and Peng (1990), are useful for the theory of contingent claim valuation, especially cases with constraints and for the theory of recursive utilities, introduced by Duffie and Epstein (1992a, 1992b).

This special issue of Mathematical Structures in Computer Science contains several contributions related to the modern field of Quantum Information and Quantum Computing. The first two papers deal with entanglement. The paper by R. Mosseri and P. Ribeiro presents a detailed description of the two- and three-qubit geometry in Hilbert space, dealing with the geometry of fibrations and discrete geometry. The paper by J.-G.Luque et al . is more algebraic and considers invariants of pure k-qubit states and their application to entanglement measurement.

This article investigates estimation of finite population totals in the presence of univariate or multivariate auxiliary information. Estimation is equivalent to attaching weights to the survey data. We focus attention on the several weighting systems that can be associated with a given amount of auxiliary information and derive a weighting system with the aid of a distance measure and a set of calibration equations. We briefly mention an application to the case in which the information consists of known marginal counts in a two- or multi-way table, known as generalized raking. The general regression estimator (GREG) was conceived with multivariate auxiliary information in mind. Ordinarily, this estimator is justified by a regression relationship between the study variable y and the auxiliary vector x. But we note that the GREG can be derived by a different route by focusing instead on the weights. The ordinary sampling weights of the kth observation is 1/πk , where πk is the inclusion probability of k. We show that the weights implied by the GREG are as close as possible, according to a given distance measure, to the 1/πk while respecting side conditions called calibration equations. These state that the sample sum of the weighted auxiliary variable values must equal the known population total for that auxiliary variable. That is, the calibrated weights must give perfect estimates when applied to each auxiliary variable. That is a consistency check that appeals to many practitioners, because a strong correlation between the auxiliary variables and the study variable means that the weights that perform well for the auxiliary variable also should perform well for the study variable. The GREG uses the auxiliary information efficiently, so the estimates are precise; however, the individual weights are not always without reproach. For example, negative weights can occur, and in some applications this does not make sense. It is natural to seek the root of the dissatisfaction in the underlying distance measure. Consequently, we allow alternative distance measures that satisfy only a set of minimal requirements. Each distance measure leads, via the calibration equations, to a specific weighting system and thereby to a new estimator. These estimators form a family of calibration estimators. We show that the GREG is a first approximation to all other members of the family; all are asymptotically equivalent to the GREG, and the variance estimator already known for the GREG is recommended for use in any other member of the family. Numerical features of the weights and ease of computation become more than anything else the bases for choosing between the estimators. The reasoning is applied to calibration on known marginals of a two-way frequency table. Our family of distance measures leads in this case to a family of generalized raking procedures, of which classical raking ratio is one.

Rényi divergence is related to Rényi entropy much like Kullback-Leibler divergence is related to Shannon's entropy, and comes up in many settings. It was introduced by Rényi as a measure of information that satisfies almost the same axioms as Kullback-Leibler divergence, and depends on a parameter that is called its order. In particular, the Rényi divergence of order 1 equals the Kullback-Leibler divergence. We review and extend the most important properties of Rényi divergence and Kullback-Leibler divergence, including convexity, continuity, limits of\(\sigma \)-algebras, and the relation of the special order 0 to the Gaussian dichotomy and contiguity. We also show how to generalize the Pythagorean inequality to orders different from 1, and we extend the known equivalence between channel capacity and minimax redundancy to continuous channel inputs (for all orders) and present several other minimax results.

We propose an assessing method of mixture model in a cluster analysis setting with integrated completed likelihood. For this purpose, the observed data are assigned to unknown clusters using a maximum a posteriori operator. Then, the integrated completed likelihood (ICL) is approximated using the Bayesian information criterion (BIC). Numerical experiments on simulated and real data of the resulting ICL criterion show that it performs well both for choosing a mixture model and a relevant number of clusters. In particular, ICL appears to be more robust than BIC to violation of some of the mixture model assumptions and it can select a number of dusters leading to a sensible partitioning of the data.

From differential equations to structured population dynamics.- Adaptive dynamics an asymptotic point of view.- Population balance equations: the renewal equation.- Population balance equations: size structure.- Cell motion and chemotaxis.- General mathematical tools.

The importance of microbial communities (MCs) cannot be overstated. MCs underpin the biogeochemical cycles of the earth's soil, oceans and the atmosphere, and perform ecosystem functions that impact plants, animals and humans. Yet our ability to predict and manage the function of these highly complex, dynamically changing communities is limited. Building predictive models that link MC composition to function is a key emerging challenge in microbial ecology. Here, we argue that addressing this challenge requires close coordination of experimental data collection and method development with mathematical model building. We discuss specific examples where model-experiment integration has already resulted in important insights into MC function and structure. We also highlight key research questions that still demand better integration of experiments and models. We argue that such integration is needed to achieve significant progress in our understanding of MC dynamics and function, and we make specific practical suggestions as to how this could be achieved.

Four human Lactobacillus acidophilus strains were tested for their ability to adhere onto human enterocyte like Caco-2 cells in culture. The LA 1 strain exhibited a high calcium independent adhesive property. This adhesion onto Caco-2 cells required a proteinaceous adhesion promoting factor, which was present in the spent bacterial broth culture supernatant. LA 1 strain also strongly bound to the mucus secreted by the homogeneous cultured human goblet cell line HT29-MTX. The inhibitory effect of LA 1 organisms against Caco-2 cell adhesion and cell invasion by a large variety of diarrhoeagenic bacteria was investigated. As a result, the following dose dependent inhibitions were obtained: (a) against the cell association of enterotoxigenic, diffusely adhering and enteropathogenic Escherichia coli, and Salmonella typhimurium; (b) against the cell invasion by enteropathogenic Escherichia coli, Yersinia pseudotuberculosis, and Salmonella typhimurium. Incubations of L acidophilus LA 1 before and together with enterovirulent E coli were more effective than incubation after infection by E coli.

Elliptic boundary value problems of the form Lu + g(x,u) in omega and u = 0 on the boundary of omega are studied where g is singular in that g(x,r) goes to infinity uniformly as r goes to zero from above. Existence of classical and generalized solutions is established and an associated nonlinear eigenvalue problem is treated. A detailed study is made of the behaviour of the solutions and their gradients near the boundary of omega. This leads to global estimates for the modulus of continuity of solutions. (Author)

Tangent spaces of a sub-Riemannian manifold are themselves sub-Riemannian manifolds. They can be defined as metric spaces, using Gromov’s definition of tangent spaces to a metric space, and they turn out to be sub-Riemannian manifolds. Moreover, they come with an algebraic structure: nilpotent Lie groups with dilations. In the classical, Riemannian, case, they are indeed vector spaces, that is, abelian groups with dilations. Actually, the above is true only for regular points. At singular points, instead of nilpotent Lie groups one gets quotient spaces G/H of such groups G . These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The subject of geometric numerical integration deals with numerical integrators that preserve geometric properties of the flow of a differential equation, and it explains how structure preservation leads to improved long-time behaviour. This article illustrates concepts and results of geometric numerical integration on the important example of the Störmer–Verlet method. It thus presents a cross-section of the recent monograph by the authors, enriched by some additional material. After an introduction to the Newton–Störmer–Verlet–leapfrog method and its various interpretations, there follows a discussion of geometric properties: reversibility, symplecticity, volume preservation, and conservation of first integrals. The extension to Hamiltonian systems on manifolds is also described. The theoretical foundation relies on a backward error analysis, which translates the geometric properties of the method into the structure of a modified differential equation, whose flow is nearly identical to the numerical method. Combined with results from perturbation theory, this explains the excellent long-time behaviour of the method: long-time energy conservation, linear error growth and preservation of invariant tori in near-integrable systems, a discrete virial theorem, and preservation of adiabatic invariants.

We consider a backward stochastic differential equation, whose data (the final condition and the coefficient) are given functions of a jump-diffusion process. We prove that under mild conditions the solution of the BSDE provides a viscosity solution of a system of parabolic integral-partial differential equations. Under an additional assumption, that system of equations is proved to have a unique solution, in a given class of continuous functions

(1995). Controle Exact De Lequation De La Chaleur. Communications in Partial Differential Equations: Vol. 20, No. 1-2, pp. 335-356.

OBJECTIVE: The presence of anxiety disorders (AD) in schizophrenia (SZ) is attracting increasing interest. However, published studies have yielded very broad variations in prevalence rates across studies. The current meta-analysis sought to (1) investigate the prevalence of co-occurring AD in SZ by reporting pooled prevalence rates and (2) identify potential sources of variations in reported rates that could guide our efforts to identify and treat these co-occurring disorders in patients with SZ. METHODS: We performed a systematic search of studies reporting prevalence of AD in SZ and related psychotic disorders. Mean prevalence rates and 95% confidence intervals (CIs) were first computed for each disorder. We then examined the impact of potential moderators related to patient sampling or to AD assessment methods on these rates. RESULTS: Fifty-two eligible studies were identified. Pooled prevalence rates and CIs were 12.1% (7.0%-17.1%) for obsessive-compulsive disorders, 14.9% (8.1%-21.8%) for social phobia, 10.9% (2.9%-18.8%) for generalized AD, 9.8% (4.3%-15.4%) for panic disorders, and 12.4% (4.0%-20.8%) for post-traumatic stress disorders. For all disorders, we found significant heterogeneity in rates across studies. This heterogeneity could at least partially be explained by the effect of moderator variables related to patient characteristics or assessment methods. CONCLUSIONS: AD are highly prevalent in SZ, but important variations in rates are observed between studies. This meta-analysis highlights several factors that affect risk for, or detection of AD in SZ, and could, thus, have an important impact on treatment and outcome of SZ patients.

In order to study weak continuity of quadratic forms on spaces of L2 solutions of systems of partial differential equations, we define defect measures on the space of positions and frequencies.A systematic use of these measures leads in particular to a compensated compactness theorem, generalizing MURAT"TARTAR's compensated compactness to variable coefficients and GOLSE"LIONS"PERTHAME"SENTIS's averaging lemma. We also obtain results on homogenization for differential operators of order I with oscillating coefficients.

(1992). Nonlinear Elliptic Equations with Right Hand Side Measures. Communications in Partial Differential Equations: Vol. 17, No. 3-4, pp. 189-258.

Abstract A novel microporous templated carbon material doped with nitrogen is synthesized by using a two‐step nanocasting process using acrylonitrile (AN) and propylene as precursors, and Na–Y zeolite as a scaffold. Liquid‐phase impregnation and in situ polymerization of the nitrogenated precursor inside the nanochannels of the inorganic scaffold, followed by gas‐phase impregnation with propylene, enables pore‐size control and functionality tuning of the resulting carbon material. The material thereby obtained has a narrow pore‐size distribution (PSD), within the micropore range, and a large amount of heteroatoms (i.e., oxygen and nitrogen). In addition, the carbon material inherits the ordered structure of the inorganic host. Such features simultaneously present in the carbon result in it being ideal for use as an electrode in a supercapacitor. Although presenting a moderately developed specific surface area ( S BET = 1680 m 2 g –1 ), the templated carbon material displays a large gravimetric capacitance (340 F g –1 ) in aqueous media because of the combined electrochemical activity of the heteroatoms and the accessible porosity. This material can operate at 1.2 V in an aqueous medium with good cycleability—‐beyond 10 000 cycles—and is extremely promising for use in the development of high‐energy‐density supercapacitors.

The criterion proved in this paper was stated by Thurston in November 1982. Thurston lectured on its proof on several occasions, notably at the NSF summer conference in Duluth, 1983, where one of the authors (JHH) was present. Using the notes of various attendants at these lectures, we have reconstructed a proof that we have made as precise as we could. Since this required a certain amount of work on our part, we thought it might be of some use to present this proof to the reader. We thank Dennis Sullivan for useful conversations, and Fritz yon Haesseler and especially Ben Wittner for help with the writing and valuable suggestions. After the first version was written, Clifford Earle pointed out that better estimates than what we had were to be found in [B].

This article is concerned with the study of approximate controllability for the semilinear heat equation in a bounded domain Ω when the control acts on any open and nonempty subset of Ω or on a part of the boundary. In the case of both an internal and a boundary control, the approximate controllability in L P (Ω) for 1 ≦ p < + ∞ is proved when the nonlinearity is globally Lipschitz with a control in L ∞ . In the case of the interior control, we also prove approximate controllability in C 0 (Ω). The proof combines a variational approach to the controllability problem for linear equations and a fixed point method. We also prove that the control can be taken to be of “quasi bang-bang” form.

Abstract. Wang & Wells [ J. Amer. Statist. Assoc. 95 (2000) 62] describe a non‐parametric approach for checking whether the dependence structure of a random sample of censored bivariate data is appropriately modelled by a given family of Archimedean copulas. Their procedure is based on a truncated version of the Kendall process introduced by Genest & Rivest [ J. Amer. Statist. Assoc. 88 (1993) 1034] and later studied by Barbe et al . [ J. Multivariate Anal. 58 (1996) 197]. Although Wang & Wells (2000) determine the asymptotic behaviour of their truncated process, their model selection method is based exclusively on the observed value of its L 2 ‐norm. This paper shows how to compute asymptotic p ‐values for various goodness‐of‐fit test statistics based on a non‐truncated version of Kendall's process. Conditions for weak convergence are met in the most common copula models, whether Archimedean or not. The empirical behaviour of the proposed goodness‐of‐fit tests is studied by simulation, and power comparisons are made with a test proposed by Shih [ Biometrika 85 (1998) 189] for the gamma frailty family.