Centre de Mathématiques Appliquées de l'École polytechnique

WINNER of a Riskbook.com Best of 2004 Book Award!During the last decade, financial models based on jump processes have acquired increasing popularity in risk management and option pricing. Much has been published on the subject, but the technical nature of most papers makes them difficult for nonspecialists to understand, and the mathematic

In a Hilbert space ℋ, discrete families of vectors {hj} with the property that f=∑j〈hj‖ f〉hj for every f in ℋ are considered. This expansion formula is obviously true if the family is an orthonormal basis of ℋ, but also can hold in situations where the hj are not mutually orthogonal and are ‘‘overcomplete.’’ The two classes of examples studied here are (i) appropriate sets of Weyl–Heisenberg coherent states, based on certain (non-Gaussian) fiducial vectors, and (ii) analogous families of affine coherent states. It is believed, that such ‘‘quasiorthogonal expansions’’ will be a useful tool in many areas of theoretical physics and applied mathematics.

This paper introduces a new class of bases, called bandelet bases, which decompose the image along multiscale vectors that are elongated in the direction of a geometric flow. This geometric flow indicates directions in which the image gray levels have regular variations. The image decomposition in a bandelet basis is implemented with a fast subband-filtering algorithm. Bandelet bases lead to optimal approximation rates for geometrically regular images. For image compression and noise removal applications, the geometric flow is optimized with fast algorithms so that the resulting bandelet basis produces minimum distortion. Comparisons are made with wavelet image compression and noise-removal algorithms.

The state of the art in real-time programming is briefly reviewed. The synchronous approach is then introduced informally and its possible impact on the design of real-time and reactive systems is discussed. The authors present and discuss the application fields and the principles of synchronous programming. The major concern of the synchronous approach is to base synchronous programming languages on mathematical models. This makes it possible to handle compilation, logical correctness proofs, and verification of real-time programs in a formal way, leading to a clean and precise methodology for design and programming.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

We solve the initial and boundary condition problem for a general first order quasilinear equation in several space variables by using a vanishing viscosity method and give a definition which characterizes the obtained solution.

As the number of single-cell transcriptomics datasets grows, the natural next step is to integrate the accumulating data to achieve a common ontology of cell types and states. However, it is not straightforward to compare gene expression levels across datasets and to automatically assign cell type labels in a new dataset based on existing annotations. In this manuscript, we demonstrate that our previously developed method, scVI, provides an effective and fully probabilistic approach for joint representation and analysis of scRNA-seq data, while accounting for uncertainty caused by biological and measurement noise. We also introduce single-cell ANnotation using Variational Inference (scANVI), a semi-supervised variant of scVI designed to leverage existing cell state annotations. We demonstrate that scVI and scANVI compare favorably to state-of-the-art methods for data integration and cell state annotation in terms of accuracy, scalability, and adaptability to challenging settings. In contrast to existing methods, scVI and scANVI integrate multiple datasets with a single generative model that can be directly used for downstream tasks, such as differential expression. Both methods are easily accessible through scvi-tools.

A scattering transform defines a locally translation invariant representation which is stable to time-warping deformation. It extends MFCC representations by computing modulation spectrum coefficients of multiple orders, through cascades of wavelet convolutions and modulus operators. Second-order scattering coefficients characterize transient phenomena such as attacks and amplitude modulation. A frequency transposition invariant representation is obtained by applying a scattering transform along log-frequency. State-the-of-art classification results are obtained for musical genre and phone classification on GTZAN and TIMIT databases, respectively.

We present a simple model of a stock market where a random communication structure between agents generically gives rise to heavy tails in the distribution of stock price variations in the form of an exponentially truncated power law, similar to distributions observed in recent empirical studies of high-frequency market data. Our model provides a link between two well-known market phenomena: the heavy tails observed in the distribution of stock market returns on one hand and herding behavior in financial markets on the other hand. In particular, our study suggests a relation between the excess kurtosis observed in asset returns, the market order flow, and the tendency of market participants to imitate each other.

We study a level-set method for numerical shape optimization of elastic structures. Our approach combines the level-set algorithm of Osher and Sethian with the classical shape gradient. Although this method is not specifically designed for topology optimization, it can easily handle topology changes for a very large class of objective functions. Its cost is moderate since the shape is captured on a fixed Eulerian mesh.

A large number of imaging problems reduce to the optimization of a cost function, with typical structural properties. The aim of this paper is to describe the state of the art in continuous optimization methods for such problems, and present the most successful approaches and their interconnections. We place particular emphasis on optimal first-order schemes that can deal with typical non-smooth and large-scale objective functions used in imaging problems. We illustrate and compare the different algorithms using classical non-smooth problems in imaging, such as denoising and deblurring. Moreover, we present applications of the algorithms to more advanced problems, such as magnetic resonance imaging, multilabel image segmentation, optical flow estimation, stereo matching, and classification.

In a financial market model with constraints on the portfolios, define the price for a claim C as the smallest real number p such that sup π E[ U ( X T x + p , π − C )]≥ sup π E[ U ( X T x , π )], where U is the negative exponential utility function and X x , π is the wealth associated with portfolio π and initial value x . We give the relations of this price with minimal entropy or fair price in the flavor of Karatzas and Kou (1996) and superreplication. Using dynamical methods, we characterize the price equation, which is a quadratic Backward SDE, and describe the optimal wealth and portfolio. Further use of Backward SDE techniques allows for easy determination of the pricing function properties.

We introduce a class of multifractal processes, referred to as multifractal random walks (MRWs). To our knowledge, it is the first multifractal process with continuous dilation invariance properties and stationary increments. MRWs are very attractive alternative processes to classical cascadelike multifractal models since they do not involve any particular scale ratio. The MRWs are indexed by four parameters that are shown to control in a very direct way the multifractal spectrum and the correlation structure of the increments. We briefly explain how, in the same way, one can build stationary multifractal processes or positive random measures.

We study the Berger-Nirenberg problem on surfaces with conical singularities, i.e. we discuss conditions under which a function on a Riemann surface is the Gaussian curvature of some conformal metric with a prescribed set of singularities of conical types.

We first present multicomponent flow models derived from the kinetic theory of gases. We then investigate the symmetric hyperbolic-parabolic structure of the resulting system of partial differential equations and discuss the Cauchy problem for smooth solutions. We also address the existence of deflagration waves also termed anchored waves. We further indicate related models which have a similar hyperbolic-parabolic structure, notably the Saint-Venant system with a temperature equation as well as the equations governing chemical equilibrium flows. We next investigate multicomponent ionized and magnetized flow models with anisotropic transport fluxes which have a different mathematical structure. We finally discuss numerical algorithms specifically devoted to complex chemistry flows, in particular the evaluation of multicomponent transport properties, as well as the impact of multicomponent transport. 1

We present a finite difference method for solving parabolic partial integro-differential equations with possibly singular kernels which arise in option pricing theory when the random evolution of the underlying asset is driven by a Lévy process or, more generally, a time-inhomogeneous jump-diffusion process. We discuss localization to a finite domain and provide an estimate for the localization error under an integrability condition on the Lévy measure. We propose an explicit-implicit finite difference scheme which can be used to price European and barrier options in such models. We study stability and convergence of the scheme proposed and, under additional conditions, provide estimates on the rate of convergence. Numerical tests are performed with smooth and nonsmooth initial conditions.

In this paper we improve an earlier result by Bukhgeim and Uhlmann We follow the general strategy of [1] but use a richer set of solutions to the Dirichlet problem. This implies a similar result for the problem of Electrical Impedance Tomography which consists in determining the conductivity of a body by making voltage and current measurements at the boundary.

We are concerned with the numerical resolution of backward stochastic differential equations. We propose a new numerical scheme based on iterative regressions on function bases, which coefficients are evaluated using Monte Carlo simulations. A full convergence analysis is derived. Numerical experiments about finance are included, in particular, concerning option pricing with differential interest rates.

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 &lt; + ∞ 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.

This paper describes some of the work on stereo that has been going on at INRIA in the last four years. The work has concentrated on obtaining dense, accurate and reliable range maps of the environment at rates compatible with the real-time constraints of such applications as the navigation of mobile vehicles in man-made or natural environments. The class of algorithms which has been selected among several is the class of algorithms which has been selected among several is the class of correlation-based stereo algorithms because they are the only ones that can produce sufficiently dense range maps with an algoritmic structure which lends itself nicely to fast implementations because of the simplicity of the underlying computation. We describe the various improvements that we have brought to the original idea, including validation and characterization of the quality of the matches, a recursive implementation of the score computation which makes the method independent of the size of the correlation window and a calibration method which does not require the use of a calibration pattern. We then describe two implementations of this algorithm on two very different pieces of hardware. The first implementation is on a board with four digital signal processors designed jointly with Matra MSII. This implementation can produce 64x64 range maps at rate varying between 200 and 400 ms, depending upon the range of disparities. The second implementation is on a board developed by DEC-PRL and can perform the cross-correlation of two 256X256 images in 140 ms. The first implementation has been integrated in the navigation system of the INRIA cart and used to correct for inertial and odometric errors in navigation experiments both indoors and outdoors on road. This is the first application of our correlation-based algorithm which is described in the paper. The second application has been done jointly with people from the french national space agence (CNES) to study the possibility of using stereo on a future planetary rover for the construction of digital elevation maps. We have shown that real time stereo is possible today at low-cost and can be applied in real applications. The algorithm that has been described is not the most sophisticated available but we have made it robust and reliable thanks to a number of improvements. Evan though each of these improvements is not earth-shattering from the pure research point of view, altogether they have allowed us to go beyond a very important threshold. This threshold measures the difference between a program that runs in the laboratory on a few images and one that works continuously for hours on a sequence of stereo pairs and produces results at such rates and of such quality that they can be used to guide a real vehicle or to produce discrete elevation maps. We believe that this threshold has only been reached in a very small number of cases.

Our focus here is on the infinitesimal model. In this model, one or several quantitative traits are described as the sum of a genetic and a non-genetic component, the first being distributed within families as a normal random variable centred at the average of the parental genetic components, and with a variance independent of the parental traits. Thus, the variance that segregates within families is not perturbed by selection, and can be predicted from the variance components. This does not necessarily imply that the trait distribution across the whole population should be Gaussian, and indeed selection or population structure may have a substantial effect on the overall trait distribution. One of our main aims is to identify some general conditions on the allelic effects for the infinitesimal model to be accurate. We first review the long history of the infinitesimal model in quantitative genetics. Then we formulate the model at the phenotypic level in terms of individual trait values and relationships between individuals, but including different evolutionary processes: genetic drift, recombination, selection, mutation, population structure, …. We give a range of examples of its application to evolutionary questions related to stabilising selection, assortative mating, effective population size and response to selection, habitat preference and speciation. We provide a mathematical justification of the model as the limit as the number M of underlying loci tends to infinity of a model with Mendelian inheritance, mutation and environmental noise, when the genetic component of the trait is purely additive. We also show how the model generalises to include epistatic effects. We prove in particular that, within each family, the genetic components of the individual trait values in the current generation are indeed normally distributed with a variance independent of ancestral traits, up to an error of order 1∕M. Simulations suggest that in some cases the convergence may be as fast as 1∕M.