This paper deals with semiparametric convolution models, where thenoise sequence has a Gaussian centered distribution, with unknownvariance. Non-parametric convolution models are concerned with the case of anentirely known distribution for the noise sequence, and they havebeen widely studied in the past decade. The main property of thosemodels is the following one: the more regular the distribution of thenoise is, the worst the rate of convergence for the estimation of thesignal's density g is [5]. Nevertheless, regularity assumptionson the signal density g improve those rates of convergence [15].In this paper, we show that whenthe noise (assumed tobe Gaussian centered) has a varianceσ2 that is unknown (actually, it is always the case inpractical applications), the rates of convergence for the estimation ofg are seriously deteriorated, whatever its regularity is supposed to be.More precisely, the minimax risk for the pointwise estimation of g over aclass of regular densities is lower bounded by a constant over log n. We construct two estimators of σ2, andmore particularly, an estimator which is consistent as soon as the signal hasa finite first order moment.We also mention as a consequence the deterioration of therate of convergence in the estimation of the parameters in the nonlinearerrors-in-variables model.

The joint estimation of both drift and diffusion coefficient parameters is treatedunder the situation where the data are discretely observed from an ergodic diffusion processand where the statistical model may or may not include the true diffusion process.We consider the minimum contrast estimator,which is equivalent to the maximum likelihood type estimator, obtained from the contrast function based on a locally Gaussian approximation of the transition density.The asymptotic normality of the minimum contrast estimator is proved.In particular, the rate of convergence for the minimum contrast estimator of diffusion coefficient parameter in a misspecified modelis different from the one in the correctly specified parametric model.

A generic control variate method is proposed to price options under stochastic volatility models by Monte Carlo simulations. This method provides a constructive way to select control variates which are martingales in order to reduce the variance of unbiased option price estimators. We apply a singular and regular perturbation analysis to characterize the variance reduced by martingale control variates. This variance analysis is done in the regime where time scales of associated driving volatility processes are well separated. Numerical results for European, Barrier, and American options are presented to illustrate the effectiveness and robustness of this martingale control variate method in regimes where these time scales are not so well separated.

It is proved that the best constant factor in the Rademacher-Gaussian tail comparison is between two explicitly defined absolute constants c 1 and c 2 such that c 2 ≈1.01 c 1.A discussion of relative merits of this result versus limit theorems is given.

In the context of self-stabilizing processes, that is processes attracted by their ownlaw, living in a potential landscape, we investigate different properties of the invariantmeasures. The interaction between the process and its law leads to nonlinear stochasticdifferential equations. In [S. Herrmann and J. Tugaut. Electron. J. Probab.15 (2010) 2087–2116], the authors proved that, for linearinteraction and under suitable conditions, there exists a unique symmetric limit measureassociated to the set of invariant measures in the small-noise limit. The aim of thisstudy is essentially to point out that this statement leads to the existence, as the noiseintensity is small, of one unique symmetric invariant measure for the self-stabilizingprocess. Informations about the asymmetric measures shall be presented too. The main keyconsists in estimating the convergence rate for sequences of stationary measures usinggeneralized Laplace’s method approximations.

Considering the centered empirical distribution function F n-F asa variable in ${\mathbb L}^p(\mu)$ , we derive non asymptotic upperbounds for the deviation of the ${\mathbb L}^p(\mu)$ -norms ofF n-F as well as central limit theorems for the empirical processindexed by the elements of generalized Sobolev balls. These resultsare valid for a large class of dependent sequences, includingnon-mixing processes and some dynamical systems.

In this paper, we are interested in estimation problem for the drift parameters matrices of m independent multivariate diffusion processes. More specifically, we consider the case where the m-parameters matrices are supposed to satisfy some uncertain constraints. Given such an uncertainty, we develop shrinkage estimators which improve over the performance of the maximum likelihood estimator (MLE). Under an asymptotic distributional quadratic risk criterion, we study the relative dominance of the established estimators. Further, we carry out simulation studies for observation periods of small and moderate lengths of time that corroborate the theoretical finding for which shrinkage estimators outperform over the MLE. The proposed method is useful in model assessment and variable selection.

Let (Ω, $\mathcal{F}$, ($\mathcal{F}_t$)t≥0, $\mathbb{P}$) be a filtered probability space satisfying the usual assumptions: it is usually not possible to extend to $\mathcal{F}_{\infty}$ (the σ-algebra generated by ($\mathcal{F}_t$)t≥0) a coherent family of probability measures ($\mathbb{Q}_t$) indexed by t ≥ 0, eachof them being defined on $\mathcal{F}_t$. It is known that for instance, on the Wiener space, this extension problem has a positive answer if one takes the filtration generated by the coordinate process, made right-continuous, but can have a negative answer if one takes its usual augmentation. On the other hand, the usual assumptions are crucial in order to obtain the existence of regular versions of paths (typically adapted and continuous or adapted and càdlàg versions) for most stochastic processes of interest, such as the local time of the standard Brownian motion, stochastic integrals, etc. For instance we shall prove that on the Wiener space, equipped with the right-continuous augmentation of the filtration generated by the canonical process, there exists no càdlàg or continuous and adapted version for the local time at level zero of the canonical process. Hence there is an incompatibility between the problem of extending a coherent family of probability measures to $\mathcal{F}_\infty$ and the classical construction of regular versions of paths, requiring the usual assumptions (this situation typically occurs in the problem of penalization of the Brownian paths or in mathematical finance), which to the best of our knowledge, has not been noticed so far.In order to fix this problem, we introduce a new property for filtrations, intermediate between the right continuityand the usual conditions. More precisely, we say that a filtration ($\mathcal{F}_t$)t≥0 satisfies the natural assumptionsif it is right-continuous and if $\mathcal{F}_0$ contains all the sets included in a countable unionof negligible sets (Bn)n≥1, such that Bn ∈ $\mathcal{F}_n$ for n ≥ 1.There is a natural way to obtain, from a given filtration ($\mathcal{F}_t$)t≥0, a new filtration which satisfies the natural assumptions: we call it the natural augmentation of ($\mathcal{F}_t$)t≥0.We show that most of the important results of the theory of stochastic processes which are generally proved under the usual augmentation, such as the existence of regular version of trajectories or the début theorem, still hold under the natural augmentation; moreover this new augmentation allows the extension of a coherent family of probability measures whenever this is possible with the original filtration. For sake of completeness, we also recall (not so well known) Parthasarathy type conditions on the underlying filtration under which the extension problem for a coherent family of probability measures has a solution. In particular, we shall see that this is always the case on the following two fundamental spaces: $\mathcal{C}$($\mathbb{R}_+$, $\mathbb{R}$), the space of continuous functions equipped with the filtration generated by the coordinate process and $\mathcal{D}$($\mathbb{R}_+$, $\mathbb{R}$), the space of càdlàg functions endowed with the filtration generated by the coordinate process.

Motivated by the well-posedness of birth-and-growth processes, a stochastic geometric differential equation and, hence, a stochastic geometric dynamical system are proposed. In fact, a birth-and-growth process can be rigorously modeled as a suitable combination, involving the Minkowski sum and the Aumann integral, of two very general set-valued processes representing nucleation and growth dynamics, respectively. The simplicity of the proposed geometric approach allows to avoid problems of boundary regularities arising from an analytical definition of the front growth. In this framework, growth is generally anisotropic and, according to a mesoscale point of view, is non local, i.e. at a fixed time instant, growth is the same at each point of the space.

In this article we tackle the problem of inverse non linear ill-posedproblems from a statistical point of view. We discuss the problemof estimating an indirectly observed function, without priorknowledge of its regularity, based on noisy observations. For this we consider two approaches: one based on the Tikhonov regularization procedure, and another one based on model selection methods for both ordered and non ordered subsets. In each case we prove consistency of the estimators and show that their rate of convergence is optimal for the given estimation procedure.

Even for a well-trained statistician the construction of a histogramfor a given real-valued data set is a difficult problem. It is evenmore difficult to construct a fully automatic procedure whichspecifies the number and widths of the bins in a satisfactory mannerfor a wide range of data sets. In this paper we compare severalhistogram construction procedures by means of a simulationstudy. The study includes plug-in methods, cross-validation,penalized maximum likelihood and the taut string procedure. Their performance on different test beds is measured by their ability to identify the peaks of an underlying density as well as by Hellinger distance.

We describe quantization designs which lead to asymptotically and order optimal functional quantizers for Gaussian processes in a Hilbert space setting. Regular variation of the eigenvalues of the covariance operator plays a crucial role to achieve these rates. For the development of a constructive quantization scheme we rely on the knowledge of the eigenvectors of the covariance operator in order to transform the problem into a finite dimensional quantization problem of normal distributions.

We investigate here the central limit theorem of the increment ratio statistic of amultifractional Brownian motion, leading to a CLT for the time varying Hurst index. Theproofs are quite simple relying on Breuer–Major theorems and an original freezingof time strategy. A simulation study shows the goodness of fit of thisestimator.

In this paper we solve the basic fractionalanalogue of the classical infinite time horizon linear-quadratic Gaussianregulator problem. For a completely observable controlled linearsystem driven by a fractional Brownian motion, we describeexplicitely the optimal control policy which minimizes anasymptotic quadratic performance criterion.

This paper is concerned with the problem of determining the typical features of a curve when it is observed with noise. It has been shown that one can characterize the Lipschitz singularities of a signal by following the propagation across scales of the modulus maxima of its continuous wavelet transform. A nonparametric approach, based on appropriate thresholding of the empirical wavelet coefficients, is proposed to estimate the wavelet maxima of a signal observed with noise at various scales. In order to identify the singularities of the unknown signal, we introduce a new tool, “the structural intensity”, that computes the “density” of the location of the modulus maxima of a wavelet representation along various scales. This approach is shown to be an effective technique for detecting the significant singularities of a signal corrupted by noise and for removing spurious estimates. The asymptotic properties of the resulting estimators are studied and illustrated by simulations. An application to a real data set is also proposed.

We want to recover a signal based on noisy inhomogeneous data (the amount of data can vary strongly on the estimation domain). We model the data using nonparametric regression with random design, and we focus on the estimation of the regression at a fixed point x 0 with little, or much data. We propose a method which adapts both to the local amount of data (the design density is unknown) and to the local smoothness of the regression function. The procedure consists of a local polynomial estimator with a Lepski type data-driven bandwidth selector, see for instance Lepski et al. [Ann. Statist.25 (1997) 929–947]. We assess this procedure in the minimax setup, over a class of function with local smoothness s >0 of Hölder type. We quantify the amount of data at x 0 in terms of a local property on the design density called regular variation, which allows situations with strong variations in the concentration of the observations. Moreover, the optimality of the procedure is proved within this framework.

We show in this article how the theory of “rough paths”allows us to construct solutions of differential equations (SDEs) driven by processes generated by divergence-form operators. For that, we use approximations of the trajectories of the stochastic process by piecewise smooth paths. A result of type Wong-Zakai follows immediately.

This paper uses the Rice method [18] to give bounds tothe distribution of the maximum of a smooth stationary Gaussianprocess. We give simpler expressions of the first two terms ofthe Rice series [3,13] for the distribution of the maximum. Our main contribution is a simpler form of the second factorial momentof the number of upcrossings which is in some sense a generalizationof Steinberg et al.'s formula ([7] p. 212). Then, we present a numerical application and asymptotic expansionsthat give a new interpretation of a result by Piterbarg [15].

In the companion paper [C. Maugis and B. Michel,A non asymptotic penalized criterion for Gaussian mixture model selection. ESAIM: P&S15 (2011) 41–68] , a penalized likelihoodcriterion is proposed to select a Gaussian mixture model among aspecific model collection. This criterion depends on unknownconstants which have to be calibrated in practical situations. A“slope heuristics” method is described and experimented to dealwith this practical problem. In a model-based clustering context,the specific form of the considered Gaussian mixtures allows us todetect the noisy variables in order to improve the data clusteringand its interpretation. The behavior of our data-driven criterionis highlighted on simulated datasets, a curve clustering exampleand a genomics application.

We study the tails of the distribution of the maximum of a stationaryGaussian process on a bounded interval of the real line. Under regularityconditions including the existence of the spectral moment of order 8, we give an additional term for this asymptotics. This widens the application of an expansion given originally by Piterbarg [CITE] for a sufficiently small interval.