We consider one-dimensional stochastic differential equations in the particular case of diffusion coefficient functions of the form|x|α , α ∈ [1/2,1). In that case, we study the rate of convergence of asymmetrized version of the Euler scheme. This symmetrized version iseasy to simulate on a computer. We prove its strong convergence and obtain the same rate ofconvergence as when the coefficients are Lipschitz.

In this paper we derive non asymptotic deviation bounds for $${\mathbb P}_\nu (|\frac 1t\int_0^t V(X_s) {\rm d}s - \int V {\rm d} \mu | \geq R)$$ where X is a μ stationary and ergodic Markov process and V is some μ integrable function. These bounds are obtained under various moments assumptions for V, and various regularity assumptions for μ. Regularity means here that μ may satisfy various functional inequalities (F-Sobolev,generalized Poincaré etc.).

We estimate the anisotropic index of an anisotropic fractional Brownian field. For all directions, we give a convergent estimator of the value of the anisotropic index in this direction, based on generalized quadratic variations. We also prove a central limit theorem. First we present a result of identification that relies on the asymptotic behavior of the spectral density of a process. Then, we define Radon transforms of the anisotropic fractional Brownian field and prove that these processes admit a spectral density satisfying the previous assumptions. Finally we use simulated fields to test the proposed estimator in different anisotropic and isotropic cases. Results show that the estimator behaves similarly in all cases and is able to detect anisotropy quiteaccurately.

In this paper, we apply the technique of decoupling to obtain some exponential inequalities for semi-bounded martingale, which extend the results of de la Peña, Ann. probab.27 (1999) 537–564.

Let (Xt, t ≥ 0) be a Lévy process started at 0, with Lévymeasure ν. We consider the first passage time T x of(Xt, t ≥ 0) to level x > 0, and Kx := XTx - x theovershoot and Lx := x- XTx- the undershoot. We first provethat the Laplace transform of the random triple (Tx,Kx,Lx )satisfies some kind of integral equation. Second, assuming thatν admits exponential moments, we show that $(\widetilde{T_x},K_x,L_x)$ converges in distribution asx → ∞, where $\widetilde{T_x}$ denotes a suitablerenormalization of T x .


In this paper we solve the basic fractional analogue of the classical linear-quadratic Gaussian regulator problem in continuous-time with partial observation. For a controlled linear system where both the state and observation processes are driven by fractional Brownian motions, we describe explicitly the optimal control policy which minimizes a quadratic performance criterion. Actually, we show that a separation principle holds, i.e., the optimal control separates into two stages based on optimal filtering of the unobservable state and optimal control of the filtered state. Both finite and infinite time horizon problems are investigated.

We elucidate the asymptotics of the Ls -quantization error induced by a sequence of Lr -optimal n-quantizers of aprobability distribution P on $\mathbb{R}^d$ when s > r. In particular we show that under natural assumptions, the optimal rate is preserved aslong as s < r+d (and for everys in the case of a compactly supported distribution). We derive some applications of these results to the error bounds for quantization based cubatureformulae in numerical integration on $\mathbb{R}^d$ and on the Wiener space.

In this paper, a very useful lemma (in two versions) is proved: itsimplifies notably the essential step to establish a Lindebergcentral limit theorem for dependent processes. Then, applying thislemma to weakly dependent processes introduced in Doukhan andLouhichi (1999), a new central limit theorem is obtained forsample mean or kernel density estimator. Moreover, by using thesubsampling, extensions under weaker assumptions of these centrallimit theorems are provided. All the usual causal or non causaltime series: Gaussian, associated, linear, ARCH(∞),bilinear, Volterra processes, ..., enter this frame.

A method is introduced to select the significant or non null mean terms among a collection of independent random variables. As an application we consider the problem of recovering thesignificant coefficients in non ordered model selection. The method is based on a convenient random centering ofthe partial sums of the ordered observations. Based onL-statistics methods we show consistency of the proposedestimator.An extension to unknown parametric distributions is considered. Simulatedexamples are included to show the accuracy of the estimator.An example of signal denoising with wavelet thresholding is also discussed.

Non-linear mixed models defined by stochastic differential equations (SDEs) are considered: the parameters of the diffusion process are random variables and vary among the individuals. A maximum likelihood estimation method based on the Stochastic Approximation EM algorithm, is proposed. This estimation method uses the Euler-Maruyama approximation of the diffusion, achieved using latent auxiliary data introduced to complete the diffusion process between each pair of measurement instants.A tuned hybrid Gibbs algorithm based on conditional Brownian bridges simulations of the unobserved process paths is included in this algorithm. The convergence is proved and the error induced on the likelihood by the Euler-Maruyama approximation is bounded as a function of the step size of the approximation. Results of a pharmacokinetic simulation study illustrate the accuracy of this estimation method. The analysis of the Theophyllin real dataset illustrates the relevance of the SDE approach relative to the deterministic approach.

A seminal paper by Rissanen, published in 1983, introduced the class of Variable Length Markov Chains and the algorithm Context which estimates the probabilistic tree generating the chain. Even if the subject was recently considered in several papers, the central question of the rate of convergence of the algorithm remained open. This is the question we address here. We provide an exponential upper bound for the probability of incorrect estimation of the probabilistic tree, as a function of the size of the sample. As a consequence we prove the almost sure consistency of the algorithm Context. We also derive exponential upper bounds for type I errors and for the probability of underestimation of the context tree. The constants appearing in the bounds are all explicit and obtained in a constructive way.

We analyze the Rosenblatt process which is a selfsimilar process with stationary increments and which appears aslimit in the so-called Non Central Limit Theorem (Dobrushin and Majòr (1979), Taqqu (1979)). This process isnon-Gaussian and it lives in the second Wiener chaos. We give its representation as a Wiener-Itô multiple integral withrespect to the Brownian motion on a finite interval and we develop a stochastic calculus with respect to it by using both pathwise type calculus and Malliavin calculus.

In a statistical mechanics model with unbounded spins, we prove uniqueness of the Gibbs measureunder various assumptions on finite volume functional inequalities. We follow Royer's approach (Royer, 1999) and obtain uniqueness by showing convergence properties of a Glauber-Langevin dynamics. The result was known when the measures on the box [-n,n]d (with free boundary conditions) satisfied the same logarithmic Sobolev inequality. We generalize this in two directions: either the constants may be allowed to grow sub-linearly in the diameter, or we may suppose a weaker inequality than log-Sobolev, but stronger than Poincaré. We conclude by giving a heuristic argument showing that this could be the right inequalities to look at.

In this paper, we give sufficient conditions to establish central limittheorems and moderate deviation principle for a class of support estimates ofempirical and Poisson point processes. The considered estimates are obtained bysmoothing some bias corrected extreme values of the point process. We show howthe smoothing permits to obtain Gaussian asymptotic limits and thereforepointwise confidence intervals. Some unidimensional and multidimensionalexamples are provided.

In this paper, we analyze the celebrated EM algorithm from the point of view of proximal point algorithms. More precisely, we study a new type of generalization of the EM procedure introduced in [Chretien and Hero (1998)] and called Kullback-proximal algorithms. The proximal framework allows us to prove new results concerning the cluster points. An essential contribution is a detailed analysis of the case where some cluster points lie on the boundary of the parameter space.

We consider a deconvolution problem of estimating a signal blurred with a random noise.The noise is assumed to be a stationary Gaussian process multiplied by a weight function function εh where h ∈ L2(R1) and ε is a small parameter. The underlying solution is assumed to be infinitely differentiable. For this model we find asymptotically minimax andBayes estimators. In the case of solutions having finite number ofderivatives similar results were obtained in [G.K. Golubev and R.Z. Khasminskii, IMS Lecture Notes Monograph Series36 (2001) 419–433].


We study a continuous-time discrete population structured by a vector of ages. Individuals reproduce asexually, age and die. The death rate takes interactions into account. Adapting the approach of Fournier and Méléard, we show that in a large population limit, the microscopic process converges to the measure-valued solution of an equation that generalizes the McKendrick-Von Foerster and Gurtin-McCamy PDEs in demography. The large deviations associated with this convergence are studied. The upper-bound is established via exponential tightness, the difficulty being that the marginals of our measure-valued processes are not of bounded masses. The local minoration is proved by linking the trajectories of the action functional's domain to the solutions of perturbations of the PDE obtained in the large population limit. The use of Girsanov theorem then leads us to regularize these perturbations. As an application, we study the logistic age-structured population. In the super-critical case, the deterministic approximation admits a non trivial stationary stable solution, whereas the stochastic microscopic process gets extinct almost surely. We establish estimates of the time during which the microscopic process stays in the neighborhood of the large population equilibrium by generalizing the works of Freidlin and Ventzell to our measure-valued setting.


We have seen in a previous article how the theory of “rough paths”allows us to construct solutions of differential equations driven by processes generated by divergence form operators. In this article, we study a convergence criterion which implies that one can interchange the integral with the limit of a family of stochastic processes generated by divergence form operators. As a corollary, we identify stochastic integrals constructed with the theory of rough paths with Stratonovich or Itô integrals already constructedfor stochastic processes generated by divergence form operators by using time-reversal techniques.

We consider a dynamical system in $\mathbb{R}$ driven by a vector field -U', where U is a multi-well potential satisfying some regularity conditions. We perturb this dynamical system by a Lévy noise of small intensity and such that the heaviest tail of its Lévy measure is regularly varying. We show that the perturbed dynamical system exhibits metastable behaviour i.e. on a proper time scale it reminds of a Markov jump process taking values in the local minima of the potential U. Due to the heavy-tailnature of the random perturbation, the results differ strongly from the well studied purely Gaussian case.

We propose a feature selection method for density estimation withquadratic loss. This method relies on the study of unidimensionalapproximation models and on the definition of confidence regions forthe density thanks to these models. It is quite general and includescases of interest like detection of relevant wavelets coefficientsor selection of support vectors in SVM. In the general case, weprove that every selected feature actually improves the performanceof the estimator. In the case where features are defined bywavelets, we prove that this method is adaptative near minimax (upto a log term) in some Besov spaces. We end the paper bysimulations indicating that it must be possible to extend theadaptation result to other features.