In this paper, we prove a Donsker theorem for one-dimensional processes generated by an operator with measurable coefficients. We construct a random walk on any grid on the state space, using the transition probabilities of the approximated process, and the conditional average times it spends on each cell of the grid. Indeed we can compute these quantities by solving some suitable elliptic PDE problems.

We investigate the small deviations under various norms for stable processes defined by the convolution of a smooth function $f : \; ]0, +\infty[ \;\to \mathbb{R}$ with a real SαS Lévy process. We show that the small ball exponent is uniquely determined by the norm and by the behaviour of f at zero, which extends the results of Lifshits and Simon, Ann. Inst. H. Poincaré Probab. Statist.41 (2005) 725–752 where this was proved for f being a power function (Riemann-Liouville processes). In the Gaussian case, the same generality as Lifshits and Simon, Ann. Inst. H. Poincaré Probab. Statist.41 (2005) 725–752 is obtained with respect to the norms, thanks to a weak decorrelation inequality due to Li, Elec. Comm. Probab.4 (1999) 111–118. In the more difficult non-Gaussian case, we use a different method relying on comparison of entropy numbers and restrict ourselves to Hölder and Lp-norms.

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 x0 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 x0 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 prove the existence and smoothness of density for the solution of a hyperbolic SPDE with free term coefficients depending on time, under hypoelliptic non degeneracy conditions. The result extends those proved in Cattiaux and Mesnager, PTRF123 (2002) 453-483 to an infinite dimensional setting.

This paper is a corrigendum to paper Toldo, ESAIM, P&S10 (2006) 141–163 where we study the stability of the solutions of Backward Stochastic Differential Equations (BSDE for short) with an almost surely finite random terminal time.

In this paper, a singular semi-linear parabolic PDE with locally periodic coefficients is homogenized. We substantially weaken previous assumptions on the coefficients. In particular, we prove new ergodic theorems. We show that in such a weak setting on the coefficients, the proper statement of the homogenization property concerns viscosity solutions, though we need a bounded Lipschitz terminal condition.

It is proved that the best constant factor in the Rademacher-Gaussian tail comparison is between two explicitly defined absolute constants c1 and c2 such that c2≈1.01 c1. A discussion of relative merits of this result versus limit theorems is given.

Variance reduction has always been a central issue in Monte Carlo experiments. Population Monte Carlo can be used to this effect, in that a mixture of importance functions, called a D-kernel, can be iteratively optimized to achieve the minimum asymptotic variance for a function of interest among all possible mixtures. The implementation of this iterative scheme is illustrated for the computation of the price of a European option in the Cox-Ingersoll-Ross model. A Central Limit theorem as well as moderate deviations are established for the D-kernel Population Monte Carlo methodology.



We introduce the fractional mixed fractional Brownian motion and fractional Brownian sheet, and investigate the small ball behavior of its sup-norm statistic. Then, we state general conditions and characterize the sufficiency part of the lower classes of some statistics of the above process by an integral test. Finally, when we consider the sup-norm statistic, the necessity part is given by a second integral test.