Kernel estimation for Lévy driven stochastic convolutions

We consider a Lévy driven stochastic convolution, also called continuous time Lévy driven moving average model X ⁢ ( t ) = ∫ 0 t a ⁢ ( t - s ) ⁢ d Z ⁢ ( s ) X(t)=\int_{0}^{t}a(t-s)\,dZ(s) , where 𝑍 is a Lévy martingale and the kernel a ( . ) a(\,{.}\,) a deterministic function square integrable on R + \mathbb{R}^{+} . Given 𝑁 i.i.d. continuous time observations ( X i ⁢ ( t ) ) t ∈ [ 0 , T ] (X_{i}(t))_{t\in[0,T]} , i = 1 , … , N i=1,\dots,N , distributed like ( X ⁢ ( t ) ) t ∈ [ 0 , T ] (X(t))_{t\in[0,T]} , we propose two types of nonparametric projection estimators of a 2 a^{2} under different sets of assumptions. We bound the L 2 \mathbb{L}^{2} -risk of the estimators and propose a data driven procedure to select the dimension of the projection space, illustrated by a short simulation study.

From regression function to diffusion drift estimation in nonparametric setting

We consider a diffusion model dXt = b(Xt)dt + σ(Xt)dWt,X0 = η, under conditions ensuring existence, stationarity and geometrical β-mixing of the process solution. We assume that we observe a sample (XkΔ)0≤k≤n+1. Our aim is to study nonparametric estimators of the drift function b(.), under general conditions. We propose projection estimators based on a least-squares type contrast and, in order to generalize existing results, we want to consider possibly non compactly supported projection bases and possibly non bounded volatility. To that aim, we relate the model with a simpler regression model, then to a more elaborate heteroscedastic model, plus some residual terms. This allows to see the role of heteroscedasticity first and the role of dependency between the variables and to present different probabilistic tools used to face each part of the problem. For each step, we try to see the “price” of each assumption. This is the developed version of the talk given in August 2018 in Dijon, Journées MAS.

Adaptive Inference for the Bivariate Mean Function in Functional Data

Inference methods are proposed for the bivariate mean function of a continuous stochastic process with a two-dimensional domain. Nonparametric bivariate estimation is facilitated by thresholded projection estimators. Estimators adapt to the sparsity of the bivariate function. Oracle inequality results are developed to describe the adaptive inference methods. The construction of nonparametric bivariate confidence bands is presented. Implementation results show the applicability of the methods in practice.

Monte Carlo algorithms for calculation of diffusive characteristics of an electron avalanche in gases

Some problems of the theory of electron transfer in gases under the action of a strong external electric field is considered in the paper. Based on the three-dimensional ELSHOW algorithm, samples of states of particles in an electron avalanche are obtained for a given time moment in order to calculate the corresponding ‘diffusion radii’ and diffusion coefficients. Randomized projection estimators and kernel estimators (for test purpose) are constructed with the use of grouped samples for evaluation of the distribution density of particles in an avalanche. Test computations demonstrate a high efficiency of projection estimators for calculation of diffusive characteristics.