The paper deals with some practical problems connected with the classical exponential smoothing in time series. The fundamental theorem of the exponential smoothing is extended to the case with missing observations and an interpolation procedure in the framework of the exponential smoothing is described. A simple method of the exponential smoothing for multivariate time series is suggested.

We consider a stationary symmetric stable bidimensional process with discrete time, having the spectral representation (1.1). We consider a general case where the spectral measure is assumed to be the sum of an absolutely continuous measure, a discrete measure of finite order and a finite number of absolutely continuous measures on several lines. We estimate the density of the absolutely continuous measure and the density on the lines.

The Kalman filter is extensively used for state estimation for linear systems under Gaussian noise. When non-Gaussian Lévy noise is present, the conventional Kalman filter may fail to be effective due to the fact that the non-Gaussian Lévy noise may have infinite variance. A modified Kalman filter for linear systems with non-Gaussian Lévy noise is devised. It works effectively with reasonable computational cost. Simulation results are presented to illustrate this non-Gaussian filtering method.

The paper solves the problem of minimization of the Kullback divergence between a partially known and a completely known probability distribution. It considers two probability distributions of a random vector $(u_1,x_1,...,u_T,x_T)$ on a sample space of $2T$ dimensions. One of the distributions is known, the other is known only partially. Namely, only the conditional probability distributions of $x_{\tau }$ given $u_1,x_1,...,u_{\tau -1},x_{\tau -1},u_{\tau }$ are known for $\tau =1,...,T$. Our objective is to determine the remaining conditional probability distributions of $u_{\tau }$ given $u_1,x_1,...,u_{\tau -1},x_{\tau -1}$ such...

Soit $X^{\epsilon }$ la solution de l’équation différentielle stochastique suivante: $X_t^{\epsilon }=x+{\sum }_{i=1}^r{\int }_0^t{\sigma }_i\left(X_s^{\epsilon }\right)d{W}_s^i+\epsilon {\sum }_{j=1}^l{\int }_0^t\sigma {̃}_j\left(X_s^{\epsilon }\right)dW{̃}_s^j+{\int }_0^{t}b\left(X_s^{\epsilon }\right)ds$, et considérons ${\phi }^{\epsilon }\varphi =\varphi \left(X^{\epsilon }\right)$. L’objectif de cet article est d’établir le principe de grandes déviations pour la famille des lois induites par $X^{\epsilon }:\epsilon >0$ pour la norme höldérienne. Par conséquent, on montre le même résultat pour la famille des lois induites par ${\phi }^{\epsilon }\varphi :\epsilon >0$. Enfin, on donne une application de ces résultats au filtrage non linéaire.

Using integration by parts on Gaussian space we construct a Stein Unbiased Risk Estimator (SURE) for the drift of Gaussian processes, based on their local and occupation times. By almost-sure minimization of the SURE risk of shrinkage estimators we derive an estimation and de-noising procedure for an input signal perturbed by a continuous-time Gaussian noise.

Using integration by parts on Gaussian space we construct a Stein Unbiased Risk Estimator (SURE) for the drift of Gaussian processes, based on their local and occupation times. By almost-sure minimization of the SURE risk of shrinkage estimators we derive an estimation and de-noising procedure for an input signal perturbed by a continuous-time Gaussian noise.