Some extensions of fractional Brownian motion and sub-fractional Brownian motion related to particle systems.
Some invariance properties (of the laws) of Ocone's martingales
Some remarks on isoperimetry of gaussian type
Some results for the quadratic analysis of Gaussian processes and applications
Some results on stochastic convolutions arising in Volterra equations perturbed by noise
Regularity of stochastic convolutions corresponding to a Volterra equation, perturbed by a white noise, is studied. Under suitable assumptions, hölderianity of the corresponding trajectories is proved.
Some thoughts about Segal's ergodic theorem
Over fifty years ago, Irving Segal proved a theorem which leads to a characterization of those orthogonal transformations on a Hilbert space which induce ergodic transformations. Because Segal did not present his result in a way which made it readily accessible to specialists in ergodic theory, it was difficult for them to appreciate what he had done. The purpose of this note is to state and prove Segal's result in a way which, I hope, will win it the recognition which it deserves.
Spectral approach for kernel-based interpolation
We describe how the resolution of a kernel-based interpolation problem can be associated with a spectral problem. An integral operator is defined from the embedding of the considered Hilbert subspace into an auxiliary Hilbert space of square-integrable functions. We finally obtain a spectral representation of the interpolating elements which allows their approximation by spectral truncation. As an illustration, we show how this approach can be used to enforce boundary conditions in kernel-based...
Spectral densities describing off-white noises
Spectral Multiplicity of Certain Gaussian Processes
Spectral type of a polynomial of stationary Gaussian process
Spherical and hyperbolic fractional Brownian motion.
Stationary gaussian random fields on hyperbolic spaces and on euclidean spheres
We recall necessary notions about the geometry and harmonic analysis on a hyperbolic space and provide lecture notes about homogeneous random functions parameterized by this space. The general principles are illustrated by construction of numerous examples analogous to Euclidean case. We also give a brief survey of the fields parameterized by Euclidean spheres. At the end we give a list of important open questions in hyperbolic case.
Stationary Gaussian random fields on hyperbolic spaces and on Euclidean spheres∗∗∗
We recall necessary notions about the geometry and harmonic analysis on a hyperbolic space and provide lecture notes about homogeneous random functions parameterized by this space. The general principles are illustrated by construction of numerous examples analogous to Euclidean case. We also give a brief survey of the fields parameterized by Euclidean spheres. At the end we give a list of important open questions in hyperbolic case.
Statistical Inference about the Drift Parameter in Stochastic Processes
In statistical inference on the drift parameter in the Wiener process with a constant drift there is a large number of options how to do it. We may, for example, base this inference on the properties of the standard normal distribution applied to the differences between the observed values of the process at discrete times. Although such methods are very simple, it turns out that more appropriate is to use the sequential methods. For the hypotheses testing about the drift parameter it is more...
Statistical inference for stochastic parabolic equations: a spectral approach.
Stochastic calculus with respect to fractional Brownian motion
Fractional Brownian motion (fBm) is a centered self-similar Gaussian process with stationary increments, which depends on a parameter called the Hurst index. In this conference we will survey some recent advances in the stochastic calculus with respect to fBm. In the particular case , the process is an ordinary Brownian motion, but otherwise it is not a semimartingale and Itô calculus cannot be used. Different approaches have been introduced to construct stochastic integrals with respect to fBm:...
Stochastic convolution in separable Banach spaces and the stochastic linear Cauchy problem
Let H be a separable real Hilbert space and let E be a separable real Banach space. We develop a general theory of stochastic convolution of ℒ(H,E)-valued functions with respect to a cylindrical Wiener process with Cameron-Martin space H. This theory is applied to obtain necessary and sufficient conditions for the existence of a weak solution of the stochastic abstract Cauchy problem (ACP) (t∈ [0,T]), almost surely, where A is the generator of a -semigroup of bounded linear operators on...
Stochastic integral of divergence type with respect to fractional brownian motion with Hurst parameter
Stochastic integration with respect to fractional brownian motion
Stochastic integration with respect to Volterra processes