Stochastic processes with independent increments,taking values in a Hilbert space

Let $\text{H}$ be a real separable Hilbert space; let X(t), t?[0, 1], be a separable, stochastically continuous, $\text{H}$-valued stochastic process with independent increments. Then a decomposition of X(t) into a uniformly convergent sum of independent processes is found. In this decomposition one of the processes is Gaussian with continuous sample functions, and the remainder of the processes have sample functions whose discontinuities correspond to those of certain real-valued Poisson processes. The decomposition of X(t) leads to a Levy-Khintchine representation of the characteristic functional of X(t). In addition, the case when X(t) has finite variance is explored, and, as a consequence of the above decomposition, a Kolmogorov-type representation of the characteristic functional of X(t) is derived.     

Nonlinear stochastic time-fractional slow and fast diffusion equations on Rd

This paper studies the nonlinear stochastic partial differential equation of fractional orders both in space and time variables: $\left({?}^{\beta }+\frac{\nu }{2}{(?\Delta )}^{\alpha ∕2}\right)u(t,x)=I_t^{\gamma }\left[\rho \left(u(t,x)\right)\stackrel{?}{W}(t,x)\right],t>0,x\in {\mathbb{R}}^d,$where $\stackrel{?}{W}$ is the space–time white noise, $\alpha \in (0,2]$, $\beta \in (0,2)$, $\gamma \ge 0$ and $\nu >0$. Fundamental solutions and their properties, in particular the nonnegativity, are derived. The existence and uniqueness of solution together with the moment bounds of the solution are obtained under Dalang’s condition: $d<2\alpha +\frac{\alpha }{\beta }min(2\gamma ?1,0)$. In some cases, the initial data can be measures. When $\beta \in (0,1]$, we prove the sample path regularity of the solution.     

The separability of the Hilbert space generated by a stochastic process

The separability of the Hilbert space generated by a stochastic process is one of the basic assumptions in the time-spectral analysis of stochastic processes. This assumption is either presupposed explicitly or, more often, obtained as a consequence of the assumption of existence of left and right limits of the process for any value of the time parameter. In this paper it is shown that the existence of a left limit only, for each value of the time parameter, is a sufficient condition for the separability of the Hilbert space generated by the process.     

Representation of Banach space valued quasimartingales by real quasimartingales

A new technique is developed which allows to study quasimartingales with values in a Banach space E via real quasimartingales. As a byproduct path compactness for a wide class of E-valued quasimartingales is proved. The first application of this technique yields the equivalence of a.s. convergence and path compactness for E-valued martingales. Furthermore local decomposability of an E-valued semimartingale into a square integrable martingale and a process of integrable variation is established. Finally, it is shown that each process of integrable variation, with values in a Banach space with Radon-Nikodym property, can be approximated by processes taking values in a finite-dimensional subspace.     

A note on computing the distribution of the norm of Hilbert space valued Gaussian random variables

Let X be a Gaussian rv with values in a separable Hilbert space H having a covariance operator R of the form $\text{R = L}{}_0{}^1\text{A}{}^1\text{AL}{}_0$, where L₀, A are linear operators on H. A method is given for computing in terms of $\text{R}{}_0\text{= L}{}_0{}^1\text{L}{}_0$ and A the distribution of |X|², |·| being the norm in H. The result is applied to the evaluation of the asymptotic distribution of Cramér-von Mises statistics when parameters are present. L₀ corresponds to the case where the true underlying parameter is known and A represents the effect of estimating the unknown parameter.     

Some theoretical properties of Silverman's method for Smoothed functional principal component analysis

Principal component analysis (PCA) is one of the key techniques in functional data analysis. One important feature of functional PCA is that there is a need for smoothing or regularizing of the estimated principal component curves. Silverman’s method for smoothed functional principal component analysis is an important approach in a situation where the sample curves are fully observed due to its theoretical and practical advantages. However, lack of knowledge about the theoretical properties of this method makes it difficult to generalize it to the situation where the sample curves are only observed at discrete time points. In this paper, we first establish the existence of the solutions of the successive optimization problems in this method. We then provide upper bounds for the bias parts of the estimation errors for both eigenvalues and eigenfunctions. We also prove functional central limit theorems for the variation parts of the estimation errors. As a corollary, we give the convergence rates of the estimations for eigenvalues and eigenfunctions, where these rates depend on both the sample size and the smoothing parameters. Under some conditions on the convergence rates of the smoothing parameters, we can prove the asymptotic normalities of the estimations.     

Algebraic polynomials and moments of stochastic integrals

We propose an algebraic method for proving estimates on moments of stochastic integrals. The method uses qualitative properties of roots of algebraic polynomials from certain general classes. As an application, we give a new proof of a variation of the Burkholder-Davis-Gundy inequality for the case of stochastic integrals with respect to real locally square integrable martingales. Further possible applications and extensions of the method are outlined.     

Behavior of the Hermite sheet with respect to theHurst index

We consider a $d$-parameter Hermite process with Hurst index $\mathbf{H}=(H_1,..,H_d)\in {\left(\frac{1}{2},1\right)}^d$ and we study its limit behavior in distribution when the Hurst parameters $H_i,i=1,..,d$ (or a part of them) converge to $\frac{1}{2}$ and/or 1. The limit obtained is Gaussian (when at least one parameter tends to $\frac{1}{2}$) and non-Gaussian (when at least one-parameter tends to 1 and none converges to $\frac{1}{2}$).