We develop a theory of Malliavin calculus for Banach space valued random variables. Using radonifying operators instead of symmetric tensor products we extend the Wiener-Itô isometry to Banach spaces. In the white noise case we obtain two sided L-estimates for multiple stochastic integrals in arbitrary Banach spaces. It is shown that the Malliavin derivative is bounded on vector-valued Wiener-I...

Suppose that the signal X to be estimated is a diffusion process in a random medium W and the signal is correlated with the observation noise. We study the historical filtering problem concerned with estimating the signal path up until the current time based upon the back observations. Using Dirichlet form theory, we introduce a filtering model for general rough signal X and establish a multipl...

Let Fn = (F1,n, ...., Fd,n), n > 1, be a sequence of random vectors such that, for every j = 1, ..., d, the random variable Fj,n belongs to a fixed Wiener chaos of a Gaussian field. We show that, as n → ∞, the components of Fn are asymptotically independent if and only if Cov(F 2 i,n, F 2 j,n) → 0 for every i 6= j. Our findings are based on a novel inequality for vectors of multiple Wiener-Itô ...

Abstract This paper is concerned with the distribution properties of the binomial aX + bXα, where X is a gamma random variable. We show in particular that aX + bXα is infinitely divisible for all α ∈ [1, 2] and a, b ∈ R+, and that for α = 2 the second order polynomial aX+bX2 is a generalized gamma convolution whose Thorin density and Wiener-gamma integral representation are computed explicitly....

In this work cylindrical Wiener processes on Banach spaces are defined by means of cylindrical stochastic processes, which are a well considered mathematical object. This approach allows a definition which is a simple straightforward extension of the real-valued situation. We apply this definition to introduce a stochastic integral with respect to cylindrical Wiener processes. Again, this defin...

The purpose of this paper is to provide a more general Cameron-Storvick theorem for the generalized analytic Feynman integral associated with Gaussian process $\mathcal Z_k$ on very Wiener space $C_{a,b}[0,T]$. $C_{a,b}[0,T]$ can be considered as set all continuous sample paths Brownian motion determined by functions $a(t)$ and $b(t)$ $[0,T]$. As an interesting application, we apply evaluate ce...

Convergence in probability and central limit laws of bipower variation for Gaussian processes with stationary increments and for integrals with respect to such processes are derived. The main tools of the proofs are some recent powerful techniques of Wiener/Itô/Malliavin calculus for establishing limit laws, due to Nualart, Peccati and others.

We deene stochastic integrals with respect to free Brownian motion, and show that they satisfy Burkholder-Gundy type inequalities in operator norm. We prove also a version of It^ o's predictable representation theorem, as well as product form and functional form of It^ o's formula. Finally we develop stochastic analysis on the free Fock space, in analogy with stochastic analysis on the Wiener s...