Related papers: A chaotic decomposition for generalized stochastic…
Hajek's stochastic comparison result is generalised to multivariate stochastic sum processes with univariate convex data functions and for univariate monoton nondecreasing convex data functions for processes with and without drift…
Statistical properties of infinite products of random isotropically distributed matrices are investigated. Both for continuous processes with finite correlation time and discrete sequences of independent matrices, a formalism that allows to…
We find logarithmic asymptotics of $L_2$-small deviation probabilities for weighted stationary Gaussian processes (both for real and complex-valued) having power-type discrete or continuous spectrum. As in the recent work by Hong, Lifshits…
We consider the problem of estimating stochastic volatility for a class of second-order parabolic stochastic PDEs. Assuming that the solution is observed at a high temporal frequency, we use limit theorems for multipower variations and…
A level-dependent L\'evy process solves the stochastic differential equation $dU(t) = dX(t)-{\phi}(U(t)) dt$, where $X$ is a spectrally negative L\'evy process. A special case is a multi-refracted L\'evy process with…
Following our previous work [68], this paper continues to investigate the evolution dynamics of local times of spectrally positive L\'evy processes with Gaussian components in the spatial direction. We prove that conditioned on the…
We introduce a theory of non-commutative $L^{p}$ spaces suitable for non-commutative probability in a non-tracial setting and use it to develop stochastic analysis of Grassmann-valued processes, including martingale inequalities, stochastic…
These notes survey some aspects of discrete-time chaotic calculus and its applications, based on the chaos representation property for i.i.d. sequences of random variables. The topics covered include the Clark formula and predictable…
We define generalized innovations associated with generalized error models having arbitrary distributions, that is, distributions that can be mixtures of continuous and discrete distributions. These models include stochastic volatility…
In this work we study the problem of constructing stochastic processes with a predetermined covariance decay by parameterizing its marginals and a given family of copulas. We show that the proposed methodology is compatibility-free and…
We generalise multivariate subordination of L\'evy processes as introduced by Barndorff-Nielsen, Pedersen, and Sato to Hilbert space valued L\'evy processes. The processes are explicitly characterised and conditions for integrability and…
The sharp asymptotics for the L^2-quantization errors of Gaussian measures on a Hilbert space and, in particular, for Gaussian processes is derived. The condition imposed is regular variation of the eigenvalues.
We show, using covariant Lyapunov vectors, that the chaotic solutions of spatially extended dissipative systems evolve within a manifold spanned by a finite number of physical modes hyperbolically isolated from a set of residual degrees of…
We study the asymptotic behaviour of stochastic processes that are generated by sums of partial sums of i.i.d. random variables and their renewals. We conclude that these processes cannot converge weakly to any nondegenerate random element…
Using a new approach, for spectrally negative L\'evy processes we find joint Laplace transforms involving the last exit time (from a semi-infinite interval), the value of the process at the last exit time and the associated occupation time,…
We develop an approach to Malliavin calculus for L\'evy processes from the perspective of expressing a random variable $Y$ by a functional $F$ mapping from the Skorohod space of c\`adl\`ag functions to $\mathbb{R}$, such that $Y=F(X)$ where…
A cylindrical Levy process does not enjoy a cylindrical version of the semi-martingale decomposition which results in the need to develop a completely novel approach to stochastic integration. In this work, we introduce a stochastic…
We define a new type of self-similarity for one-parameter families of stochastic processes, which applies to a number of important families of processes that are not self-similar in the conventional sense. This includes a new class of…
Let $T$ be an underlying space with a non-atomic measure $\sigma$ on it (e.g. $T=\mathbb R^d$ and $\sigma$ is the Lebesgue measure). We introduce and study a class of non-commutative generalized stochastic processes, indexed by points of…
Recently a generalized master equation was derived that extends the Lindblad theory to highly non-Markovian quantum processes (H.-P. Breuer, Phys. Rev. A \textbf{75}, 022103 (2007)). We perform a stochastic unravelling of this master…