Related papers: Generalized Fock space and moments

We present a systematic study of moment evolution in multidimensional stochastic difference systems, focusing on characterizing systems whose low-order moments diverge in the neighborhood of a stable fixed point. We consider systems with a…

A new class of fractional-order stochastic evolution equations of the form $(\partial_t + A)^\gamma X(t) = \dot{W}^Q(t)$, $t\in[0,T]$, $\gamma \in (0,\infty)$, is introduced, where $-A$ generates a $C_0$-semigroup on a separable Hilbert…

We develop several results on hitting probabilities of random fields which highlight the role of the dimension of the parameter space. This yields upper and lower bounds in terms of Hausdorff measure and Bessel--Riesz capacity,…

We introduce the concept of numerical Gaussian processes, which we define as Gaussian processes with covariance functions resulting from temporal discretization of time-dependent partial differential equations. Numerical Gaussian processes,…

We introduce a notion of vague convergence for random marked metric measure spaces. Our main result shows that convergence of the moments of order $k \ge 1$ of a random marked metric measure space is sufficient to obtain its vague…

Locally stationary Hawkes processes have been introduced in order to generalise classical Hawkes processes away from stationarity by allowing for a time-varying second-order structure. This class of self-exciting point processes has…

In this paper we study a large class of nonlinear stochastic wave equations that arise in laser generation models and models for propagation in random media in a unified mathematical framework. Continuous and pulse-wave propagation models,…

Motivated by applications, we introduce a general and new framework for operator valued positive definite kernels. We further give applications both to operator theory and to stochastic processes. The first one yields several dilation…

This paper deals with the expectation of monomials with respect to the stochastic area integral $A_{1,2}(t,t+h)=\int_{t}^{t+h}\int_{t}^{s}{\rm d} W_{1}(r){\rm d} W_{2}(s) -\int_{t}^{t+h}\int_{t}^{s}{\rm d} W_{2}(r){\rm d} W_{1}(s)$ and the…

This survey is a preliminary version of a chapter of the forthcoming book "Stochastic Analysis for Poisson Point Processes: Malliavin Calculus, Wiener-It\^o Chaos Expansions and Stochastic Geometry" edited by Giovanni Peccati and Matthias…

A white noise quantum stochastic calculus is developped using classical measure theory as mathematical tool. Wick's and Ito's theorems have been established. The simplest quantum stochastic differential equation has been solved, unicity and…

We illustrate a counter-intuitive effect of an additive stochastic force, which acts independently on each element of an ensemble of globally coupled oscillators. We show numerically and semi-analytically that a very small white noise is…

A new notion of stochastic germs for quantum processes is introduced and a characterisation of the stochastic differentials for positive definite (PD) processes is found in terms of their germs for arbitrary Ito algebra. A representation…

In the article, integration of temporal functions in (possibly non-UMD) Banach spaces with respect to (possibly non-Gaussian) fractional processes from a finite sum of Wiener chaoses is treated. The family of fractional processes that is…

In this paper, we consider discrete-time non-linear stochastic dynamical systems with additive process noise in which both the initial state and noise distributions are uncertain. Our goal is to quantify how the uncertainty in these…

This paper generalizes the integration theory for volatility modulated Brownian-driven Volterra processes onto the space G* of Potthoff-Timpel distributions. Sufficient conditions for integrability of generalized processes are given,…

We introduce a new Gaussian process, a generalization of both fractional and subfractional Brownian motions, which could serve as a good model for a larger class of natural phenomena. We study its main stochastic properties and some…

We consider stochastic equations of the form $X_k = \phi_k(X_{k+1}) Z_k$, $k \in \mathbb{N}$, where $X_k$ and $Z_k$ are random variables taking values in a compact group $G_k$, $\phi_k: G_{k+1} \to G_k$ is a continuous homomorphism, and the…

This paper develops power series expansions of a general class of moment functions, including transition densities and option prices, of continuous-time Markov processes, including jump--diffusions. The proposed expansions extend the ones…

The generalization of fractional Brownian motion in infinite-dimensional white and grey noise spaces has been recently carried over, following the Mandelbrot-Van Ness representation, through Riemann-Liouville type fractional operators. Our…