Stochastic Volterra Equations of Nonscalar Type in Hilbert Space (original) (raw)

Stochastic Volterra equations of nonscalar type

2005

In the paper stochastic Volterra equations of nonscalar type are studied using resolvent approach. The aim of this note is to provide some results on stochastic convolution and integral mild solutions to those Volterra equations. The motivation of the paper comes from a model of aging viscoelastic materials.

A series approach to stochastic Volterra equations of convolution time

arXiv: Probability, 2012

In the paper stochastic Volterra equations with noise terms driven by series of independent scalar Wiener processes are considered. In our study we use the resolvent approach to the equations under consideration. We give sufficient condition for the existence of strong solution to the class of stochastic Volterra equations of convolution type. We provide regularity of stochastic convolution, as well.

Strong solutions to stochastic Volterra equations

2005

In this paper stochastic Volterra equations admitting exponentially bounded resolvents are studied. After obtaining convergence of resolvents, some properties for stochastic convolutions are studied. Our main result provide sufficient conditions for strong solutions to stochastic Volterra equations.

Stochastic Volterra equations under perturbations

Electronic Communications in Probability, 2014

We study stochastic perturbed Volterra equations of convolution type in an infinite dimensional case. Our interest is directed towards the existence and regularity of stochastic convolutions connected to the equations considered under some kind of perturbations. We use an operator theoretical method for the representation of solutions.

A Series Approach to Perturbed Stochastic Volterra Equations of Convolution Type

Advances in Pure Mathematics, 2015

In the paper, perturbed stochastic Volterra Equations with noise terms driven by series of independent scalar Wiener processes are considered. In the study, the resolvent approach to the equations under consideration is used. Sufficient conditions for the existence of strong solution to the class of perturbed stochastic Volterra Equations of convolution type are given. Regularity of stochastic convolution is supplied, as well.

Some results on stochastic convolutions arising in Volterra equations perturbed by noise

Analisi matematica.-Some results on stochastic convolutions arising in Volterra equations perturbed by noise. Nota di PHILIPPE CLéMENT e GIUSEPPE DA PRATO, presentata!") dal Corrisp. G. Da Prato. ABSTRACT.-Regularity of stochastic convolutions corresponding to a Volterra equation, perturbed by a white noise, is studied. Under suitable assumptions, hòlderianity of the corresponding trajectories is proved. KEY WORDS: Stochastic convolution; Volterra equations; Completely positive kernels. RIASSUNTO.-Sulle convoluzioni stocastiche relative a equazioni di Volterra perturbate da un rumore. Si dimostra la regolarità della convoluzione stocastica relativa a equazioni di Volterra perturbate da un rumore bianco. Sotto opportune ipotesi viene provata l'hòlderianità delle traiettorie corrispondenti.

A weak solution theory for stochastic Volterra equations of convolution type

The Annals of Applied Probability, 2021

We obtain general weak existence and stability results for stochastic convolution equations with jumps under mild regularity assumptions, allowing for non-Lipschitz coefficients and singular kernels. Our approach relies on weak convergence in L p spaces. The main tools are new a priori estimates on Sobolev-Slobodeckij norms of the solution, as well as a novel martingale problem that is equivalent to the original equation. This leads to generic approximation and stability theorems in the spirit of classical martingale problem theory. We also prove uniqueness and path regularity of solutions under additional hypotheses. To illustrate the applicability of our results, we consider scaling limits of nonlinear Hawkes processes and approximations of stochastic Volterra processes by Markovian semimartingales. Contents 1 Introduction and main results 2 2 Sobolev-Slobodeckij a priori estimate 8

Temporal and spatial regularity of solutions to stochastic Volterra equations of convolution type

In the paper regularity of solutions to stochastic Volterra equations in a separable Hilbert space is studied. Sufficient conditions for the temporal and spatial regularity of stochastic convolutions corresponding to the equations under consideration are provided. The results obtained generalize some well-known regularity results for solutions to stochastic differential equations. The paper is a continuation of previous author's papers concerning stochastic Volterra equations.

Properties of convolutions arising in stochastic Volterra equations

2004

The aim of this note is to provide some results for stochastic convolutions corresponding to stochastic Volterra equations in separable Hilbert space. We study convolution of the form WPsi(t):=int0tS(t−tau)Psi(tau)dW(tau)W^{\Psi}(t):=\int_0^t S(t-\tau)\Psi(\tau)dW(\tau)WPsi(t):=int0tS(t−tau)Psi(tau)dW(tau), tgeq0t\geq 0tgeq0, where S(t),tgeq0S(t), t\geq 0S(t),tgeq0, is so-called {\em resolvent} for Volterra equation considered,$\Psi$ is an appropriate process and WWW is a cylindrical Wiener process.

Regularity of solutions to stochastic Volterra equations of convolution type

Integral Transforms and Special Functions, 2009

In this article we give necessary and sufficient conditions providing regularity of solutions to stochastic Volterra equations with infinite delay on a d-dimensional torus. The harmonic analysis techniques and stochastic integration in function spaces are used. The work applies to both the stochastic heat and wave equations.