Non-exponential stability of scalar stochastic Volterra equations (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.
Stochastic Analysis and Applications, 2006
In this note we address the question of how large a stochastic perturbation an asymptotically stable linear functional differential system can tolerate without losing the property of being pathwise asymptotically stable. In particular, we investigate noise perturbations which are either independent of the state or influenced by the current and past states. For perturbations independent of the state, we prove that the assumed rate of fading for the noise is optimal.
On a random solution of a nonlinear perturbed stochastic integral equation of the Volterra type
Bulletin of the Australian Mathematical Society, 1973
where io € ft , the supporting set of the complete probability measure space (ft, A, \i). We are concerned with the existence and uniqueness of a random solution to the above equation. A random solution, x(t; u) , of the above equation is defined to be a vector random variable which satisfies the equation y almost everywhere.
Comparison of Stochastic Volterra Equations
Bernoulli, 2000
A pathwise comparison theorem for a class of one-dimensional stochastic Volterra equations driven by continuous semimartingales is proved under suitable conditions. The result is applied to equations appearing in applications.
On the Limit Measure to Stochastic Volterra Equations
Journal of Integral Equations and Applications, 2003
The paper is concerned with a limit measure of stochastic Volterra equation driven by a spatially homogeneous Wiener process with values in the space of real tempered distributions. Necessary and sufficient conditions for the existence of the limit measure are provided and a form of any limit measure is given as well.
arXiv (Cornell University), 2013
The almost sure rate of exponential-polynomial growth or decay of affine stochastic Volterra and affine stochastic finite-delay equations is investigated. These results are achieved under suitable smallness conditions on the intensities of the deterministic and stochastic perturbations diffusion, given that the asymptotic behaviour of the underlying deterministic resolvent is determined by the zeros of its characteristic equation. The results rely heavily upon a stochastic variant of the admissibility theory for linear Volterra operators.
On numerical solutions to stochastic Volterra equations
2004
In the first part we recall some apparently well-known results concerning the Volterra equations under consideration. In the second one we describe a numerical algorithm used and next present some examples of numerical solutions in order to illustrate the pertinent features of the technique used in the paper.