Solution space characterisation of perturbed linear discrete and continuous stochastic Volterra convolution equations: the $\ell^p$ and $L^p$ cases

TL;DR

This paper characterizes when solutions to perturbed linear stochastic Volterra equations are almost surely p-summable or p-integrable, revealing differences between discrete and continuous cases and analyzing asymptotic behaviors.

Contribution

It provides necessary and sufficient conditions for p-summability and p-integrability of solutions in discrete and continuous stochastic Volterra equations, including asymptotic analysis.

Findings

Discrete case: p-summability of perturbations ensures p-summable solutions.

Continuous case: solutions can be p-integrable even with non-integrable perturbations.

Asymptotic behavior analysis shows conditions for convergence to zero.

Abstract

In this article, we are concerned with characterising when solutions of perturbed linear stochastic Volterra summation equations are almost surely $p$-summable and when their continuous time counterparts, perturbed linear stochastic Volterra integro-differential equations, are almost surely $p$-integrable. In the discrete case, we find it necessary and sufficient that perturbing functions are $p$-summable in order to ensure paths of the discrete equation are almost surely $p$-summable, while in the continuous case, it transpires one can have almost surely $p$-integrable sample paths with non-integrable perturbation functions. For the continuous equation, the main converse is clinched by considering an appropriate discretisation and applying results from the discrete case. We also conduct a thorough study of the asymptotic behaviour of the trajectories of solutions to the continuous equation in the regime of $p$-integrable paths and provide a characterisation of almost sure convergence to zero in the case of diagonal noise. Additionally, we highlight how all proof methods can be applied to obtain stronger results for stochastic functional differential equations.