Global solutions to the stochastic Volterra Equation driven by L\'evy noise

TL;DR

This paper establishes conditions for the existence and uniqueness of solutions to a stochastic Volterra equation driven by Lévy noise, with applications in viscoelasticity.

Contribution

It provides new theoretical results on global solutions for stochastic Volterra equations with Lévy noise, including conditions on coefficients and an applied example.

Findings

Proved existence and uniqueness of solutions under specified conditions.

Derived conditions on memory function and nonlinear mappings.

Applied results to a linear viscoelasticity model.

Abstract

In this article we investigate the existence and uniqueness of the stochastic Volterra equation driven by a \levy noise of pure jump type. In particular, we consider the following type of equation $ du(t) = ( A\int_0 ^t b(t-s) u(s)\,ds) \, dt + F(t,u(t))\,dt+ \int_ZG(t,u(t), z) \tilde \eta(dz,dt) + \int_{Z_L}G_L(t,u(t), z) \eta_L(dz,dt) ;\, t\in (0,T], $, $u(0)=u_0$, where $Z$ and $Z_L$ are Banach spaces, $\tilde \eta$ is a time-homogeneous compensated Poisson random measure on $Z$ with \levy measure $\nu$ capturing the small jumps, and $\eta_L$ is a time-homogeneous Poisson random measure on $Z_L$ with finite \levy measure $\nu_L$ capturing the large jumps. Here, $A$ is a selfadjoint operator on a Hilbert space $H$, $b$ is a scalar memory function and $F$, $G$ and $G_L$ are nonlinear mappings. We provide conditions on $b$, $F$ $G$ and $G_L$ under which a unique global solution exists. Finally, we present an example from the theory of linear viscoelasticity where our result is applicable.