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

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.