Global solutions to the stochastic Volterra Equation driven by Lévy 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 $du(t)=(A{\int }_0^{t}b(t-s)u(s)ds)dt+F(t,u(t))dt+{\int }_{Z}G(t,u(t),z)\tilde{\hat{I}}\cdot (dz,dt)+{\int }_{Z_L}{G}_L(t,u(t),z)\hat{I}{\cdot }_L(dz,dt);t\in (0,T],$, $u(0)=u_0$, where $Z$ and $Z_L$ are Banach spaces, $\tilde{\hat{I}}\cdot$ is a time-homogeneous compensated Poisson random measure on $Z$ with \levy measure $\hat{I}½$ capturing the small jumps, and $\hat{I}{\cdot }_L$ is a time-homogeneous Poisson random measure on $Z_L$ with finite \levy measure $\hat{I}{½}_L$ 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.