Global solutions to stochastic Volterra equations driven by Lévy noise

Abstract

Abstract In this paper we investigate the existence and uniqueness of semilinear stochastic Volterra equations driven by multiplicative Lévy noise of pure jump type. In particular, we consider the equation $$\begin{array}{} \left\{ \begin{aligned} du(t) & = \left( A\int_0 ^t b(t-s) u(s)\,ds\right) \, dt + F(t,u(t))\,dt \\ & {} + \int_ZG(t,u(t), z) \tilde \eta(dz,dt) + \int_{Z_L}G_L(t,u(t), z) \eta_L(dz,dt),\, t\in (0,T],\\ u(0)&=u_0, \end{aligned} \right. \end{array} $$ where Z and ZL are Banach spaces, η̃ is a time-homogeneous compensated Poisson random measure on Z with intensity measure ν (capturing the small jumps), and ηL is a time-homogeneous Poisson random measure on ZL independent to η̃ with finite intensity measure νL (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 GL are nonlinear mappings. We provide conditions on b, F G and GL under which a unique global solution exists. We also present an example from the theory of linear viscoelasticity where our result is applicable. The specific kernel b(t) = cρtρ−2, 1 < ρ < 2, corresponds to a fractional-in-time stochastic equation and the nonlinear maps F and G can include fractional powers of A.