GLOBAL SOLUTIONS TO STOCHASTIC VOLTERRA EQUATIONS DRIVEN BY LEVY NOISE

In this paper we investigate the existence and uniqueness of semilinear stochastic Volterra equations driven by multiplicative Levy noise of pure jump type. In particular, we consider the equation {du(t) = (A integral(t)(0) b(t - s)u(s) ds) dt vertical bar F(t, u(t)) dt + integral(Z) G(t, u(t), z) (eta) over tilde (dz, dt) + integral(ZL) G(L)(t, u(t), z)eta(L)(dz, dt), t is an element of (0, T], u(0) = u(0), where Z and Z(L) are Banach spaces, (eta) over tilde is a time-homogeneous compensated Poisson random measure on Z with intensity measure. (capturing the small jumps), and eta(L) is a time-homogeneous Poisson random measure on ZL independent to (eta) over tilde. with finite intensity 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 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) = c(rho)t(rho-2), 1 < rho < 2, corresponds to a fractional-in-time stochastic equation and the nonlinear maps F and G can include fractional powers of A.