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.
Optimal control problems constrained by the stochastic Navier–Stokes equations with multiplicative Lévy noise
