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Global solutions to stochastic Volterra equations driven by Lévy noise. (English) Zbl 1436.60064

Summary: 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 \[\begin{cases} du(t) = \bigg( A \int_0^t b(t-s) u(s)\,ds \bigg )dt + F(t,u(t))\,dt + \int_Z G(t,u(t), z) \tilde \eta (dz,dt) + \int_{Z_L} G_L(t,u(t), z) \eta_L(dz,dt),\, t \in (0,T],\\ u(0) =u_0,\end{cases}\] where \(Z\) and \(Z_L\) are Banach spaces, \(\eta\) is a time-homogeneous compensated Poisson random measure on \(Z\) with intensity measure \(\nu\) (capturing the small jumps), and \(\eta_L\) is a time-homogeneous Poisson random measure on \(Z_L\) independent to \(\eta\) 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 \(G_L\) 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 \(A\).

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60G57 Random measures
45D05 Volterra integral equations
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