Global solutions to stochastic Volterra equations driven by Lévy noise
Artikel i vetenskaplig tidskrift, 2018

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 du(t) = A t 0 b(t -s)u(s) ds dt + F(t, u(t)) dt + Z G(t, u(t), z) η(dz, dt) + ZL GL(t, u(t), z)ηL(dz, dt), t ∈ (0, T], u(0) = u0, where Z and ZL are Banach spaces, ∼η is a time-homogeneous compensated Poisson random measure on Z with intensity measure (capturing the small jumps), and ηL is a time-homogeneous Poisson random measure on ZL independent to ∼η with finite intensity measure 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 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.

stochastic partial differential equations of fractional order

stochastic Volterra equation

global solution

Poisson random measure

Lévy noise

stochastic integral of jump type


Erika Hausenblas

Montanuniversität Leoben

Mihaly Kovacs

Chalmers, Matematiska vetenskaper, Tillämpad matematik och statistik

Fractional Calculus and Applied Analysis

1311-0454 (ISSN) 1314-2224 (eISSN)

Vol. 21 5 1170-1202



Sannolikhetsteori och statistik

Matematisk analys



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