Existence, uniqueness and regularity for a class of semilinear stochastic Volterra equations with multiplicative noise

By Boris Baeumer, Matthias Geissert, Mihály Kovács. 2014

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Abstract We consider a class of semilinear Volterra type stochastic evolution equation driven by multiplicative Gaussian noise. The memory kernel, not necessarily analytic, is such that the deterministic linear equation exhibits a parabolic character. Under appropriate Lipschitz-type and linear growth assumptions on the nonlinear terms we show that the unique mild solution is mean- p Holder continuous with values in an appropriate Sobolev space depending on the kernel and the data. In particular, we obtain pathwise space–time (Sobolev–Holder) regularity of the solution together with a maximal type bound on the spatial Sobolev norm. As one of the main technical tools we establish a smoothing property of the derivative of the deterministic evolution operator family.

Subfields: Control and Systems Engineering, Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/df473e5d7cd6f0cc49521c06c22c05e7715cdabe