Integration theory for infinite dimensional volatility modulated Volterra processes. (English) Zbl 1341.60047

Summary: We treat a stochastic integration theory for a class of Hilbert-valued, volatility-modulated, conditionally Gaussian Volterra processes. We apply techniques from Malliavin calculus to define this stochastic integration as a sum of a Skorohod integral, where the integrand is obtained by applying an operator to the original integrand, and a correction term involving the Malliavin derivative of the same altered integrand, integrated against the Lebesgue measure. The resulting integral satisfies many of the expected properties of a stochastic integral, including an Itô formula. Moreover, we derive an alternative definition using a random field approach and relate both concepts. We present examples related to fundamental solutions to partial differential equations.


60H05 Stochastic integrals
60H07 Stochastic calculus of variations and the Malliavin calculus
60G15 Gaussian processes
60G60 Random fields
60H30 Applications of stochastic analysis (to PDEs, etc.)
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