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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.

MSC:

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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