I know there are Itô formulas for cylindrical Brownian motions with values in a Hilbert space and Itô formulas for Lévy processes in $\mathbb{R}^d$. My question is: does there exist an Itô formula for a Lévy process with values in a Hilbert space?
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Yes, there is an Itô formula for Hilbert space valued semimartingales. On this matter, see Appendix D of
S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations with Levy Noise, Encyclopedia of Mathematics, Cambridge University Press, 2007.

$\begingroup$ Great! But I was wondering if there is an Ito formula for a function of time and space similar to Theorem 4.32 of "Stochastic Equations in Infinite DimensionsCambridge University Press (2014)" but with Levy Noise instead of Brownian Noise $\endgroup$ – AIM_BLB Sep 4 '16 at 15:20