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*?



Itô Formula for Hilbert space-valued Lévy processes
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.
> 
> 
> 


