Let

- $(\Omega,\mathcal A,\operatorname P)$ be a complete probability space
- $T>0$
- $I:=(0,T]$
- $(\mathcal F_t)_{t\in\overline I}$ be a complete and right-continuous filtration on $(\Omega,\mathcal A,\operatorname P)$
- $U$ be a separable $\mathbb R$-Hilbert space
- $W$ be a $Q$-Wiener process on $(\Omega,\mathcal A,\mathcal F,\operatorname P)$
- $H$ be a separable $\mathbb R$-Hilbert space

Assume $X:\Omega\times\overline I\to H$ satisfies $$X_t=X_0+\int_0^t\varphi_s\:{\rm d}s+\int_0^t\Phi_s\:{\rm d}W_s\;\;\;\text{for all }t\in\overline I\tag1$$ for some $\varphi,\Phi$ such that the integrals are well-defined. Moreover, assume $f:\Omega\times\overline I\times H\to\mathbb R$ satisfies $$f(t,x)=f(0,x)+\int_0^tg(s,x)\:{\rm d}s+\int_0^th(s,x)\:{\rm d}W_s\;\;\;\text{for all }t\in\overline I\text{ and }x\in H\tag2$$ for some $g,h$ such that the integrals are well-defined.

Are we able to prove an Itō formula for the process $f(t,X_t)$, $t\in\overline I$?

For $H=\mathbb R^d$, $d\in\mathbb N$, the answer is yes and the resulting formula is known as the Itō-Wentzell formula. Proofs can be found in the books of Kunita and Rozovskii.

Unfortunately, these books seem to be the only references on the topic at all.