# Is there an Itō formula for random functions in infinite-dimensions?

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.

The case when the drift and diffusion coefficients for $X$ are bounded is easy-- the following hand wavy proof is almost a proof. Because of the isometries between all seperable Hilbert spaces and the fact that the ito integral is defined in mean square allows the fact that the Ito formula is true for each of the basis functions, then by a two or three epsilon proof it should be true for arbitrary functions from $I \times H$ to $\mathbb{R}$
• I don't think that it's easy even when we assume that $\varphi$ and $\Phi$ are independent of time and take only finitely many values. My problem is $f$. In the "ordinary" Itō formula, $f$ is differentiable in time and twice differentiable in space. Here, for each fixed $x$, $f(\;\cdot\;,x)$ is a semimartingale (of a special type; sometimes called Itō process). Now the first problem is that for each fixed $x$ the Itō integral $\int_0^th(s,x)\:{\rm d}W_s$ is only defined up to a set of measure 0. And clearly, the union of all these sets doesn't need to be of measure zero. – 0xbadf00d May 15 '18 at 19:16
• And even when this problem is solved, we need conditions that, for example, at least ensure that outside a set of measure zero, $f(t,\;\cdot\;)$ is Fréchet differentiable for all $t$. If you know how we obtain these things, please let me know. – 0xbadf00d May 15 '18 at 19:16