Let
- $d\in\mathbb N$
- $\Lambda\subseteq\mathbb R^d$ be open
- $(\Omega,\mathcal A,\operatorname P)$ be a probability space
- $(Y_t)_{t\ge0}$ be an $L^2(\Lambda)$-valued process on $(\Omega,\mathcal A,\operatorname P)$
- $k\in\mathbb N$
- $(W_t)_{t\ge0}$ be an $\mathbb R^k$-valued Wiener process on $(\Omega,\mathcal A,\operatorname P)$ with covariance operator $\operatorname{id}_{\mathbb R^k}$
- $Q$ be a linear operator from $\mathbb R^k$ to $L^2(\Lambda)$
Let $(X_t)_{t\ge0}$ be an $L^2(\Lambda)$-valued process on $(\Omega,\mathcal A,\operatorname P)$ with $$\langle X_t,\varphi\rangle_{L^2(\Lambda)}=\langle X_0,\varphi\rangle_{L^2(\Lambda)}+\int_0^t\langle Y_s,\varphi\rangle_{L^2(\Lambda)}\:{\rm d}s+\langle QW_t,\varphi\rangle_{L^2(\Lambda)}\tag1$$ for all $t\ge0$ almost surely for all $\varphi\in C_c^\infty(\Lambda)$.
Now let $f\in C^2(\Lambda)$. Are we able to deduce an Itō formula for the process $f(X)$? At least for the choice $f(x):=\left\|x\right\|_{L^2(\Lambda)}^2$ for $x\in L^2(\Lambda)$?
Note that the ordinary Itō formula can be stated in the following way: If $U,H$ are separable $\mathbb R$-Hilbert spaces, $(\mathcal F_t)_{t\ge0}$ is a complete right-continuous filtration on $(\Omega,\mathcal A,\operatorname P)$, $W$ is a $U$-valued $\mathcal F$-Wiener process on $(\Omega,\mathcal A,\operatorname P)$ with covariance operator $Q\in\mathfrak L_1(U)$, $X_0$ is a $\mathcal F_0$-measurable $H$-valued random variable $X_0$ and $\varphi,\Phi$ are $H$-valued $\mathcal F$-predictable processes which are suitably integrable, then $$X_t:=\int_0^t\varphi_s\:{\rm d}s+\int_0^t\Phi_s\:{\rm d}W_s\;\;\;\text{for }t\ge0$$ is well-defined and if $f:[0,\infty)\times H\to\mathbb R$ is Fréchet differentiable in the first and twice Fréchet differentiable in the second variable, then \begin{equation}\begin{split}f(t,X_t)&=f(0,X_0)+\int_0^t\frac{\partial f}{\partial t}(s,X_s)\:{\rm d}s\\&+\frac12\int_0^t{\rm D}_x^2f(s,X_s)\:{\rm d}[\![X]\!]_s+\int_0^t{\rm D}_xf(s,X_s)\:{\rm d}X_s,\end{split}\tag2\end{equation} where $[\![M]\!]$ denotes the tensor-quadratic variation of $M$.
