# Operator-valued stochastic integral and quadratic variation for operator-valued processes

Let $U$ be a separable $\mathbb R$-Hilbert space and $W$ be a $Q$-Wiener process on a complete and right-continuous filtered probability space. Let $H$ be a separable $\mathbb R$-Hilbert space and $X$ be a $H$-valuedcontinuous adapted process with $$X_t=X_0+\int_0^t\varphi_s\:{\rm d}s+\int_0^t\Phi_s\:{\rm d}W_s$$ for some $H$-valued processes $\varphi$ and $\Phi$ such that the integrals are well-defined. The Itō formula, as presented by Da Prato and Zabczyk in Theorem 4.32 of their book Stochastic Equations in Infinite Dimensions, yields $$f(t,X_t)-f(0,X_0)=\int_0^t\frac{\partial f}{\partial t}(s,X_s)\:{\rm d}s+\frac12\int_0^t{\rm D}_x^2f(s,x_s)\:{\rm d}[\![X]\!]_s+\int_0^t{\rm D}_xf(s,X_s)\:{\rm d}X_s$$ for all $t$ almost surely for suitable $f:[0,\infty)\times H\to\mathbb R$. Above $[\![X]\!]$ denotes the tensor-quadratic variation of $X$. In the second integral, the second Fréchet derivative is considered as taking values in $(H\:\hat\otimes_\pi\:H)'$.

If $H=\mathbb R$, we're able to apply the Itō formula once again to obtain that $$Z_t:={\rm D}_xf(t,X_t)=\frac{\partial f}{\partial x}(t,X_t)\;\;\;\text{for }t\ge0$$ is again a semimartingale with $$[Z,X]_t=\left[\frac{\partial ^2f}{\partial x^2}(\;\cdot\;,X)\cdot X,X\right]_t=\int_0^t\frac{\partial ^2f}{\partial x^2}(s,X_s)\:{\rm d}[X]_s$$ for all $t$, where $[\;\cdot\;,\;\cdot\;]$ denotes the scalar-quadratic covariation and $\frac{\partial ^2f}{\partial x^2}(\;\cdot\;,X)\cdot X$ denotes the Itō integral process.

I would like to obtain a similar result in the General case. However, there are two major problems:

1. How do we need to define the stochastic integral $\int_0^t{\rm D}_x^2f(s,X_s)\:{\rm d}X_s$? (The problem being that the integration theory as presented by Da Prato and Zabczyk only deals with Hilbert space valued integrals, but here the integrand is takes values in $\mathfrak L(H,\mathfrak L(H,\mathbb R))$)
2. How do we need to define the tensor-quadratic covariation $[\![X,Z]\!]$? (The problem being that $Z$ is not Hilbert space valued, but $\mathfrak L(H,\mathbb R)$-valued; I know that we're in the special Situation where $\mathfrak L(H,\mathbb R)\cong H$, but I would like to come up with a solution which works for general operator-valued processes)

Last remark: I know that there is a stochastic integration theorey for Banach space valued processes. However, since above we're only dealing with processes which take values in spaces of operators between Hilbert spaces, I hope that there is an easy generalization available which is based on the Hilbert space valued theory.