May I ask why the following operator between Banach spaces is continuous please?

Question

I am reading the paper "On the Initial Value Problem of Stochastic Evolution Equations in Hilbert Spaces" (Here is the link for electronic version). But when reading it, I meet a problem and would like to ask for help:
Background:
1.The event space is $\Omega$ with sigma algebra and probability measure $P$.
2.$H$ and $K$ are two real Hilberts.
3.$L^2(\Omega,H)$ denotes the collection of strongly measurable, square-integrable, $H$-valued random variables and it is a Banach space with norm: for any element $u$, $\|u(.)\|_{L^2}=(E\|u(.,\omega)\|)^{1/2}$, where the expectation $E$ is defined by $Eu=\int_{\Omega}u(\omega)dP$, $\omega\in\Omega$.
4.$W_{t}$ denotes the cylindrical Wiener process valued in $K$ with covariance operator $Q$.
5.Let $J$ denotes some closed subinterval in $[0,\infty)$, $C(J,L^2(\Omega,H))$ denotes the collection of continuous processes from $J$ to $L^2(\Omega,H)$ such that $sup_{t\in J}E\|u(t)\|^2<\infty$. It is a Banach space with the norm $\|u\|_{C}=(sup_{t \in J}E\|u\|^2)^{1/2}$.
6.$A$ is an operator on $H$ which generates a family of compact operators $(S_{t})_{t>0}$.
Problem:
Assume $f:J\times H \to L_{HS}(K,H)$ is continuous nonlinear map, where $L_{HS}(K,H)$ denotes the Hilbert-Schmidt operators; the set $\{E\|f(t,u(t))\|^2:t \in J \}$ is bounded for any bounded $E\|u(t) \|^2$.
Then consider a subinterval $[t_0,t_0+h]$ of $J$,
the map $(Fu)(t) = \int_{to}^tS(t-s)f(s,u(s))dW_s$ is continuous from $C(J,L^2(\Omega,H))$ to itself.
(The above map is different from formula (7) in the paper, but the essential should be the same. ) For its proof, it is just pointed out that due to continuity of $f$, map $F$ is continuous, I could not figure out why.
Furthermore, if we put randomness on $f$, for instance $f$ is a stochastic process valued in the space of continuous maps between $H$ and $L_{HS}(K,H)$ with continuous sample paths, will the same argument hold please?
Thank you very much!