$\sum_{i=1}^\infty\sqrt{\lambda_i}<\infty$. The following result can be found in [@maslowski2003evolution Proposition 2.1].

\[itofbm\][ Assume that $\sum_{i=1}^\infty\sqrt{\lambda_i}<\infty.$ Then there exists $\Omega_1 \subset \Omega$, $\mathbb{P}(\Omega_1)=1$, such that the pathwise integral Eq. (\[infbm\]) is well-defined on $\Omega_1$ for each $G:\Omega_1\times [0,T]\to L(V)$ satisfying $G(\omega,\cdot)e_i\in W_{\alpha,1}(0,T;V),$ for $   \omega\in \Omega_1,$ such that $\sup_{i\in\mathbb{N}}|G(\omega,\cdot)e_i|_{\alpha,1}<\infty,$ for $\omega\in\Omega_1.$ In addition $$\begin{aligned}
    \Big|\int_0^TG(s) \mathrm{d} B^H_s\Big| \le \Lambda_{\alpha,B^H}^{0,T}\sup_{i\in\mathbb{N}} |G(\cdot)e_i|_{\alpha,1}, \qquad           
    \omega \in \Omega_1,\end{aligned}$$ where $\Lambda_{\alpha,B^H}^{0,T}:=\sum_{i=1}^\infty\sqrt{\lambda_i} \Lambda_\alpha^{0,T}(\beta_i^H)$. Note that $\Lambda_{\alpha,B^H}^{0,T}$ is finite a.s. ]{}

We recall the following two auxiliary technical lemmas from [@GarridoAtienza2010dcds].

\[inq-s\] For any positive constants $a, d$, if $a+d-1>0$ and $a<1 $, one has $$\begin{aligned}
\int_{0}^{r}(r-s)^{-a}(t-s)^{-d} d s &\leq (t-r)^{1-a-d} B(1-a, d+a-1),\\
\int_{r}^{t}(s-r)^{-a}(t-s)^{-d} d s &\leq (t-r)^{1-a-d}B(1-a, d+a-1),\end{aligned}$$ where $r \in(0, t)$ and $B$ is the Beta Function.

\[inq-rho\] For any non-negative a and $d$ such that $a+d<1,$ and for any $\rho \geq 1$, there exists a positive constant $C$ such that $$\begin{aligned}
    \int_{0}^{t} e^{-\rho(t-r)}(t-r)^{-a} r^{-d} d r \leq C \rho^{a+d-1}.
    \end{aligned}$$ In addition, for $d \leq 0$ and $0 \leq a<1,$ and for any $\rho \geq 1$, we have $$\begin{aligned}
    \int_{0}^{t} e^{-\rho(t-r)}(t-r)^{-a} r^{-d} d r \leq \Gamma(1-a) t^{-d} \rho^{a-1}.
    \end{aligned}$$

Please note that $C$ and $C_\ast$ denote certain positive constants that may depend on the parameters 