[@Zahle1998], for $h\in W^{\alpha,1}(0,T;V), 0 \le s<t \le T$, we define the generalized Stieltjes integral $$\begin{aligned}
\label{g-s}
    \int_0^T h(r) \mathrm{d}l(r)=&(-1)^\alpha\int_0^TD_{0+}^{\alpha}h(r)D_{T-}^{1-\alpha}l_{T-}(r)\mathrm{d}r,\\
    \int_s^t h(r)\mathrm{d}l(r)=&\int_0^T h(r) \mathbf{1}_{(s,t)}\mathrm{d}l(r),\end{aligned}$$ where, in general, for $0\le a < c\le T, l_{c-}(r):=l(r)-l(c),$ and for $a<t<c$ the Weyl derivatives are given by $$\begin{aligned}
    D_{\alpha+}^\alpha h(t)&=&\frac{1}{\Gamma(1-\alpha)}\Big(\frac{h(t)}{(t-a)^\alpha}+\alpha\int_a^t\frac{h(t)-h(\zeta)}{(t-\zeta)^{\alpha-1}}\mathrm{d}\zeta\Big),\cr
    D_{c-}^{1-\alpha} l_{c-} (t)&=&\frac{(-1)^{1-\alpha}}{\Gamma(\alpha)}\Big(\frac{l(t)-l(c)}{(c-t)^{1-\alpha}}+(1-\alpha) \int_t^c\frac{l(t)-l(\zeta)}{(\zeta-t)^{2-\alpha}}\mathrm{d}\zeta\Big),\end{aligned}$$ where $\Gamma$ denotes the Gamma function. It can be proved that the integral (\[g-s\]) exists and that the following crucial inequality holds $$\begin{aligned}
\Big|\int_0^T h(t) \mathrm{d}l(t)\Big| \leq \frac{\|l\|_{\alpha,0,T}}{\Gamma(1-\alpha)\Gamma(\alpha)} |h|_{\alpha,1},\end{aligned}$$ where $$\|l\|_{\alpha,0,T}:=\sup_{0\le s<t\le T}\Big(\frac{|l(t)-l(s)|}{(t-s)^{1-\alpha}}+\int_s^t\frac{|l(\zeta)-l(s)|}{(\zeta-s)^{2-\alpha}}\mathrm{d}\zeta\Big)<\infty.$$

For the sake of shortness, we denote $\Lambda_\alpha^{0,T}(l):=\frac{\|l\|_{\alpha,0,T}}{\Gamma(1-\alpha)\Gamma(\alpha)}.$

Let $H\in(\frac{1}{2},1)$, take a parameter $\alpha \in( 1-H,\frac{1}{2})$ which will be fixed througout this paper. For $h\in W^{\alpha,1}(0,T;V)$ the integral $$\int_{0}^{T}h(s)\mathrm{d}\beta^H(s)$$ will be understood in the sense of definition (\[g-s\]) pathwise, which makes sense because $\Lambda_\alpha^{0,T}(\beta^H)<\infty$ a.s. (cf. [@Rascanu2002]).

Let $L(V)$ denote the space of linear bounded operators on $V$ and let $G:\Omega \times[0,T]\to L(V)$ be an operator valued map such that $G(\omega,\cdot) e_i\in W^{\alpha,1}(0,T;V)$ for each $i\in \mathbb{N}$ and almost $\omega\in\Omega$. We define $$\begin{aligned}
\label{infbm}
\int_0^TG(s)\mathrm{d} B_s^H:=\sum_{i=1}^\infty \sqrt{\lambda_i} \int_0^TG(s)e_i \mathrm{d} \beta_i^H(s),\end{aligned}$$ where the convergence of the sums in (\[infbm\]) is understood as $\mathbb{P}$-a.s. convergence in $V$.

From now on, to make the pathwise integral (\[infbm\]) well-defined, we assume 