memory. Note that if $H\neq \frac{1}{2}$, an fBm is not a semimartingale nor a Markov process.

Now, we recall the definition of an infinite-dimensional fBm following [@GarridoAtienza2010dcds; @maslowski2003evolution]. Let $(V,  \langle \cdot,\cdot \rangle)$ be a separable Hilbert space. Its norm is denoted by $ |\cdot|$. For a sequences $\{\lambda_{i}\}_{i \in \mathbb{N}}$ of positive real numbers with $\sum_{i=1}^{\infty} {\lambda_{i}}<\infty$ and an orthonormal basis $\{e_{i}\}_{i \in \mathbb{N}}$ of $V$, a $V$-valued fBm $B^{H}$ is defined by $$B^{H}_t=\sum_{i=1}^{\infty} \sqrt{\lambda_{i}} e_{i} \beta_{i}^{H}(t), \qquad t \ge 0,$$ where $\left\{\beta_{i}^{H}\right\}_{i \in \mathbb{N}}$ is a sequence of independent one-dimensional fBm’s. It is known that the right hand side is convergent in $L^{2}$ for every $t$ and has a continuous modification in $t$.

Let $(\Omega, \mathcal{F}, \{\mathcal{F}_t\}_{t\ge 0}, \mathbb{P})$ be a suitable probability space with a filtration satisfying the usual condition. We assume that a $V$-valued $\{\mathcal{F}_t\}$-Brownian motion $W$ and $\{\mathcal{F}_t\}$-adapted one-dimensional fBm’s $\beta^H_i, i \in {\mathbb N},$ are defined on this probability space. We further assume that $W$ and $\beta^H_i, i \in {\mathbb N},$ are all independent. (For the definition of a $V$-valued Brownian motion, see [@da2014stochastic Proposition 4.3] where is called $Q$-Wiener process.)

From now on we assume $H \in (\frac12, 1)$ and work on the 