Let us consider the fractional brownian motion $B_t^H, H \in (0,1)$ with mean zero and covariance function given by $$ \frac{1}{2} (t^{2H} + s^{2H} - |t-s|^{2H}), $$ taking values in $\mathbb{R}$.
Let us consider a random variable $X$ independent from the fBm with, for instance, uniform law. Let us define the following process $\tilde{B}_t^H = B_t^H + X$.
I need to compute the following quantity for a Borel function $f$ \begin{equation} \mathbb{E}\left[\left \vert \int_0^t f (\tilde{B}_s^H) \ ds\right \vert^2\right]. \end{equation} My first attempt was to obtain a kernel, essentially the joint law of the process $\tilde{B}_t^H$ such that $$ \mathbb{E}\left[\left \vert \int_0^t f (\tilde{B}_s^H) \ ds\right \vert^2\right]= \int_\mathbb{R}\int_{\mathbb{R}}\int_0^t \int_0^t f (x) f(y) K(x,y,t,s) \ dx \ dy \ ds \ dt. $$ However, I wanted to use a property of stationarity for this process, but in the case $H= \frac{1}{2}$ this is false.
Which can be a good approach or what are some way to estimate the quantity above.