Consider the SDE driven by the fractional Brownian motion with Hurst parameter $H \in (0.5, 1)$ $$\mathrm{d}Y_t = aY_t\mathrm{d}B^H_t + bY_t\mathrm{d}t, \quad Y_0 = \xi$$ Then they state in this paper on page 22 of the pdf that this has as unique solution $$Y_t = \xi\exp(aB^H_t + bt)$$ I am interested in the case that there is no drift term ($b\equiv0$), see also this question I posted today.
The paper states that this solution follows directly from the change of variables formula given in $(31)$: \begin{equation*} F(f(y)) - F(f(a)) = \int_a^y F'(f(t))\mathrm{d}f(t) \end{equation*} I take that it should be $f(t) = B^H_t$, but then I have no clue how they get to the result mentioned above.
I know it's probably trivial, but it has been bugging for far too long.
Any help, complete solutions or hints are greatly appreciated.