Your approach is correct. But unfortunately it is not applicable. In the case i.e. $\frac{1}{2}<H<1$, Dai and Heyde have defined a stochastic integral as limit of Riemann sums.Their approach does not satisfy the property $E[\int_{0}^{t}f(s)dB_{H}s]=0$ (Why?!!). For this reason, Duncan have introduced a new stochastic integral with zero mean which is the limit of Riemann sums defined by means of the Wick product. Nowadays, Since the fBm is a Gaussian process, one can apply the stochastic calculus of and introduce the stochastic integral as the divergence operator,that is, the adjoint of the derivative operator variations.Using the notions of fractional integral and derivative, Zhahle has introduced
a path-wise stochastic integral. The integral of a process $f(s)$ on a time interval $[0, T]$ is defined as
$$\int_{0}^{T}f(s)dB_{H}(s)=(-1)^{\beta}\int_{0}^{T} D_{0^{t}}^{\beta}D_{T-}^{1-\beta}B_{H}(s)f(s)ds+f(0+)B_{H}(T)$$
where $D^{\beta}$ denotes the fractional derivative of order $\beta$, and assuming $\beta>1−H$ and $f$ belongs to the space $I_{0+}^{\beta}(L^1([0,T])$.
I should mention when process $f$ has $\lambda$-Holder continuous paths with $\lambda>1 − H$, then this integral can be interpreted as a Riemann_Stieltjes integral.