I am currently interested in fractional Brownian motion (fBm) and possible ways to define an integral with respect to some fBm. Now I have stumbled upon the notion of a forward integral defined as the ucp limit of $$\int_0^t Y(s) \frac{X(s+\epsilon)-X(s)}{\epsilon}\mathrm{d}s$$ as $\epsilon\rightarrow 0^+$, provided that the limit exists. Now assuming that $X$ is a fBm I'm wondering what are suitable conditions under which existence is ensured? I presume something like $Y\in \mathcal{L}^1_{loc}$ is enough but even if I suppose that $Y\in\mathcal{L}^\infty$ I got confused. We know that the fBm is Hölder-continuous with for some $0<\alpha < H$ with $H<1$. But then it was not possible to bound the sequence with easy estimations since $$\left|\int_0^t Y(s) \frac{X(s+\epsilon)-X(s)}{\epsilon}\mathrm{d}s\right| \leq C \int_0^t \left|\frac{X(s+\epsilon)-X(s)}{\epsilon}\right|\mathrm{d}s \leq Ct\epsilon^{\alpha-1} \rightarrow \infty$$ as $\epsilon\rightarrow 0$.
Is there something I am missing? How can I find suitable classes of functions for which this limits exists (with reasonable effort :D)? Does anyone have a tip?
Best regards and many thanks in advance!
When you talk about defining an integral with respect to FBM, isn't that a stochastic integral? I.e. $$\int_{0}^{t}Y_{s}dX_{s}$$ In this case you need $X$ is a Martingal bounded in $L^2$ and that $Y$ is a progressive process of the form $$Y_{s}(w)=\sum_{i=0}^{p-1}Y_{i}(w)\mathbf{1}_{(t_{i},t_{i+1}]}(s)$$ where $0=t_{0}< \cdots < t_{p}$ and for every $ i \in \{0,1,...,p-1\}$,$Y_{i}$ is a bounded and $\mathscr{F}_{t_{i}}$-measurable random variable.