Rate of convergence for discretization of integrals with respect to Fractional Brownian motion

In this article, an uniform discretization of stochastic integrals ${\int }_0^1{f}_{-}^{\mathrm{\prime }}(B_t)\text{ud}{B}_t$, with respect to fractional Brownian motion with Hurst parameter $H\in (1\mathrm{/}2,1)$, for a large class of convex functions $f$ is considered. In Statistics & Decisions, 27, 129-143, for any convex function $f$, the almost sure convergence of uniform discretization to such stochastic integral is proved. Here we prove $L^r$- convergence of uniform discretization to stochastic integral. In addition, we obtain a rate of convergence. It turns out that the rate of convergence can be brought as closely as possible to $H - 1/2$.