Optimal approximation rate of certain stochastic integrals

Given an increasing function $H:[0,1)\to [0,\infty)$ and $$A_n(H):=\inf_{\tau\in \mathcal{T}_n}(\sum_{i=1}^n \int_{t_{i-1}}^{t_i} (t_i-t)H^2(t)dt)^{{1/2}},$$ where $\mathcal{T}_n:=\{\tau=(t_i)_{i=0}^n: 0=t_0<t_1<...<t_n=1\}$, we characterize the property $A_n(H)\leq \frac{c}{\sqrt{n}}$, and give conditions for $A_n(H)\leq \frac{c}{\sqrt{n^\beta}}$ and $A_n(H)\geq \frac{1}{c\sqrt{n^\beta}}$ for $\beta\in (0,1)$, both in terms of integrability properties of $H$. These results are applied to the approximation of certain stochastic integrals.