Rate of Convergence for Discretization of Integrals with Respect to Fractional Brownian Motion

In this article, a uniform discretization of stochastic integrals $$\int _{0}^{1} f^{\prime }_-(B_t)\mathrm d B_t$$ ∫ 0 1 f − ′ ( B t ) d B t , where $$B$$ B denotes the fractional Brownian motion with Hurst parameter $$H \in (\frac{1}{2},1)$$ H ∈ ( 1 2 , 1 ) , is considered for a large class of convex functions $$f$$ f . In Azmoodeh et al. (Stat Decis 27:129–143, 2010), for any convex function $$f$$ f , the almost sure convergence of uniform discretization to such stochastic integral is proved. Here, we prove $$L^r$$ 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 arbitrarily close to $$H - \frac{1}{2}$$ H − 1 2 .