Integrals Along Rough Paths via Fractional Calculus

Using fractional calculus, we introduce an integral along β-Holder rough paths for any β ∈ (0,1]. This is a natural generalization of the Riemann–Stieltjes integral along smooth curves. We prove that, under suitable conditions on the integrand, this integral is a continuous functional with respect to the Holder topology. As a result, this provides an alternative definition of the first level path of the rough integral along geometric Holder rough paths.