Rough Path Analysis Via Fractional Calculus

Using fractional calculus we define integrals of the form ∫ b a f(xt)dyt, where x and y are vector-valued Hölder continuous functions of order β ∈ ( 1 3 , 1 2 ) and f is a continuously differentiable function such that f ′ is λ-Höldr continuous for some λ > 1 β − 2. Under some further smooth conditions on f the integral is a continuous functional of x, y, and the tensor product x ⊗ y with respect to the Hölder norms. We derive some estimates for these integrals and we solve differential equations driven by the function y. We discuss some applications to stochastic integrals and stochastic differential equations.