Rough path analysis via fractional calculus

Abstract

Using fractional calculus we define integrals of the form ∫ a b f ( x t ) d y t \int _{a}^{b}f(x_{t})dy_{t} , where x x and y y are vector-valued Hölder continuous functions of order β ∈ ( 1 3 , 1 2 ) \beta \in (\frac {1}{3}, \frac {1}{2}) and f f is a continuously differentiable function such that f ′ f^{\prime } is λ \lambda -Hölder continuous for some λ > 1 β − 2 \lambda >\frac {1}{\beta }-2 . Under some further smooth conditions on f f the integral is a continuous functional of x x , y y , and the tensor product x ⊗ y x\otimes y with respect to the Hölder norms. We derive some estimates for these integrals and we solve differential equations driven by the function y y . We discuss some applications to stochastic integrals and stochastic differential equations.