Rough path analysis via fractional calculus

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