ROUGH PATH ANALYSIS VIA FRACTIONAL CALCULUS

Using fractional calculus we define integrals of the form integral(b)(a)f(x(t))dy(t), where x and y are vector-valued Holder continuous functions of order beta is an element of (1/3, 1/2) and f is a continuously differentiable function such that f' is lambda-Holder continuous for some lambda > 1/beta - 2. Under some further smooth conditions on f the integral is a continuous functional of x, y, and the tensor product x circle times y with respect to the Holder 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.