Fractional Ito calculus

Abstract

We derive Itô-type change of variable formulas for smooth functionals of irregular paths with non-zero $p-$th variation along a sequence of partitions where $p\ge 1$ is arbitrary, in terms of fractional derivative operators, extending the results of the Föllmer-Ito calculus to the general case of paths with 'fractional' regularity. In the case where $p$ is not an integer, we show that the change of variable formula may sometimes contain a non-zero a 'fractional' Itô remainder term and provide a representation for this remainder term. These results are then extended to paths with non-zero $\varphi -$variation and multi-dimensional paths. Finally, we derive an isometry property for the pathwise Föllmer integral in terms of $\varphi$ variation.

References in corpus (2)

• Pathwise integration with respect to paths of finite quadratic variation
• On pathwise quadratic variation for cadlag functions