Fractional Ito calculus

TL;DR

This paper extends Itô calculus to irregular paths with fractional regularity, deriving change of variable formulas involving fractional derivatives and remainders, applicable to paths with non-zero p-variation and multi-dimensional cases.

Contribution

It introduces fractional Itô calculus formulas for paths with fractional regularity, including fractional derivatives, remainders, and extensions to multi-dimensional paths.

Findings

Derived fractional Itô change of variable formulas.

Identified conditions for non-zero fractional remainders.

Established an isometry property for pathwise Föllmer integral.

Abstract

We derive It\^o-type change of variable formulas for smooth functionals of irregular paths with non-zero $p-$th variation along a sequence of partitions where $p \geq 1$ is arbitrary, in terms of fractional derivative operators, extending the results of the F\"ollmer-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' It\^o remainder term and provide a representation for this remainder term. These results are then extended to paths with non-zero $\phi-$variation and multi-dimensional paths. Finally, we derive an isometry property for the pathwise F\"ollmer integral in terms of $\phi$ variation.