Fractional Ito calculus - Cont, R; Jin, R (American Mathematical Society, 2024)
We derive Itô–type change of variable formulas for smooth functionals of irregular paths with nonzero pth variation along a sequence of partitions, where p ≥ 1 is arbitrary, in terms of fractional derivative operators. Our results extend the results of the Föllmer–Itô 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 nonzero ‘fractional’ Itô remainder term and provide a representation for this remainder term. These results are then extended to functionals of paths with nonzero φ-variation and multidimensional paths. Using these results, we derive an isometry property for the pathwise Föllmer integral in terms of φ-variation.