2021/11/27 by Rama Cont, Cont, Rama, Ruhong Jin +1 · 1 citation
Mathematics · #Advanced Harmonic Analysis Research #Nonlinear Partial Differential Equations #Advanced Mathematical Physics Problems
paper · pdf · doi:10.48550/arxiv.2111.13979
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 ≥ 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 ϕ-variation and multi-dimensional paths. Finally, we derive an isometry property for the pathwise Föllmer integral in terms of ϕ variation.