Rough calculus: pathwise calculus for functionals of irregular paths Abstract

Hans Foellmer showed that the Ito formula holds pathwise, for functions paths with finite quadratic variation along a sequence of partitions. We build on Foellmer's insight to construct a pathwise calculus for smooth functionals of continuous paths with regularity defined in terms of the p-th variation along a sequence of time partitions for arbitrary large p >0. We construct a pathwise integral, defined as a pointwise limit of compensated Riemann sums and show that it satisfies a change of variable formula and an isometry formula. Results for functions are extended to path-dependent functionals using the concept of vertical derivative of a functional. Finally, we obtain a "signal plus noise" decomposition for regular functionals of paths with strictly increasing p-th variation. Our results apply to sample paths of semi-martingales as well as fractional Brownian motion with arbitrary Hurst parameter H>0.

Based on joint work with: Anna Ananova (Oxford), Henry Chiu (Imperial College London) and Nicholas Perkowski (Humboldt).

https://www.fm.mathematik.uni-muenchen.de/teaching/teaching_summer_term_2019/seminars/oberseminar_finanz_2019/cont/index.html