Three concepts of local times for deterministic càdlàg paths are developed and the corresponding pathwise Tanaka–Meyer formulae are provided. For semimartingales, it is shown that their sample paths a.s. satisfy all three pathwise definitions of local times and that all coincide with the classical semimartingale local time. In particular, this demonstrates that each definition constitutes a legit pathwise counterpart of probabilistic local times. The last pathwise construction presented in the paper expresses local times in terms of normalized numbers of interval crossings and does not depend on the choice of the sequence of grids. This is a new result also for càdlàg semimartingales, which may be related to previous results of Nicole El Karoui  and Marc Lemieux .
This project was generously supported by the European Research Council under (FP7/2007-2013)/ERC Grant agreement no. 335421. The research of RMŁ was partially supported by the National Science Centre (Poland) under the grant agreements no. 2016/21/B/ST1/0148 and no. 2019/35/B/ST1/0429.
JO is grateful to St John’s College Oxford for their support, and to the Sydney Mathematical Research Institute, where the final stages of this research were completed, for their hospitality.
"Local times and Tanaka–Meyer formulae for càdlàg paths." Electron. J. Probab. 26 1 - 29, 2021. https://doi.org/10.1214/21-EJP638