Open AccessCausal functional calculus
Author(s) -
Chiu Henry,
Cont Rama
Publication year - 2022
Publication title -
transactions of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.43
H-Index - 7
ISSN - 2052-4986
DOI - 10.1112/tlm3.12050
Subject(s) - mathematics , quadratic variation , martingale (probability theory) , partition (number theory) , pure mathematics , space (punctuation) , calculus (dental) , combinatorics , computer science , medicine , statistics , dentistry , brownian motion , operating system
We construct a new topology on the space of stopped paths and introduce a calculus for causal functionals on generic domains of this space. We propose a generic approach to pathwise integration without any assumption on the variation index of a path and obtain functional change of variable formulae which extend the results of Föllmer [Séminaire de probabilités 15 (1981), 143–150] and Cont and Fournié [J. Funct. Anal. 259 (2010), no. 4, 1043–1072] to a larger class of functionals, including Föllmer's pathwise integrals. We show that a class of smooth functionals possess a pathwise analogue of the martingale property. For paths that possess finite quadratic variation, our approach extends the Föllmer–Ito calculus and removes previous restriction on the time partition sequence. We introduce a foliation structure on this path space and show that harmonic functionals may be represented as pathwise integrals of closed 1‐forms.