Causal functional calculus. (8th October 2022)

• Record Type: Journal Article
• Authors: Chiu, Henry; Cont, Rama
• Is Part Of: Transactions of the London Mathematical Society. Volume 9:Number 1(2022)
• Page Start: 237
• Page End: 269
• Publication Date: 2022-10-08
• Subjects: Mathematics -- Periodicals; 510.5
• DOI: 10.1112/tlm3.12050
• Languages: English
• ISSNs: 2052-4986

Abstract

Abstract: 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.