Causal functional calculus

We construct a new topology on the space of stopped paths and introduce calculus for causal functionals generic domains this space. propose approach to pathwise integration without any assumption variation index path obtain functional change variable formulae which extend results Föllmer [Séminaire de probabilités 15 (1981), 143–150] Cont Fournié [J. Funct. Anal. 259 (2010), no. 4, 1043–1072] larger class functionals, including Föllmer's integrals. show that smooth possess analogue martingale property. For finite quadratic variation, our extends Föllmer–Ito removes previous restriction time partition sequence. foliation structure harmonic may be represented as integrals closed 1-forms.