Causal functional calculus
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...
| Main Authors: | , |
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| Format: | Journal article |
| Language: | English |
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London Mathematical Society
2022
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| _version_ | 1797112556760858624 |
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| author | Henry Chiu Rama Cont |
| author_facet | Henry Chiu Rama Cont |
| author_sort | Henry Chiu |
| collection | OXFORD |
| description | 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. |
| first_indexed | 2024-03-07T08:27:17Z |
| format | Journal article |
| id | oxford-uuid:2d62854e-2cdc-4bf4-b823-9997436cfe45 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T08:27:17Z |
| publishDate | 2022 |
| publisher | London Mathematical Society |
| record_format | dspace |
| spelling | oxford-uuid:2d62854e-2cdc-4bf4-b823-9997436cfe452024-02-22T14:53:20ZCausal functional calculusJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:2d62854e-2cdc-4bf4-b823-9997436cfe45EnglishSymplectic ElementsLondon Mathematical Society2022Henry ChiuRama ContWe 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. |
| spellingShingle | Henry Chiu Rama Cont Causal functional calculus |
| title | Causal functional calculus |
| title_full | Causal functional calculus |
| title_fullStr | Causal functional calculus |
| title_full_unstemmed | Causal functional calculus |
| title_short | Causal functional calculus |
| title_sort | causal functional calculus |
| work_keys_str_mv | AT henrychiu causalfunctionalcalculus AT ramacont causalfunctionalcalculus |