Fractional Ito calculus
We derive Itô–type change of variable formulas for smooth functionals of irregular paths with nonzero pth variation along a sequence of partitions, where p ≥ 1 is arbitrary, in terms of fractional derivative operators. Our results extend the results of the Föllmer–Itô calculus to the general case of...
| Main Authors: | , |
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| Format: | Journal article |
| Language: | English |
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American Mathematical Society
2024
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| _version_ | 1824458783750881280 |
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| author | Cont, R Jin, R |
| author_facet | Cont, R Jin, R |
| author_sort | Cont, R |
| collection | OXFORD |
| description | We derive Itô–type change of variable formulas for smooth functionals of irregular paths with nonzero pth variation along a sequence of partitions, where p ≥ 1 is arbitrary, in terms of fractional derivative operators. Our results extend the results of the Föllmer–Itô calculus to the general case of paths with ‘fractional’ regularity. In the case where p is not an integer, we show that the change of variable formula may sometimes contain a nonzero ‘fractional’ Itô remainder term and provide a representation for this remainder term. These results are then extended to functionals of paths with nonzero φ-variation and multidimensional paths. Using these results, we derive an isometry property for the pathwise Föllmer integral in terms of φ-variation. |
| first_indexed | 2025-02-19T04:31:23Z |
| format | Journal article |
| id | oxford-uuid:4bfd5473-f82a-49a1-a59f-1f81cb3bb118 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2025-02-19T04:31:23Z |
| publishDate | 2024 |
| publisher | American Mathematical Society |
| record_format | dspace |
| spelling | oxford-uuid:4bfd5473-f82a-49a1-a59f-1f81cb3bb1182025-01-10T12:10:51ZFractional Ito calculusJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:4bfd5473-f82a-49a1-a59f-1f81cb3bb118EnglishSymplectic ElementsAmerican Mathematical Society2024Cont, RJin, RWe derive Itô–type change of variable formulas for smooth functionals of irregular paths with nonzero pth variation along a sequence of partitions, where p ≥ 1 is arbitrary, in terms of fractional derivative operators. Our results extend the results of the Föllmer–Itô calculus to the general case of paths with ‘fractional’ regularity. In the case where p is not an integer, we show that the change of variable formula may sometimes contain a nonzero ‘fractional’ Itô remainder term and provide a representation for this remainder term. These results are then extended to functionals of paths with nonzero φ-variation and multidimensional paths. Using these results, we derive an isometry property for the pathwise Föllmer integral in terms of φ-variation. |
| spellingShingle | Cont, R Jin, R Fractional Ito calculus |
| title | Fractional Ito calculus |
| title_full | Fractional Ito calculus |
| title_fullStr | Fractional Ito calculus |
| title_full_unstemmed | Fractional Ito calculus |
| title_short | Fractional Ito calculus |
| title_sort | fractional ito calculus |
| work_keys_str_mv | AT contr fractionalitocalculus AT jinr fractionalitocalculus |