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 |
| Published: |
American Mathematical Society
2024
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| Summary: | 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. |
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