Quadratic variation and quadratic roughness
We study the concept of quadratic variation of a continuous path along a sequence of partitions and its dependence with respect to the choice of the partition sequence. We define the concept of quadratic roughness of a path along a partition sequence and show that, for H\"older-continuous paths...
| Main Authors: | , |
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| Format: | Journal article |
| Language: | English |
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Bernoulli Society for Mathematical Statistics and Probability
2022
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| _version_ | 1826308624569335808 |
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| author | Cont, R Das, P |
| author_facet | Cont, R Das, P |
| author_sort | Cont, R |
| collection | OXFORD |
| description | We study the concept of quadratic variation of a continuous path along a
sequence of partitions and its dependence with respect to the choice of the
partition sequence. We define the concept of quadratic roughness of a path
along a partition sequence and show that, for H\"older-continuous paths
satisfying this roughness condition, the quadratic variation along balanced
partitions is invariant with respect to the choice of the partition sequence.
Typical paths of Brownian motion are shown to satisfy this quadratic roughness
property almost-surely along any partition with a required step size condition.
Using these results we derive a formulation of F\"ollmer's pathwise integration
along paths with finite quadratic variation which is invariant with respect to
the partition sequence. |
| first_indexed | 2024-03-07T07:22:05Z |
| format | Journal article |
| id | oxford-uuid:f969025e-f0f6-46ee-92fb-064e5381d988 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T07:22:05Z |
| publishDate | 2022 |
| publisher | Bernoulli Society for Mathematical Statistics and Probability |
| record_format | dspace |
| spelling | oxford-uuid:f969025e-f0f6-46ee-92fb-064e5381d9882022-10-19T10:50:51ZQuadratic variation and quadratic roughnessJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:f969025e-f0f6-46ee-92fb-064e5381d988EnglishSymplectic ElementsBernoulli Society for Mathematical Statistics and Probability2022Cont, RDas, PWe study the concept of quadratic variation of a continuous path along a sequence of partitions and its dependence with respect to the choice of the partition sequence. We define the concept of quadratic roughness of a path along a partition sequence and show that, for H\"older-continuous paths satisfying this roughness condition, the quadratic variation along balanced partitions is invariant with respect to the choice of the partition sequence. Typical paths of Brownian motion are shown to satisfy this quadratic roughness property almost-surely along any partition with a required step size condition. Using these results we derive a formulation of F\"ollmer's pathwise integration along paths with finite quadratic variation which is invariant with respect to the partition sequence. |
| spellingShingle | Cont, R Das, P Quadratic variation and quadratic roughness |
| title | Quadratic variation and quadratic roughness |
| title_full | Quadratic variation and quadratic roughness |
| title_fullStr | Quadratic variation and quadratic roughness |
| title_full_unstemmed | Quadratic variation and quadratic roughness |
| title_short | Quadratic variation and quadratic roughness |
| title_sort | quadratic variation and quadratic roughness |
| work_keys_str_mv | AT contr quadraticvariationandquadraticroughness AT dasp quadraticvariationandquadraticroughness |