On the uniqueness of higher order Gubinelli derivatives and an analogue of the Doob – Meyer theorem for rough paths of the arbitrary positive Holder index
In this paper, we investigate the features of higher order Gubinelli derivatives of controlled rough paths having an arbitrary positive Holder index. There is used a notion of the (α, β)-rough map on the basis of which the sufficient conditions are given for the higher order Gubinelli derivatives un...
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| Format: | Article |
| Language: | Belarusian |
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Belarusian State University
2022-07-01
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| Series: | Журнал Белорусского государственного университета: Математика, информатика |
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| Online Access: | https://journals.bsu.by/index.php/mathematics/article/view/4362 |
| _version_ | 1811282375001243648 |
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| author | Maksim M. Vaskovskii |
| author_facet | Maksim M. Vaskovskii |
| author_sort | Maksim M. Vaskovskii |
| collection | DOAJ |
| description | In this paper, we investigate the features of higher order Gubinelli derivatives of controlled rough paths having an arbitrary positive Holder index. There is used a notion of the (α, β)-rough map on the basis of which the sufficient conditions are given for the higher order Gubinelli derivatives uniqueness. Using the theorem on the uniqueness of higher order Gubinelli derivatives an analogue of the Doob – Meyer theorem for rough paths with an arbitrary positive Holder index is proved. In the final section of the paper, we prove that the law of the local iterated logarithm for fractional Brownian motion allows using all the main results of this paper for integration over the multidimensional fractional Brownian motions of the arbitrary Hurst index. The examples demonstrating the connection between the rough path integrals and the Ito and Stratonovich integrals are represented. |
| first_indexed | 2024-04-13T01:51:27Z |
| format | Article |
| id | doaj.art-23a8cd972f624ab982aa67c01883d005 |
| institution | Directory Open Access Journal |
| issn | 2520-6508 2617-3956 |
| language | Belarusian |
| last_indexed | 2024-04-13T01:51:27Z |
| publishDate | 2022-07-01 |
| publisher | Belarusian State University |
| record_format | Article |
| series | Журнал Белорусского государственного университета: Математика, информатика |
| spelling | doaj.art-23a8cd972f624ab982aa67c01883d0052022-12-22T03:07:52ZbelBelarusian State UniversityЖурнал Белорусского государственного университета: Математика, информатика2520-65082617-39562022-07-01261410.33581/2520-6508-2022-2-6-144362On the uniqueness of higher order Gubinelli derivatives and an analogue of the Doob – Meyer theorem for rough paths of the arbitrary positive Holder indexMaksim M. Vaskovskii0https://orcid.org/0000-0001-5769-3678Belarusian State University, 4 Niezaliežnasci Avenue, Minsk 220030, BelarusIn this paper, we investigate the features of higher order Gubinelli derivatives of controlled rough paths having an arbitrary positive Holder index. There is used a notion of the (α, β)-rough map on the basis of which the sufficient conditions are given for the higher order Gubinelli derivatives uniqueness. Using the theorem on the uniqueness of higher order Gubinelli derivatives an analogue of the Doob – Meyer theorem for rough paths with an arbitrary positive Holder index is proved. In the final section of the paper, we prove that the law of the local iterated logarithm for fractional Brownian motion allows using all the main results of this paper for integration over the multidimensional fractional Brownian motions of the arbitrary Hurst index. The examples demonstrating the connection between the rough path integrals and the Ito and Stratonovich integrals are represented.https://journals.bsu.by/index.php/mathematics/article/view/4362rough pathsgubinelli derivativedoob – meyer expansionfractional brownian motion |
| spellingShingle | Maksim M. Vaskovskii On the uniqueness of higher order Gubinelli derivatives and an analogue of the Doob – Meyer theorem for rough paths of the arbitrary positive Holder index Журнал Белорусского государственного университета: Математика, информатика rough paths gubinelli derivative doob – meyer expansion fractional brownian motion |
| title | On the uniqueness of higher order Gubinelli derivatives and an analogue of the Doob – Meyer theorem for rough paths of the arbitrary positive Holder index |
| title_full | On the uniqueness of higher order Gubinelli derivatives and an analogue of the Doob – Meyer theorem for rough paths of the arbitrary positive Holder index |
| title_fullStr | On the uniqueness of higher order Gubinelli derivatives and an analogue of the Doob – Meyer theorem for rough paths of the arbitrary positive Holder index |
| title_full_unstemmed | On the uniqueness of higher order Gubinelli derivatives and an analogue of the Doob – Meyer theorem for rough paths of the arbitrary positive Holder index |
| title_short | On the uniqueness of higher order Gubinelli derivatives and an analogue of the Doob – Meyer theorem for rough paths of the arbitrary positive Holder index |
| title_sort | on the uniqueness of higher order gubinelli derivatives and an analogue of the doob meyer theorem for rough paths of the arbitrary positive holder index |
| topic | rough paths gubinelli derivative doob – meyer expansion fractional brownian motion |
| url | https://journals.bsu.by/index.php/mathematics/article/view/4362 |
| work_keys_str_mv | AT maksimmvaskovskii ontheuniquenessofhigherordergubinelliderivativesandananalogueofthedoobmeyertheoremforroughpathsofthearbitrarypositiveholderindex |