Malliavin Calculus for Functionals with Generalized Derivatives and Some Applications to Stable Processes
We introduce the notion of a generalized derivative of a functional on a probability space with respect to some formal differentiation. We establish a sufficient condition for the existence of the distribution density of a functional in terms of its generalized derivative. This result is used for th...
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| Date: | 2002 |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2002
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/4057 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510187372150784 |
|---|---|
| author | Kulik, A. M. Кулик, О. М. |
| author_facet | Kulik, A. M. Кулик, О. М. |
| author_sort | Kulik, A. M. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:20:24Z |
| description | We introduce the notion of a generalized derivative of a functional on a probability space with respect to some formal differentiation. We establish a sufficient condition for the existence of the distribution density of a functional in terms of its generalized derivative. This result is used for the proof of the smoothness of the distribution of the local time of a stable process. |
| first_indexed | 2026-03-24T02:53:01Z |
| format | Article |
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| id | umjimathkievua-article-4057 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:53:01Z |
| publishDate | 2002 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/fc/aa41ec96662ab476849faa334dbc66fc.pdf |
| spelling | umjimathkievua-article-40572020-03-18T20:20:24Z Malliavin Calculus for Functionals with Generalized Derivatives and Some Applications to Stable Processes Числення Маллявена для функціоналів із узагальненими похідними та деякі застосування до стійких процесів Kulik, A. M. Кулик, О. М. We introduce the notion of a generalized derivative of a functional on a probability space with respect to some formal differentiation. We establish a sufficient condition for the existence of the distribution density of a functional in terms of its generalized derivative. This result is used for the proof of the smoothness of the distribution of the local time of a stable process. Введено поняття узагальненої похідної функціонала на ймовірнісному просторі відносно формального диференціювання. Отримано достатню умову існування щільності розподілу функціонала у термінах йою узагальненої похідної. Ці результати використано для доведення гладкості розподілу локального часу від стійкого процесу. Institute of Mathematics, NAS of Ukraine 2002-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4057 Ukrains’kyi Matematychnyi Zhurnal; Vol. 54 No. 2 (2002); 216-226 Український математичний журнал; Том 54 № 2 (2002); 216-226 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/4057/4824 https://umj.imath.kiev.ua/index.php/umj/article/view/4057/4825 Copyright (c) 2002 Kulik A. M. |
| spellingShingle | Kulik, A. M. Кулик, О. М. Malliavin Calculus for Functionals with Generalized Derivatives and Some Applications to Stable Processes |
| title | Malliavin Calculus for Functionals with Generalized Derivatives and Some Applications to Stable Processes |
| title_alt | Числення Маллявена для функціоналів із узагальненими похідними та деякі застосування до стійких процесів |
| title_full | Malliavin Calculus for Functionals with Generalized Derivatives and Some Applications to Stable Processes |
| title_fullStr | Malliavin Calculus for Functionals with Generalized Derivatives and Some Applications to Stable Processes |
| title_full_unstemmed | Malliavin Calculus for Functionals with Generalized Derivatives and Some Applications to Stable Processes |
| title_short | Malliavin Calculus for Functionals with Generalized Derivatives and Some Applications to Stable Processes |
| title_sort | malliavin calculus for functionals with generalized derivatives and some applications to stable processes |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4057 |
| work_keys_str_mv | AT kulikam malliavincalculusforfunctionalswithgeneralizedderivativesandsomeapplicationstostableprocesses AT kulikom malliavincalculusforfunctionalswithgeneralizedderivativesandsomeapplicationstostableprocesses AT kulikam čislennâmallâvenadlâfunkcíonalívízuzagalʹnenimipohídnimitadeâkízastosuvannâdostíjkihprocesív AT kulikom čislennâmallâvenadlâfunkcíonalívízuzagalʹnenimipohídnimitadeâkízastosuvannâdostíjkihprocesív |