A complete characterization of local martingales which are functions of Brownian motion and its maximum
We prove the max-martingale conjecture given in recent article with Marc Yor. We show that for a continuous local martingale $(N\_t:t\ge 0)$ and a function $H:R x R\_+\to R$, $H(N\_t,\sup\_{s\leq t}N\_s)$ is a local martingale if and only if there exists a locally integrable function $f$ such that $...
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| Format: | Journal article |
| Language: | English |
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2005
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| _version_ | 1797080044538953728 |
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| author | Obloj, J |
| author_facet | Obloj, J |
| author_sort | Obloj, J |
| collection | OXFORD |
| description | We prove the max-martingale conjecture given in recent article with Marc Yor. We show that for a continuous local martingale $(N\_t:t\ge 0)$ and a function $H:R x R\_+\to R$, $H(N\_t,\sup\_{s\leq t}N\_s)$ is a local martingale if and only if there exists a locally integrable function $f$ such that $H(x,y)=\int\_0^y f(s)ds-f(y)(x-y)+H(0,0)$. This implies readily, via Levy's equivalence theorem, an analogous result with the maximum process replaced by the local time at 0. |
| first_indexed | 2024-03-07T00:54:29Z |
| format | Journal article |
| id | oxford-uuid:878e6d60-a160-4333-87d9-c40777f777b8 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T00:54:29Z |
| publishDate | 2005 |
| record_format | dspace |
| spelling | oxford-uuid:878e6d60-a160-4333-87d9-c40777f777b82022-03-26T22:11:28ZA complete characterization of local martingales which are functions of Brownian motion and its maximumJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:878e6d60-a160-4333-87d9-c40777f777b8EnglishSymplectic Elements at Oxford2005Obloj, JWe prove the max-martingale conjecture given in recent article with Marc Yor. We show that for a continuous local martingale $(N\_t:t\ge 0)$ and a function $H:R x R\_+\to R$, $H(N\_t,\sup\_{s\leq t}N\_s)$ is a local martingale if and only if there exists a locally integrable function $f$ such that $H(x,y)=\int\_0^y f(s)ds-f(y)(x-y)+H(0,0)$. This implies readily, via Levy's equivalence theorem, an analogous result with the maximum process replaced by the local time at 0. |
| spellingShingle | Obloj, J A complete characterization of local martingales which are functions of Brownian motion and its maximum |
| title | A complete characterization of local martingales which are functions of
Brownian motion and its maximum |
| title_full | A complete characterization of local martingales which are functions of
Brownian motion and its maximum |
| title_fullStr | A complete characterization of local martingales which are functions of
Brownian motion and its maximum |
| title_full_unstemmed | A complete characterization of local martingales which are functions of
Brownian motion and its maximum |
| title_short | A complete characterization of local martingales which are functions of
Brownian motion and its maximum |
| title_sort | complete characterization of local martingales which are functions of brownian motion and its maximum |
| work_keys_str_mv | AT oblojj acompletecharacterizationoflocalmartingaleswhicharefunctionsofbrownianmotionanditsmaximum AT oblojj completecharacterizationoflocalmartingaleswhicharefunctionsofbrownianmotionanditsmaximum |