Anticipative integrals with respect to a filtered Lévy process and Lévy-Itô decomposition
A filtered process $X^k$ is defined as an integral of a deterministic kernel $k$ with respect to a stochastic process $X$. One of the main problems to deal with such processes is to define a stochastic integral with respect to them. When $X$ is a Brownian motion one can use the Gaussian properties o...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2017 |
| País: | España |
| Institución: | Universidad de Barcelona |
| Repositorio: | Dipòsit Digital de la UB |
| OAI Identifier: | oai:diposit.ub.edu:2445/124835 |
| Acceso en línea: | https://hdl.handle.net/2445/124835 |
| Access Level: | acceso abierto |
| Palabra clave: | Anàlisi estocàstica Processos estocàstics Analyse stochastique Stochastic processes |
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Anticipative integrals with respect to a filtered Lévy process and Lévy-Itô decompositionSavy, NicolasVives i Santa Eulàlia, Josep, 1963-Anàlisi estocàsticaProcessos estocàsticsAnalyse stochastiqueStochastic processesA filtered process $X^k$ is defined as an integral of a deterministic kernel $k$ with respect to a stochastic process $X$. One of the main problems to deal with such processes is to define a stochastic integral with respect to them. When $X$ is a Brownian motion one can use the Gaussian properties of $X^k$ to define an integral intrinsically. When $X$ is a jump process or a Levy process, this is not possible. Alternatively, we can use the integrals defined by means of the so called $\mathcal{S}$-transform or by means of the integral with respect to the process $X$ and a linear operator $\mathcal{K}$ constructed from $k$. The usual fact that even for predictable $Y$, $K^{\ast}(Y)$ may not be predictable forces us to consider only anticipative integrals. The aim of this paper is, on the one hand, to clarify the links between these integrals for a given $X$ and on the other hand, to investigate how the Lévy-Itô decomposition of a Levy process $L$, roughly speaking $L=B+J$, where $B$ is a Brownian motion and $J$ is a pure jump Lévy process, behaves with respect to these integrals.Serials Publications2017info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionapplication/pdfhttps://hdl.handle.net/2445/124835Articles publicats en revistes (Matemàtiques i Informàtica)reponame:Dipòsit Digital de la UBinstname:Universidad de BarcelonaInglésReproducció del document publicat a: https://www.math.lsu.edu/cosa/11-1-05[543].pdfCommunications on Stochastic Analysis, 2017, vol. 11, num. 1, p. 63-85(c) Serials Publications, 2017info:eu-repo/semantics/openAccessoai:diposit.ub.edu:2445/1248352026-05-27T06:46:51Z |
| dc.title.none.fl_str_mv |
Anticipative integrals with respect to a filtered Lévy process and Lévy-Itô decomposition |
| title |
Anticipative integrals with respect to a filtered Lévy process and Lévy-Itô decomposition |
| spellingShingle |
Anticipative integrals with respect to a filtered Lévy process and Lévy-Itô decomposition Savy, Nicolas Anàlisi estocàstica Processos estocàstics Analyse stochastique Stochastic processes |
| title_short |
Anticipative integrals with respect to a filtered Lévy process and Lévy-Itô decomposition |
| title_full |
Anticipative integrals with respect to a filtered Lévy process and Lévy-Itô decomposition |
| title_fullStr |
Anticipative integrals with respect to a filtered Lévy process and Lévy-Itô decomposition |
| title_full_unstemmed |
Anticipative integrals with respect to a filtered Lévy process and Lévy-Itô decomposition |
| title_sort |
Anticipative integrals with respect to a filtered Lévy process and Lévy-Itô decomposition |
| dc.creator.none.fl_str_mv |
Savy, Nicolas Vives i Santa Eulàlia, Josep, 1963- |
| author |
Savy, Nicolas |
| author_facet |
Savy, Nicolas Vives i Santa Eulàlia, Josep, 1963- |
| author_role |
author |
| author2 |
Vives i Santa Eulàlia, Josep, 1963- |
| author2_role |
author |
| dc.subject.none.fl_str_mv |
Anàlisi estocàstica Processos estocàstics Analyse stochastique Stochastic processes |
| topic |
Anàlisi estocàstica Processos estocàstics Analyse stochastique Stochastic processes |
| description |
A filtered process $X^k$ is defined as an integral of a deterministic kernel $k$ with respect to a stochastic process $X$. One of the main problems to deal with such processes is to define a stochastic integral with respect to them. When $X$ is a Brownian motion one can use the Gaussian properties of $X^k$ to define an integral intrinsically. When $X$ is a jump process or a Levy process, this is not possible. Alternatively, we can use the integrals defined by means of the so called $\mathcal{S}$-transform or by means of the integral with respect to the process $X$ and a linear operator $\mathcal{K}$ constructed from $k$. The usual fact that even for predictable $Y$, $K^{\ast}(Y)$ may not be predictable forces us to consider only anticipative integrals. The aim of this paper is, on the one hand, to clarify the links between these integrals for a given $X$ and on the other hand, to investigate how the Lévy-Itô decomposition of a Levy process $L$, roughly speaking $L=B+J$, where $B$ is a Brownian motion and $J$ is a pure jump Lévy process, behaves with respect to these integrals. |
| publishDate |
2017 |
| dc.date.none.fl_str_mv |
2017 |
| dc.type.none.fl_str_mv |
info:eu-repo/semantics/article info:eu-repo/semantics/publishedVersion |
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article |
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publishedVersion |
| dc.identifier.none.fl_str_mv |
https://hdl.handle.net/2445/124835 |
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https://hdl.handle.net/2445/124835 |
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Inglés |
| language_invalid_str_mv |
Inglés |
| dc.relation.none.fl_str_mv |
Reproducció del document publicat a: https://www.math.lsu.edu/cosa/11-1-05[543].pdf Communications on Stochastic Analysis, 2017, vol. 11, num. 1, p. 63-85 |
| dc.rights.none.fl_str_mv |
(c) Serials Publications, 2017 info:eu-repo/semantics/openAccess |
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(c) Serials Publications, 2017 |
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openAccess |
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application/pdf |
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Serials Publications |
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Serials Publications |
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Articles publicats en revistes (Matemàtiques i Informàtica) reponame:Dipòsit Digital de la UB instname:Universidad de Barcelona |
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Universidad de Barcelona |
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Dipòsit Digital de la UB |
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Dipòsit Digital de la UB |
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15,301629 |