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The required ansatz to construct Lie point transformations and the symmetries of a first-order stochastic differential equation
In this thesis we demonstrate how to obtain the required ansatz to determine Lie point transformations of evolution-type equations from the contact transformation approach. We indicate that the Lie point transformations of the Fokker-Planck equation (FPE), which is a second-order linear parabolic pa...
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| Format: | Thesis |
| Language: | English |
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Department of Mathematics and Applied Mathematics
2015
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| _version_ | 1867613205228945408 |
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| access_status_str | Open Access |
| author | Masike, Kakanyo Knowledge |
| author2 | Fredericks, Ebrahim |
| author_browse | Fredericks, Ebrahim Masike, Kakanyo Knowledge |
| author_facet | Fredericks, Ebrahim Masike, Kakanyo Knowledge |
| author_sort | Masike, Kakanyo Knowledge |
| collection | Thesis |
| description | In this thesis we demonstrate how to obtain the required ansatz to determine Lie point transformations of evolution-type equations from the contact transformation approach. We indicate that the Lie point transformations of the Fokker-Planck equation (FPE), which is a second-order linear parabolic partial differential equation (PDE), are projectable by using the ansatz. We further obtain the symmetries of a stochastic ordinary differential equation (SODE) which corresponds to those of the FPE. This is possible because there exists a relationship between an SODE and the associated (deterministic) FPE. The study of SODEs is an interesting and applicable concept in the real world and one of the building factors to this study is an Ito integral. These Ito integrals are of much use, for instance, in the field of mathematical finance whereby its use has shown the relationship between call options and their non-deterministic underlying stock prices. Wiener processes must be considered in finding an approximation of these integrals. Acclimatization of Sophus Lie's work to SODEs has been done by (Gaeta and Quintero [2]; Wafo Soh and Mahomed [41]; Unal [42]; Fredericks and Mahomed [43]). The determining equations for the first-order SODEs are derived in an Ito calculus context and are non-stochastic. Consequently, symmetries of an SODE are obtained without the consultation of its corresponding FPE. |
| format | Thesis |
| id | oai:open.uct.ac.za:11427/14145 |
| institution | University of Cape Town (South Africa) |
| language | eng |
| last_indexed | 2026-06-10T12:32:26.116Z |
| license_str | Not specified — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from UCTD — University of Cape Town Open Access Repository |
| publishDate | 2015 |
| publishDateRange | 2015 |
| publishDateSort | 2015 |
| publisher | Department of Mathematics and Applied Mathematics |
| publisherStr | Department of Mathematics and Applied Mathematics |
| record_format | dspace |
| source_str | UCTD — University of Cape Town Open Access Repository |
| spelling | oai:open.uct.ac.za:11427/14145 The required ansatz to construct Lie point transformations and the symmetries of a first-order stochastic differential equation Masike, Kakanyo Knowledge Fredericks, Ebrahim Ebobisse Bille, Francois Mathematics and Applied Mathematics In this thesis we demonstrate how to obtain the required ansatz to determine Lie point transformations of evolution-type equations from the contact transformation approach. We indicate that the Lie point transformations of the Fokker-Planck equation (FPE), which is a second-order linear parabolic partial differential equation (PDE), are projectable by using the ansatz. We further obtain the symmetries of a stochastic ordinary differential equation (SODE) which corresponds to those of the FPE. This is possible because there exists a relationship between an SODE and the associated (deterministic) FPE. The study of SODEs is an interesting and applicable concept in the real world and one of the building factors to this study is an Ito integral. These Ito integrals are of much use, for instance, in the field of mathematical finance whereby its use has shown the relationship between call options and their non-deterministic underlying stock prices. Wiener processes must be considered in finding an approximation of these integrals. Acclimatization of Sophus Lie's work to SODEs has been done by (Gaeta and Quintero [2]; Wafo Soh and Mahomed [41]; Unal [42]; Fredericks and Mahomed [43]). The determining equations for the first-order SODEs are derived in an Ito calculus context and are non-stochastic. Consequently, symmetries of an SODE are obtained without the consultation of its corresponding FPE. 2015-10-06T14:15:37Z 2015-10-06T14:15:37Z 2011 Master Thesis Masters MSc http://hdl.handle.net/11427/14145 eng application/pdf Department of Mathematics and Applied Mathematics Faculty of Science University of Cape Town |
| spellingShingle | Mathematics and Applied Mathematics Masike, Kakanyo Knowledge The required ansatz to construct Lie point transformations and the symmetries of a first-order stochastic differential equation |
| thesis_degree_str | Master's |
| title | The required ansatz to construct Lie point transformations and the symmetries of a first-order stochastic differential equation |
| title_full | The required ansatz to construct Lie point transformations and the symmetries of a first-order stochastic differential equation |
| title_fullStr | The required ansatz to construct Lie point transformations and the symmetries of a first-order stochastic differential equation |
| title_full_unstemmed | The required ansatz to construct Lie point transformations and the symmetries of a first-order stochastic differential equation |
| title_short | The required ansatz to construct Lie point transformations and the symmetries of a first-order stochastic differential equation |
| title_sort | required ansatz to construct lie point transformations and the symmetries of a first order stochastic differential equation |
| topic | Mathematics and Applied Mathematics |
| url | http://hdl.handle.net/11427/14145 |
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