{"id":{"repo_id":"cape-town","oai_identifier":"oai:open.uct.ac.za:11427/14145"},"canonical_url":"https://search.dev.ndltd.org/etd/cape-town/oai:open.uct.ac.za:11427/14145","repository":{"repo_id":"cape-town","name":"University of Cape Town","base_url":"https://open.uct.ac.za/oai/request"},"display":{"title":"The required ansatz to construct Lie point transformations and the symmetries of a first-order stochastic differential equation","abstract":"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.","abstract_html":"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.","abstract_has_math":false,"creators":["Masike, Kakanyo Knowledge"],"institution":"Department of Mathematics and Applied Mathematics","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Fredericks, Ebrahim","Ebobisse Bille, Francois"],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011","date_published":"2011","updated_at":"2026-07-22T22:22:50Z","subjects":[],"languages":["eng"],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/11427/14145","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Fredericks, Ebrahim","Ebobisse Bille, Francois"]},{"key":"dc:creator","label":"Author","values":["Masike, Kakanyo Knowledge"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2015-10-06T14:15:37Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2015-10-06T14:15:37Z"]},{"key":"dc:date.issued","label":"Date","values":["2011"]},{"key":"dc:publisher.department","label":"Dc Publisher Department","values":["Department of Mathematics and Applied Mathematics"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["University of Cape Town"]},{"key":"dc:type","label":"Dc Type","values":["Master Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["Masters"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["MSc"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["eng"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/11427/14145"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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."]},{"key":"dc:title","label":"Title","values":["The required ansatz to construct Lie point transformations and the symmetries of a first-order stochastic differential equation"]}]}],"canonical_facts":{"dc:contributor.advisor":["Fredericks, Ebrahim","Ebobisse Bille, Francois"],"dc:creator":["Masike, Kakanyo Knowledge"],"dc:date.accessioned":["2015-10-06T14:15:37Z"],"dc:date.available":["2015-10-06T14:15:37Z"],"dc:date.issued":["2011"],"dc:description.abstract":["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."],"dc:identifier.uri":["http://hdl.handle.net/11427/14145"],"dc:language.iso":["eng"],"dc:publisher.department":["Department of Mathematics and Applied Mathematics"],"dc:publisher.institution":["University of Cape Town"],"dc:title":["The required ansatz to construct Lie point transformations and the symmetries of a first-order stochastic differential equation"],"dc:type":["Master Thesis"],"dc:type.qualificationlevel":["Masters"],"dc:type.qualificationname":["MSc"]},"updated_at":"2026-07-22T22:22:50Z"}