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Statistical study of the stochastic area
Thesis (MSc)--Stellenbosch University, 2023.
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| Format: | Thesis |
| Language: | en_ZA |
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Stellenbosch : Stellenbosch University
2023
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| _version_ | 1867613901661667328 |
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| access_status_str | Open Access |
| author | Mnyulwa, Thamu |
| author2 | Touchette, Hugo |
| author_browse | Mnyulwa, Thamu Touchette, Hugo |
| author_facet | Touchette, Hugo Mnyulwa, Thamu |
| author_sort | Mnyulwa, Thamu |
| collection | Thesis |
| dc_rights_str_mv | Stellenbosch University |
| description | Thesis (MSc)--Stellenbosch University, 2023. |
| format | Thesis |
| id | oai:scholar.sun.ac.za:10019.1/127121 |
| institution | Stellenbosch University (South Africa) |
| language | en_ZA |
| last_indexed | 2026-06-10T12:43:30.888Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from SUNScholar — Stellenbosch University Repository |
| publishDate | 2023 |
| publishDateRange | 2023 |
| publishDateSort | 2023 |
| publisher | Stellenbosch : Stellenbosch University |
| publisherStr | Stellenbosch : Stellenbosch University |
| record_format | dspace |
| source_str | SUNScholar — Stellenbosch University Repository |
| spelling | oai:scholar.sun.ac.za:10019.1/127121 Statistical study of the stochastic area Mnyulwa, Thamu Touchette, Hugo Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Stochastic analysis Brownian motion processes Stochastic processes -- Mathematical models Stochastic differential equations UCTD Thesis (MSc)--Stellenbosch University, 2023. ENGLISH SUMMARY: We study in this thesis the properties of a stochastic line integral, known as the stochastic area or area loop, which measures the amount of area enclosed or covered by the trajectory of a stochastic differential equation ( SDE). This quantity was studied by Paul Lévy in the 1950s for Brownian motion in two dimensions and was rediscovered recently in physics in the context of linear SDEs modelling nonequilibrium systems. In this thesis, we review these works and re-derive in particular an explicit formula for linear SDEs for the long-time expectation of the stochastic area. We then study the definition of this quantity to show that its value does not depend on the definition of the stochastic integral or stochastic calculus used. Finally, we study the convergence and statistics of an estimator of the stochastic area in simulation for linear SDEs. AFRIKAANSE OPSOMMING: Ons bestudeer in hierdie tesis die eienskappe van ’n stogastiese lynintegraal, bekend as die stogastiese area of area lus, wat die hoeveelheid area meet wat ingesluit of gedek word deur die trajektorie van ’n stogastiese differensiaalvergelyking (SDE). Hierdie hoeveelheid is in die 1950’s deur Paul Lévy bestudeer vir Brown se beweging in twee dimensies en is onlangs in fisikah rontdek in die konteks van lineere SDE’s wat nie-ewewigstelsels modelleer. In hierdie tesis hersien ons hierdie werk en herlei ons veral ’n eksplisiete formule vir lineere SDE’s vir die lang tyd verwagting van die stogastiese area. Ons bestudeer dan die definisie van hierdie kwantiteit om te wys dat die waarde daarvan nie afhang van die definisie van die stogastiese integraal, of stogastiese calculus wat gebruik word nie. Laastens bestudeer ons die konvergensie en statistieke van ’n beramer van die stogastiese area in simulasie vir lineere SDE’s. Masters 2023-02-24T13:45:09Z 2023-05-18T07:05:20Z 2023-02-24T13:45:09Z 2023-05-18T07:05:20Z 2023-03 Thesis http://hdl.handle.net/10019.1/127121 en_ZA Stellenbosch University viii, 62 pages application/pdf Stellenbosch : Stellenbosch University |
| spellingShingle | Stochastic analysis Brownian motion processes Stochastic processes -- Mathematical models Stochastic differential equations UCTD Mnyulwa, Thamu Statistical study of the stochastic area |
| title | Statistical study of the stochastic area |
| title_full | Statistical study of the stochastic area |
| title_fullStr | Statistical study of the stochastic area |
| title_full_unstemmed | Statistical study of the stochastic area |
| title_short | Statistical study of the stochastic area |
| title_sort | statistical study of the stochastic area |
| topic | Stochastic analysis Brownian motion processes Stochastic processes -- Mathematical models Stochastic differential equations UCTD |
| url | http://hdl.handle.net/10019.1/127121 |
| work_keys_str_mv | AT mnyulwathamu statisticalstudyofthestochasticarea |