{"id":{"repo_id":"uwo","oai_identifier":"oai:uwo.scholaris.ca:20.500.14721/33152"},"canonical_url":"https://search.dev.ndltd.org/etd/uwo/oai:uwo.scholaris.ca:20.500.14721/33152","repository":{"repo_id":"uwo","name":"Western University","base_url":"https://uwo.scholaris.ca/server/oai/request"},"display":{"title":"The Statistical Exploration in the $G$-expectation Framework: The Pseudo Simulation and Estimation of Variance Uncertainty","abstract":"The $G$-expectation framework, motivated by problems with emph{uncertainty}, is a new generalization of the classical probability framework. Similar to the Choquet expectation, the $G$-expectation can be represented as the supremum of a class of linear expectations. In the past two decades, it has developed into a complete stochastic structure connected with a large family of nonlinear PDEs. Nonetheless, to apply it to real-world problems with uncertainty, it is fundamentally necessary to build up the associated statistical methodology. This thesis explores the emph{computation, simulation, and estimation} of the $G$-normal distribution (a typical distribution with variance uncertainty) by constructing a new substructure called the emph{Semi-$G$-normal distribution} which provides the transition from classical normal to $G$-normal distribution. Interestingly, it also gives an efficient iterative scheme to stochastically solve the nonlinear emph{Black-Scholes-Barenblatt equation with volatility uncertainty}. This thesis is the theoretical and technical preparation for the future industrial application of $G$-framework.","abstract_html":"The $G$-expectation framework, motivated by problems with emph{uncertainty}, is a new generalization of the classical probability framework. Similar to the Choquet expectation, the $G$-expectation can be represented as the supremum of a class of linear expectations. In the past two decades, it has developed into a complete stochastic structure connected with a large family of nonlinear PDEs. Nonetheless, to apply it to real-world problems with uncertainty, it is fundamentally necessary to build up the associated statistical methodology. This thesis explores the emph{computation, simulation, and estimation} of the $G$-normal distribution (a typical distribution with variance uncertainty) by constructing a new substructure called the emph{Semi-$G$-normal distribution} which provides the transition from classical normal to $G$-normal distribution. Interestingly, it also gives an efficient iterative scheme to stochastically solve the nonlinear emph{Black-Scholes-Barenblatt equation with volatility uncertainty}. This thesis is the theoretical and technical preparation for the future industrial application of $G$-framework.","abstract_has_math":true,"creators":["Li, Yifan"],"institution":"The University of Western Ontario","degree_name":"M Sc","degree_level":null,"degree_discipline":"Statistics and Actuarial Sciences","degree_department":null,"school":null,"contributors":[],"advisors":["Kulperger, Reg"],"committee_chairs":[],"committee_members":[],"year":2018,"date_issued":"2018-07-27","date_published":"2018-07-27","updated_at":"2026-07-27T21:56:07Z","subjects":["Uncertainty","$G$-expectation framework","Semi-$G$-normal distribution","Sublinear expectation","Statistical theory","Black-Scholes-Barenblatt equation"],"languages":["en_ca"],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/20.500.14721/33152","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Kulperger, Reg"]},{"key":"dc:creator","label":"Author","values":["Li, Yifan"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2025-07-10T19:51:02Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2025-07-10T19:51:02Z"]},{"key":"dc:date.issued","label":"Date","values":["2018-07-27"]},{"key":"dc:publisher","label":"Institution","values":["The University of Western Ontario"]},{"key":"dc:type","label":"Dc Type","values":["thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Statistics and Actuarial Sciences"]},{"key":"thesis:degree_name","label":"Degree Name","values":["M Sc"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Uncertainty","$G$-expectation framework","Semi-$G$-normal distribution","Sublinear expectation","Statistical theory","Black-Scholes-Barenblatt equation"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en_ca"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/20.500.14721/33152"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["The thesis cover page in the PDF document includes references to Western University’s previous institutional repository platform, known as Scholarship@Western, and links to that platform (beginning with ir.lib.uwo.ca). In citing or referring to this thesis, use the DOI or handle from this page instead. Sample citation: Author name, \"Thesis title.\" (Year). Western University Open Repository. https://doi.org/10.71858/123456."]},{"key":"dc:description.abstract","label":"Abstract","values":["The $G$-expectation framework, motivated by problems with emph{uncertainty}, is a new generalization of the classical probability framework. Similar to the Choquet expectation, the $G$-expectation can be represented as the supremum of a class of linear expectations. In the past two decades, it has developed into a complete stochastic structure connected with a large family of nonlinear PDEs. Nonetheless, to apply it to real-world problems with uncertainty, it is fundamentally necessary to build up the associated statistical methodology. This thesis explores the emph{computation, simulation, and estimation} of the $G$-normal distribution (a typical distribution with variance uncertainty) by constructing a new substructure called the emph{Semi-$G$-normal distribution} which provides the transition from classical normal to $G$-normal distribution. Interestingly, it also gives an efficient iterative scheme to stochastically solve the nonlinear emph{Black-Scholes-Barenblatt equation with volatility uncertainty}. This thesis is the theoretical and technical preparation for the future industrial application of $G$-framework."]},{"key":"dc:title","label":"Title","values":["The Statistical Exploration in the $G$-expectation Framework: The Pseudo Simulation and Estimation of Variance Uncertainty"]}]}],"canonical_facts":{"dc:contributor.advisor":["Kulperger, Reg"],"dc:creator":["Li, Yifan"],"dc:date.accessioned":["2025-07-10T19:51:02Z"],"dc:date.available":["2025-07-10T19:51:02Z"],"dc:date.issued":["2018-07-27"],"dc:description":["The thesis cover page in the PDF document includes references to Western University’s previous institutional repository platform, known as Scholarship@Western, and links to that platform (beginning with ir.lib.uwo.ca). In citing or referring to this thesis, use the DOI or handle from this page instead. Sample citation: Author name, \"Thesis title.\" (Year). Western University Open Repository. https://doi.org/10.71858/123456."],"dc:description.abstract":["The $G$-expectation framework, motivated by problems with emph{uncertainty}, is a new generalization of the classical probability framework. Similar to the Choquet expectation, the $G$-expectation can be represented as the supremum of a class of linear expectations. In the past two decades, it has developed into a complete stochastic structure connected with a large family of nonlinear PDEs. Nonetheless, to apply it to real-world problems with uncertainty, it is fundamentally necessary to build up the associated statistical methodology. This thesis explores the emph{computation, simulation, and estimation} of the $G$-normal distribution (a typical distribution with variance uncertainty) by constructing a new substructure called the emph{Semi-$G$-normal distribution} which provides the transition from classical normal to $G$-normal distribution. Interestingly, it also gives an efficient iterative scheme to stochastically solve the nonlinear emph{Black-Scholes-Barenblatt equation with volatility uncertainty}. This thesis is the theoretical and technical preparation for the future industrial application of $G$-framework."],"dc:identifier.uri":["https://hdl.handle.net/20.500.14721/33152"],"dc:language.iso":["en_ca"],"dc:publisher":["The University of Western Ontario"],"dc:subject":["Uncertainty","$G$-expectation framework","Semi-$G$-normal distribution","Sublinear expectation","Statistical theory","Black-Scholes-Barenblatt equation"],"dc:title":["The Statistical Exploration in the $G$-expectation Framework: The Pseudo Simulation and Estimation of Variance Uncertainty"],"dc:type":["thesis"],"thesis:degree_discipline":["Statistics and Actuarial Sciences"],"thesis:degree_name":["M Sc"]},"updated_at":"2026-07-27T21:56:07Z"}