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[{"key": "dc.contributor.advisor", "value": "Geiss, Stefan", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Koivu, Jesse", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2021-12-09T07:05:07Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2021-12-09T07:05:07Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2021", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/78919", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4ss\u00e4 tutkielmassa k\u00e4sittelemme odotusarvokent\u00e4llisi\u00e4 stokastisia differentiaaliyht\u00e4l\u00f6it\u00e4, mitk\u00e4 ovat yleistys klassisille stokastisille differentiaaliyht\u00e4l\u00f6ille. Odotusarvokent\u00e4llisen stokastisen differentiaaliyht\u00e4l\u00f6n kerroinfunktiot saattavat riippua ylim\u00e4\u00e4r\u00e4isest\u00e4 mittakomponentista ratkaisun jakauman muodossa. K\u00e4sittelemme heikkojen ratkaisujen olemassaoloa t\u00e4llaisille yht\u00e4l\u00f6ille olettaen, ett\u00e4 kerroinfunktiot ovat rajoitettuja ja jatkuvia, miss\u00e4 jatkuvuus mittakomponentin suhteen ymm\u00e4rret\u00e4\u00e4n jatkuvuutena 2-Wasserstein metriikan suhteen. Seuraamme artikkelia Li, J. ja Min, H. Weak solutions of mean-field stochastic differential equations (2017). Aloitamme palauttamalla mieliimme joitakin keskeisi\u00e4 k\u00e4sitteit\u00e4 stokastisesta analyysist\u00e4. T\u00e4m\u00e4n j\u00e4lkeen esittelemme polkuavaruuden, klassisen lokaalin martingaaliongelman ja funktionaaliset stokastiset differentiaaliyht\u00e4l\u00f6t. Lis\u00e4ksi esittelemme Wassersteinin mittojen avaruudet ja funktioiden differentioituvuuden mittakomponentin suhteen. Lopuksi osoitamme heikkojen ratkaisujen olemassaolon odotusarvokent\u00e4llisille stokastisille differentiaaliyht\u00e4l\u00f6ille olettaen, ett\u00e4 kerroinfunktiot ovat rajoitettuja, mitallisia ja jatkuvia. T\u00e4m\u00e4 tehd\u00e4\u00e4n n\u00e4ytt\u00e4m\u00e4ll\u00e4, ett\u00e4 vastaavalla lokaalilla martingaaliongelmalla on olemassa ratkaisu.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.abstract", "value": "In this thesis we consider mean-field stochastic differential equations, which are an extension of classical stochastic differential equations, where the coefficients may depend on an additional measure component in the law of the solution. We consider the existence of weak solutions of such equations under the assumption that the coefficients are bounded and continuous, where the continuity is understood in the 2-Wasserstein metric in the measure component. We follow the treatment given in the article of Li, J. and Min, H., Weak solutions of mean-field stochastic differential equations (2017). We start by recalling some fundamental notions from stochastic analysis. Then we introduce the path space, along with the classical local martingale problem and functional stochastic differential equations. Furthermore we introduce the Wasserstein spaces of measures and how to differentiate functions depending on a measure variable. Finally we show the existence of weak solutions to mean-field stochastic differential equations under bounded, measurable and continuous coefficients by showing that there exists a solution to the corresponding local martingale problem.", "language": "en", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2021-12-09T07:05:07Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2021-12-09T07:05:07Z (GMT). No. of bitstreams: 0\n Previous issue date: 2021", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "43", "language": "", "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.format.mimetype", "value": "application/pdf", "language": null, "element": "format", "qualifier": "mimetype", "schema": "dc"}, {"key": "dc.language.iso", "value": "eng", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": "en", "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.subject.other", "value": "stochastics", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "stochastic differential equations", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "mean-field", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Existence of weak solutions of mean-field stochastic differential equations", "language": "", "element": "title", "qualifier": null, "schema": "dc"}, {"key": "dc.type", "value": "master thesis", "language": null, "element": "type", "qualifier": null, "schema": "dc"}, {"key": "dc.identifier.urn", "value": "URN:NBN:fi:jyu-202112095908", "language": "", "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu -tutkielma", "language": "fi", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master\u2019s thesis", "language": "en", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Matemaattis-luonnontieteellinen tiedekunta", "language": "fi", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Faculty of Sciences", "language": "en", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Matematiikan ja tilastotieteen laitos", "language": "fi", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Department of Mathematics and Statistics", "language": "en", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "Jyv\u00e4skyl\u00e4n yliopisto", "language": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": "en", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Stokastiikka ja todenn\u00e4k\u00f6isyysteoria", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Stochastics and Probability", "language": "en", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "yvv.contractresearch.funding", "value": "0", "language": "", "element": "contractresearch", "qualifier": "funding", "schema": "yvv"}, {"key": "dc.type.coar", "value": "http://purl.org/coar/resource_type/c_bdcc", "language": null, "element": "type", "qualifier": "coar", "schema": "dc"}, {"key": "dc.rights.accesslevel", "value": "openAccess", "language": null, "element": "rights", "qualifier": "accesslevel", "schema": "dc"}, {"key": "dc.type.publication", "value": "masterThesis", "language": null, "element": "type", "qualifier": "publication", "schema": "dc"}, {"key": "dc.subject.oppiainekoodi", "value": "4041", "language": "", "element": "subject", "qualifier": "oppiainekoodi", "schema": "dc"}, {"key": "dc.subject.yso", "value": "differentiaaliyht\u00e4l\u00f6t", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "matematiikka", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "differential equations", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "mathematics", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.format.content", "value": "fulltext", "language": null, "element": "format", "qualifier": "content", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}, {"key": "dc.type.okm", "value": "G2", "language": null, "element": "type", "qualifier": "okm", "schema": "dc"}]
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