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[{"key": "dc.contributor.advisor", "value": "Geiss, Stefan", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.advisor", "value": "Laukkarinen, Eija", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Nyk\u00e4nen, Jani", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2020-01-24T09:56:30Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2020-01-24T09:56:30Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2020", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/67506", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4ss\u00e4 tutkielmassa tutustutaan McKeanin-Vlasovin stokastisiin differentiaaliyht\u00e4l\u00f6ihin, jotka yleist\u00e4v\u00e4t tavalliset stokastiset differentiaaliyht\u00e4l\u00f6t lis\u00e4\u00e4m\u00e4ll\u00e4 kerroinfunktioihin riippuvuuden tuntemattoman prosessin jakaumasta tietyll\u00e4 ajanhetkell\u00e4. P\u00e4\u00e4asiallisena l\u00e4hteen\u00e4 seurataan K. Bahlalin, M. Mezerdin ja B. Mezerdin artikkelia \\textit{Stability of Mckean-Vlasov stochastic differential equations and applications}.\n\nTutkielmassa k\u00e4yd\u00e4\u00e4n l\u00e4pi tarvittavia esitietoja todenn\u00e4k\u00f6isyysteoriasta ja tavallisista stokastisista differentiaaliyht\u00e4l\u00f6ist\u00e4. Kerroinfunktioiden jatkuvuuden ja mitallisuuden m\u00e4\u00e4ritt\u00e4miseksi esitell\u00e4\u00e4n Wassersteinin et\u00e4isyys, joka on metriikka \u00e4\u00e4rellismomenttisten reaaliavaruuden todenn\u00e4k\u00f6isyysmittojen avaruudessa. Metriikan avulla saadaan yleistetty\u00e4 lause, joka takaa ratkaisun olemassaolon ja yksik\u00e4sitteisyyden, kun kerroinfunktiot ovat Lipschitz-jatkuvia ja toteuttavat lineaarisen kasvuehdon. Lis\u00e4ksi osoitetaan, ett\u00e4 yksik\u00e4sitteisyys on voimassa er\u00e4\u00e4ll\u00e4 Lipschitz-jatkuvuutta heikommalla ehdolla.\n\nNumeerisessa ratkaisemisessa voidaan hy\u00f6dynt\u00e4\u00e4 tulosta, jossa konstruoidaan iteroitu jono prosesseja, jotka suppenevat kohti yksik\u00e4sitteist\u00e4 ratkaisua. Lopuksi tarkastellaan ratkaisuprosessien stabiiliutta erikseen alkuarvon, kerroinfunktioiden ja integroivan prosessin suhteen.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.abstract", "value": "In this thesis we introduce McKean-Vlasov stochastic differential equations, which are a generalization of ordinary stochastic differential equations, but now the coefficients depend on the distribution of the unknown process. In our main results we follow K. Bahlali, M. Mezerdi and B. Mezerdi's article \\textit{Stability of Mckean-Vlasov stochastic differential equations and applications}.\n\nWe start by giving preliminary theory required to understand our main results. To define continuity and measurability of the coefficient functions, we introduce the Wasserstein distance, which is a metric in the space of probability measures on the real line with finite moments. With the metric we generalize a theorem that states that a unique solution exists provided that the coefficients are Lipschitz continuous and satisfy the linear growth condition. In addition we show that in a specific case the uniqueness holds even if the coefficients satisfy a condition weaker than Lipschitz continuity.\n\nIn numerics one can use a result that provides a way to approximate the solution with a sequence of iterated processes converging to the unique solution. In the last part we consider stability of the solution with respect to the initial value, the coefficients and the driving process.", "language": "en", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Miia Hakanen (mihakane@jyu.fi) on 2020-01-24T09:56:30Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2020-01-24T09:56:30Z (GMT). No. of bitstreams: 0\n Previous issue date: 2020", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "79", "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": "stochastic differential equations", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "probability theory", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "stochastic calculus", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "On the uniqueness of a solution and stability of McKean-Vlasov 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-202001241782", "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": "stokastiset prosessit", "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": "stochastic processes", "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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