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[{"key": "dc.contributor.advisor", "value": "Parviainen, Mikko", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Taipalus, Janne", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2023-08-04T05:43:44Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2023-08-04T05:43:44Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2023", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/88498", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4ss\u00e4 tutkielmassa tutustumme k\u00f6ydenvetopeliin satunnaiskohinalla. Kyseinen peli on kahden pelaajan stokastinen peli, jossa kukin pelaaja yritt\u00e4\u00e4 saavuttaa alueen reunan sellaisesta kohdasta, joka on h\u00e4nelle edullinen. Pelin lopuksi toinen pelaaja maksaa toiselle pelaajalle sen verran rahaa, kuin alueen reunalla m\u00e4\u00e4ritelty funktio siin\u00e4 kohdassa m\u00e4\u00e4r\u00e4\u00e4.\n\nJokaisella kierroksella on t\u00e4ysin sattumasta kiinni, kumman pelaajan vuoro on p\u00e4\u00e4tt\u00e4\u00e4 minne liikutaan ja lis\u00e4ksi t\u00e4ss\u00e4 on mukana satunnaiskohina, joka voi vied\u00e4 pelaajat satunnaiseen pisteeseen. \nPelaajien ottamien askelten pituus on oltava pienemp\u00e4\u00e4 kuin \\(\\varepsilon > 0\\) ja samoin satunnaisen pisteen et\u00e4isyys nykyisest\u00e4 pelitilanteesta on oltava pienemp\u00e4\u00e4 kuin \\(\\varepsilon\\).\n\nKyseisen k\u00f6ydenvetopelin avulla m\u00e4\u00e4rittelemme \\((p,\\varepsilon)\\)-harmoniset funktiot jotka suppenevat \\(p\\)-harmoniseen funktioon, kun pelin askelpituus \\(\\varepsilon\\) l\u00e4hestyy nollaa. Todistaaksemme t\u00e4m\u00e4n suppenemisen meid\u00e4n pit\u00e4\u00e4 osoittaa s\u00e4\u00e4nn\u00f6llisyys lokaalisti ja alueen reunalla. Tutkielmassa annamme peliteoreettiset todistukset n\u00e4ille s\u00e4\u00e4nn\u00f6llisyystuloksille. \n\nTutkielman alussa k\u00e4ymme l\u00e4pi osittaisdifferentiaaliyht\u00e4l\u00f6iden ja stokastiikan osalta tarvittavia k\u00e4sitteit\u00e4 ja tuloksia. Tutkielmassa k\u00e4sittelemme peliteoreettista (tunnetaan my\u00f6s normalisoituna) \\(p\\)-Laplacen yht\u00e4l\u00f6\u00e4. M\u00e4\u00e4ritell\u00e4ksemme t\u00e4m\u00e4n tutustumme ensin Laplacen yht\u00e4l\u00f6\u00f6n ja \\(\\infty\\)-Laplacen yht\u00e4l\u00f6\u00f6n. K\u00e4sittelemme t\u00e4ss\u00e4 tutkielmassa viskositeettiratkaisuja ja k\u00e4y ilmi, ett\u00e4 peliteoreettisen \\(p\\)-Laplacen yht\u00e4l\u00f6n ja \\(p\\)-Laplacen yht\u00e4l\u00f6n viskositeettiratkaisut ovat samat. Joten voimme tutkia \\(p\\)-Laplacen yht\u00e4l\u00f6n ratkaisuja tutkimalla peliteoreettisen \\(p\\)-Laplacen yht\u00e4l\u00f6n ratkaisuja.\n\nStokastiikasta tarvitsemme erityisesti pys\u00e4htymisaikaa ja valinnaisen pys\u00e4htymisen lausetta. N\u00e4it\u00e4 varten tutustumme moniin stokastiikan perusk\u00e4sitteisiin ja er\u00e4isiin stokastisiin prosesseihin, joita kutsutaan martingaaleiksi.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2023-08-04T05:43:44Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2023-08-04T05:43:44Z (GMT). No. of bitstreams: 0\n Previous issue date: 2023", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "60", "language": "", "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.language.iso", "value": "fin", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": null, "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.subject.other", "value": "p-Laplacen yht\u00e4l\u00f6", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "k\u00f6ydenvetopeli satunnaiskohinalla", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "viskositeettiratkaisut", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "p-harmoniset funktiot", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "K\u00f6ydenvetopeli satunnaiskohinalla ja p-Laplacen yht\u00e4l\u00f6", "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-202308044613", "language": "", "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master\u2019s thesis", "language": "en", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu -tutkielma", "language": "fi", "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": "Matematiikka", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Mathematics", "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.copyright", "value": "\u00a9 The Author(s)", "language": null, "element": "rights", "qualifier": "copyright", "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": "matematiikka", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "yht\u00e4l\u00f6t", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "osittaisdifferentiaaliyht\u00e4l\u00f6t", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "funktiot", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "peliteoria", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}]
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