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[{"key": "dc.contributor.advisor", "value": "Geiss, Christel", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Kotkajuuri, Jimi", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2023-08-02T09:00:20Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2023-08-02T09:00:20Z", "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/123456789/88485", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4m\u00e4n tutkielman tarkoituksena on tarkastella er\u00e4st\u00e4 versiota stokastisen integroinnin avaintuloksesta nimelt\u00e4\u00e4n It\u00f4n kaava, jolla on t\u00e4rke\u00e4 rooli niin stokastiikan teorian kuin sen erin\u00e4isten sovellusten kannalta. It\u00f4n kaavoja voidaan johtaa perustuen useille eri oletuksille sek\u00e4 tilanteille. T\u00e4ss\u00e4 tutkimuksessa oletamme p\u00e4\u00e4tuloksena esittett\u00e4v\u00e4n It\u00f4n kaavassa k\u00e4ytett\u00e4v\u00e4n stokastisen prosessin olevan L\u00e9vy-prosessi, joka toteuttaa rajallisen vaihtelun ehdon ja vastaavasti kaavassa k\u00e4ytett\u00e4v\u00e4n funtion oletamme jatkuvaksi ja heikosti derivoituvaksi.\n\nTulemme k\u00e4sittelem\u00e4\u00e4n oleellisimmat stokastiikan sek\u00e4 analyysin esitiedot p\u00e4\u00e4tuloksena olevaa It\u00f4n kaavan todistamista varten. Stokastisten prosessien osalta k\u00e4sittelemme yleisimpi\u00e4 esimerkkej\u00e4 L\u00e9vy-prosesseista sek\u00e4 esittelemme niiden t\u00e4rkeimpi\u00e4 perusominaisuuksia. M\u00e4\u00e4rittelemme my\u00f6s Poisson satunnaismitan, jonka t\u00e4rke\u00e4n\u00e4 erikoistapauksena on muun muuassa hyppymitta. Lis\u00e4ksi esittelemme joitain kuuluisia stokastiikan tuloksia kuten L\u00e9vy-It\u00f4-hajotelma sek\u00e4 L\u00e9vy-Khintchine-kaava.\n\nLis\u00e4ksi t\u00e4rke\u00e4n\u00e4 osana It\u00f4n kaavaa m\u00e4\u00e4rittelemme ja konstruoimme tarkasti stokastisen integraalin alkaen yksinkertaisista prosesseista ja lopulta yleist\u00e4en sen koskemaan laajempaa osaa prosesseista. Jatkona stokastiseen integrointiin tarkastelemme viel\u00e4 l\u00e4hemmin er\u00e4st\u00e4 stokastisen integraalin laajennusta Poisson satunnaismitan suhteen. Lopuksi esittelemme ja todistamme er\u00e4\u00e4n version It\u00f4n kaavasta, joka k\u00e4ytt\u00e4\u00e4 oletuksinaan rajallisen vaihtelun ehdon toteuttavaa prosessia, mutta p\u00e4\u00e4tuloksesta poiketen olettaa funktioiden olevan heikosti derivoituvuuden sijaan ainoastaan jatkuvasti differentioituvia.\n\nJohtuen erityisesti heikosti derivoituvuuden oletuksesta k\u00e4ymme lis\u00e4ksi l\u00e4pi joitain reaali- ja funktionaalianalyysin perustuloksia. Erityisen\u00e4 huomion kohteena ovat tulokset koskien distribuutioteoriaa ja heikkoa derivoituvuutta. Lopuksi n\u00e4it\u00e4 esitietoja k\u00e4ytt\u00e4en ja oletukset tarkasti m\u00e4\u00e4ritellen todistamme yksityiskohtaisesti tutkielman p\u00e4\u00e4tuloksena olevan version It\u00f4n kaavasta tapauksessa, jossa dimensio on 1.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.abstract", "value": "In this thesis we examine a version of the integral result of stochastic integration called It\u00f4's formula which plays an important role both in terms of theory of stochastic and also its various applications. It\u00f4's formulas can be derived based on several different circumstances and situations. In this thesis, we assume that the stochastic process used in It\u00f4's formula presented as the main result is a L\u00e9vy process, which fulfills the condition of finite variation, and in addition to this we assume the function used in the formula to be continuous and weakly differentiable.\n\nWe will introduce the most essential stochastic and analysis prerequisites for proving the It\u00f4 formula as the main result. Regarding stochastic processes, we discuss the most common examples of L\u00e9vy processes and present their most important basic properties. We also define the Poisson random measure, whose important special case is jump measure. In addition, we present some famous stochastic results such as the L\u00e9vy-It\u00f4 decomposition and the L\u00e9vy-Khintchine formula.\n\nFurthermore, as an important part of It\u00f4's formula, we precisely define and construct the stochastic integral starting from simple processes and finally generalizing it into a wider range of processes. As a continuation of stochastic integration, we will take a closer look at an extension of the stochastic integral in terms of the Poisson random measure. Finally, we present and prove a version of It\u00f4's formula, which uses as its assumptions a process fulfilling the finite variation condition, but, in contrast to the main result, assumes that the functions are weakly differentiable instead of only continuously differentiable.\n\nDue to the assumption of weak differentiability, we also review some of the basic results of real and functional analysis. Particularly important for the main result are the results regarding distribution theory and weak differentiability. Finally, using this preliminary information and precisely specifying the assumptions, we prove in detail the version of It\u00f4's formula, which is the main result of the thesis, in the case where the dimension is 1.", "language": "en", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2023-08-02T09:00:20Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2023-08-02T09:00:20Z (GMT). No. of bitstreams: 0\n Previous issue date: 2023", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "54", "language": "", "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.language.iso", "value": "eng", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": null, "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.title", "value": "It\u00f4\u2019s formula for finite variation L\u00e9vy processes", "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-202308024600", "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": "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.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": "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": "todenn\u00e4k\u00f6isyyslaskenta", "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.subject.yso", "value": "probability calculation", "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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