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[{"key": "dc.contributor.advisor", "value": "Geiss, Stefan", "language": null, "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Luuri, Eetu", "language": null, "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2024-02-26T06:29:09Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2024-02-26T06:29:09Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2024", "language": null, "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/93650", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4m\u00e4n tutkielman aiheena ovat erilaiset kaoottiset hajotelmat L\u00e9vy prosessien funktionaaleille. N\u00e4ill\u00e4 hajotelmilla pyrit\u00e4\u00e4n esitt\u00e4m\u00e4\u00e4n kyseiset funktionaalit iteroitujen integraalien summana tietyn, kesken\u00e4\u00e4n ortogonaalisten martingaalien joukon suhteen.\n\nEnsimm\u00e4isen\u00e4 k\u00e4ymme l\u00e4pi hieman teoriaa, jonka pohjalle my\u00f6hemmin tutkielmassa esiintyv\u00e4t hajotelmat pohjautuvat. Esittelemme joukon m\u00e4\u00e4ritelmi\u00e4, jotka ovat tarpeen t\u00e4ss\u00e4 tutkielmassa esiintyv\u00e4n teorian ymm\u00e4rt\u00e4miseksi. N\u00e4ihin m\u00e4\u00e4ritelmiin lukeutuu muun muassa L\u00e9vy processit, martingaalit ja stokastiset integraalit. Lis\u00e4ksi esittelemme my\u00f6hemmiss\u00e4 todistuksissa tarvittavia ep\u00e4yht\u00e4l\u00f6it\u00e4, lemmoja ja lauseita.\n\nKun olemme k\u00e4sitelleet tarvittavat esitiedot, siirrymme kohti tutkielman keskeisint\u00e4 lausetta. T\u00e4t\u00e4 lausetta varten esittelemme niin kutsutut Teugelin martingaalit. N\u00e4m\u00e4 martingaalit ovat k\u00e4yt\u00e4nn\u00f6ss\u00e4 L\u00e9vy prosessin kompensoituja hyppyprosesseja. N\u00e4ist\u00e4 Teugelin martingaaleista muodostamme kesken\u00e4\u00e4n ortogonaalisen joukon, jota k\u00e4yt\u00e4mme kaoottisen hajotelman m\u00e4\u00e4rittelemiseen. T\u00e4m\u00e4 teoria ja kaoottinen hajotelma pohjautuvat David Nualartin ja Win Schoutensin artikkeliin Chaotic and predictable representations for L\u00e9vy processes. K\u00e4yt\u00e4mme t\u00e4t\u00e4 tutkielmamme keskeisimp\u00e4n\u00e4 l\u00e4hteen\u00e4, jossa esiintyvi\u00e4 lauseita ja todistuksia tutkimme yksityiskohtaisemmin. Lis\u00e4ksi esittelemme ja k\u00e4sittelemme muita kirjallisuudessa esiintyvi\u00e4 kaoottisia hajotelmia.\n\nYksi n\u00e4ist\u00e4 hajotelmista on Kyoshi It\u00f4n ortogonaalinen hajotelma, jonka h\u00e4n esitteli artikkelissaan Spectral Type of the Shift Transformation of Differential Processes With stationary increments. T\u00e4m\u00e4 lause hy\u00f6dynt\u00e4\u00e4 Wiener integraaleja L\u00e9vy prosessin avulla m\u00e4\u00e4ritellyn kahdesti integroituvien satunnaismuuttujien avaruuden ortogonaalisen hajotelman m\u00e4\u00e4rittelyss\u00e4. T\u00e4m\u00e4n hajotelman todistuksen k\u00e4ymme l\u00e4pi ykityiskohtaisesti, jonka j\u00e4lkeen hy\u00f6dynn\u00e4mme sit\u00e4 toisen hajotelman todistamiseen.\n\nLopuksi esittelemme viel\u00e4 hieman yleisemp\u00e4\u00e4n tapaukseen soveltuvan hajotelman. Paolo Di Tellan ja Haus-Juergen Engelbertin, artikkelissa The Chaotic Representation of Compensated-Covariation Stable Families of Martingales, esittelem\u00e4 hajotelma sopeutuu funktionaalejen esitt\u00e4miseen iteroitujen Wiener integraalien avulla suhteessa ortogonaaliseen ja kompensoidun kovarianssin suhteen vakaiden martingaalien joukkoon.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.abstract", "value": "In the present thesis, we will study the chaotic representation properties for functionals on L\u00e9vy processes. These chaotic representation properties are a way to represent square integrable random variables as a sum of iterated integrals with respect to a certain set of orthogonal martingales.\n\nWe will first go over the basic settings and some preliminary theory we need in order to understand L\u00e9vy processes, martingale theory, stochastic integrals and the chaotic representation properties following later in the thesis. These preliminaries include some inequalities, lemmas and theorems used in the proofs of this thesis as well as the basic definitions.\n\nThe main result of this thesis characterizes a chaotic representation property using a pairwise strongly orthogonal family of so-called Teugels martingales. These Teugels martingales are, in fact, the compensated power jump processes of a L\u00e9vy process. This theorem covering the chaotic representation property for Teugels martingales was explored by David Nualart and Wim Schoutens in their article Chaotic and predictable representations for L\u00e9vy processes. We use this article as our main source for this thesis and expand upon it by providing more details and exploring alternative versions of chaotic representation properties found in the literature.\n\nOne of the chaotic representation properties we examine and prove in detail after our main theorem is It\u00f4's orthogonal decomposition introduced in Spectral Type of the Shift Transformation of Differential Processes With stationary increments by Kyoshi It\u00f4. This theorem uses multiple Wiener integrals to define an orthogonal decomposition of the space of square integrable random variables. After the proof, we use this theorem to formulate another, different orthogonal decomposition.\n\nFinally we conclude our thesis by going over a more general decomposition. This chaotic representation property uses iterated integrals with respect to a family of compensated-covariance stable martingales. This property has been covered by Paolo Di Tella and Hans-Juergen Engelbert in The Chaotic Representation of Compensated-Covariation Stable Families of Martingales.", "language": "en", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2024-02-26T06:29:09Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2024-02-26T06:29:09Z (GMT). No. of bitstreams: 0\n Previous issue date: 2024", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "46", "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": "L\u00e9vy processes", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "chaotic decomposition", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "chaotic representation property", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "stochastic integrals", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Teugels martingales", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Chaotic decompositions of the L\u00e9vy-It\u00f4 space", "language": null, "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-202402262115", "language": null, "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": 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