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Author:Tikanmäki, Johanna
Title:Stokastinen Pettisin integrointi Banachin avaruuksissa
Publication type:Master's thesis
Publication year:2004
Pages:60      Language:   fin
Department/School:Teknillisen fysiikan ja matematiikan osasto
Main subject:Matematiikka   (Mat-1)
Supervisor:Londen, Stig-Olof
Instructor:
OEVS:
Electronic archive copy is available via Aalto Thesis Database.
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Location:P1 Ark TF80     | Archive
Keywords:stochastic integration
Pettis integral
Banach space
Hilbert space
stokoastinen integrointi
Pettisin integraali
Banachin avaruus
Hilbertin avaruus
Abstract (eng): This thesis is about stochastic integrals.
We start by defining the integral with respect to Brownian motion on the real axis.
After that we continue through n-dimensional real spaces and separable Hilbert spaces to Banach spaces and integrals of Banach space valued operators.

We deal basically with the integrals and the definitions associated with it.
The actual computation of stochastic integrals is not considered.

The treatment is based on numerous results.
We introduce nuclear and Hilbert-Schmidt operators, Wiener processes and cylindrical Wiener processes, reproducing kernel Hilbert spaces, gamma-radonifying operators and, last but not least, Pettis integration.

Stochastic integration theory in finite dimensional real spaces is presented in a multitude of books.
About ten years ago Da Prato and Zabczyk [DPZ92] published a comprehensive monograph on stochastic equations in Hilbert spaces.
The treatment of Banach space theory in this thesis is based on the recent works of van Neerven and Weis [NW03].

The reader is assumed to be familiar with the basic concepts of stochastic analysis.
Some familiarity with Hilbert and Banach space setting is required.

The thesis does not contain new results.
However, no earlier treatment of this sort, from the Ito theory to a modern Banach space approach, is known to us.
Abstract (fin): Tässä työssä tutkitaan stokastista integraalia.
Käsittely aloitetaan määrittelemällä integraali ensin aivan tavalliselle reaaliakselille.
Tämän jälkeen siirrytään n-ulotteisen reaaliavaruuden ja separoituvan Hilbertin avaruuden kautta integroimaan Banachin avaruuksissa arvonsa saavia operaattoreita.

Työ keskittyy nimenomaan integraalin ja integroituvien funktioiden määrittelemiseen.
Integraalien käytännön laskemiseen ei puututa.

Luonnollisesti työssä rakennetaan myös määritelmien tueksi tarvittavaa teoriaa.
Tarpeellisia työkaluja ovat muun muassa ydinoperaattorit, Hilbertin-Schmidtin operaattorit, Wienerin prosessit sekä sylinteriprosessit, monistuvan ytimen avaruudet, gamma-radonisoivat operaattorit sekä tietenkin Pettisin integraali.

Äärellisulotteisten reaaliavaruuksien stokastisen integroinnin teoriaa voi opiskella lähes mistä tahansa stokastisen analyysin oppikirjasta.
Hilbertin avaruuksien stokastisista yhtälöistä ovat Da Prato ja Zabczyk [DPZ92] kirjoittaneet reilut kymmenen vuotta sitten kattavan teoksen.
Esitetty Banachin avaruuksien teoria perustuu van Neervenin ja Weisin [NW03] tuoreisiin tuloksiin.

Lukijan oletetaan tuntevan stokastiikan perusteet sekä annos Hilbertin ja Banachin avaruuksien teoriaa.

Työ ei sisällä varsinaisesti uusia tuloksia.
Tällaista rakennelmaa Iton 1940-luvulla julkaisemista tuloksista aivan uusimpiin näkökulmiin ei kuitenkaan tiettävästi ole aiemmin julkaistu.
ED:2004-07-13
INSSI record number: 25428
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