# Stochastic Filtering

 dc.contributor Helsingin yliopisto, matemaattis-luonnontieteellinen tiedekunta, matematiikan ja tilastotieteen laitos fi dc.contributor Helsingfors universitet, matematisk-naturvetenskapliga fakulteten, institutionen för matematik och statistik sv dc.contributor University of Helsinki, Faculty of Science, Department of Mathematics and Statistics en dc.contributor.author Yazigi, Adil fi dc.date.accessioned 2010-11-25T12:12:34Z dc.date.available 2010-11-25T12:12:34Z dc.date.issued 2010-10-05 fi dc.identifier.uri URN:NBN:fi-fe201011032663 fi dc.identifier.uri http://hdl.handle.net/10138/21308 dc.description.abstract The stochastic filtering has been in general an estimation of indirectly observed states given observed data. This means that one is discussing conditional expected values as being one of the most accurate estimation, given the observations in the context of probability space. In my thesis, I have presented the theory of filtering using two different kind of observation process: the first one is a diffusion process which is discussed in the first chapter, while the third chapter introduces the latter which is a counting process. The majority of the fundamental results of the stochastic filtering is stated in form of interesting equations, such the unnormalized Zakai equation that leads to the Kushner-Stratonovich equation. The latter one which is known also by the normalized Zakai equation or equally by Fujisaki-Kallianpur-Kunita (FKK) equation, shows the divergence between the estimate using a diffusion process and a counting process. I have also introduced an example for the linear gaussian case, which is mainly the concept to build the so-called Kalman-Bucy filter. As the unnormalized and the normalized Zakai equations are in terms of the conditional distribution, a density of these distributions will be developed through these equations and stated by Kushner Theorem. However, Kushner Theorem has a form of a stochastic partial differential equation that needs to be verify in the sense of the existence and uniqueness of its solution, which is covered in the second chapter. en dc.language.iso en fi dc.publisher Helsingin yliopisto fi dc.publisher Helsingfors universitet sv dc.publisher University of Helsinki en dc.rights Julkaisu on tekijänoikeussäännösten alainen. Teosta voi lukea ja tulostaa henkilökohtaista käyttöä varten. Käyttö kaupallisiin tarkoituksiin on kielletty. fi dc.rights This publication is copyrighted. You may download, display and print it for Your own personal use. Commercial use is prohibited. en dc.rights Publikationen är skyddad av upphovsrätten. Den får läsas och skrivas ut för personligt bruk. Användning i kommersiellt syfte är förbjuden. sv dc.title Stochastic Filtering en dc.type.ontasot Pro gradu fi dc.type.ontasot Master's thesis en dc.type.ontasot Pro gradu sv dc.ths Gasbarra, Dario fi dc.type.dcmitype Text fi

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