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On Galerkin Approximations for the Zakai Equation with Diffusive and Point Process Observations

We are interested in a nonlinear filtering problem motivated by an
information-based approach for modelling the dynamic evolution of a
portfolio of credit risky securities.
We solve this
problem by `change of measure method\\\' and show the existence of the
density of the unnormalized conditional distribution which is a
solution to the Zakai equation. Zakai equation is a linear SPDE
which, in general, cannot be solved analytically. We apply Galerkin
method to solve it numerically and show the convergence of Galerkin
approximation in mean square. Lastly, we design an adaptive Galerkin
filter with a basis of Hermite polynomials and we present numerical
examples to illustrate the effectiveness of the proposed method. The
work is closely related to the paper Frey and Schmidt (2010).

Identiferoai:union.ndltd.org:DRESDEN/oai:qucosa.de:bsz:15-qucosa-65103
Date16 February 2011
CreatorsXu, Ling
ContributorsLing Xu, Fakultät für Mathematik und Informatik., Professor Rüdiger Frey, Professor Thorsten Schmidt
PublisherUniversitätsbibliothek Leipzig
Source SetsHochschulschriftenserver (HSSS) der SLUB Dresden
LanguageEnglish
Detected LanguageEnglish
Typedoc-type:doctoralThesis
Formatapplication/pdf

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