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 condit...
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| Format: | Doctoral Thesis |
| Language: | English |
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Universitätsbibliothek Leipzig
2011
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| Online Access: | http://nbn-resolving.de/urn:nbn:de:bsz:15-qucosa-65103 http://nbn-resolving.de/urn:nbn:de:bsz:15-qucosa-65103 http://www.qucosa.de/fileadmin/data/qucosa/documents/6510/Galerkin%20approximation.pdf |
| Summary: | 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). |
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