Hostname: page-component-848d4c4894-r5zm4 Total loading time: 0 Render date: 2024-06-25T00:17:24.911Z Has data issue: false hasContentIssue false
  • English
  • Français

The Kalman-Bucy filter for integrable Lévy processes with infinite second moment

Part of: Inference from stochastic processes Stochastic analysis Stochastic processes Stochastic systems and control

Published online by Cambridge University Press:  30 March 2016

David Applebaum  and
Stefan Blackwood
David Applebaum*
Affiliation:
The University of Sheffield
Stefan Blackwood
Affiliation:
The University of Sheffield
*
∗ Postal address: School of Mathematics and Statistics, The University of Sheffield, Hicks Building, Hounsfield Road, Sheffield S3 7RH, UK.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We extend the Kalman-Bucy filter to the case where both the system and observation processes are driven by finite dimensional Lévy processes, but whereas the process driving the system dynamics is square-integrable, that driving the observations is not; however it remains integrable. The main result is that the components of the observation noise that have infinite variance make no contribution to the filtering equations. The key technique used is approximation by processes having bounded jumps.

Keywords

Lévy processRiccati equationKalman-Bucy filter

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes
Secondary: 93E11: Filtering 60H10: Stochastic ordinary differential equations 60G35: Signal detection and filtering 62M20: Prediction
Type
Research Papers
Information
Journal of Applied Probability , Volume 52 , Issue 3 , September 2015 , pp. 636 - 648
Copyright
Copyright © 2015 by the Applied Probability Trust 

Footnotes

∗∗∗

Current address: The Floow Ltd., Oxo House, 4 Joiner Street, Sheffield S3 8GW, UK.

References

Ahn, H. and Feldman, R. E. (2000). Optimal filtering of a Gaussian signal in the presence of Lévy noise. SIAM J. Appl. Math. 60 359-369.Google Scholar
Applebaum, D. (2009). Extending stochastic resonance for neuron models to general Lévy noise. IEEE Trans. Neural Networks 20 1993-1995.Google Scholar
Applebaum, D. (2009). Lévy Processes and Stochastic Calculus, 2nd edn. Cambridge University Press.Google Scholar
Bain, A. and Crisan, D. (2009). Fundamentals of Stochastic Filtering (Stoch. Modelling Appl. Prob. 60), Springer, New York.Google Scholar
Bartumeus, F. (2007). Lévy processes in animal movement: an evolutionary hypothesis. Fractals 15 151-162.CrossRefGoogle Scholar
Blackwood, S. (2014). Lévy processes and filtering theory. Doctoral thesis, The University of Sheffield.Google Scholar
Cipra, B. (1993). Engineers look to Kalman filtering for guidance. SIAM News 26 32-50.Google Scholar
Cont, R. and Tankov, P. (2004). Financial Modelling with Jump Processes. Chapman & Hall/CRC, Boca Raton, FL.Google Scholar
Davis, M. H. A. (1977). Linear Estimation and Stochastic Control. Chapman & Hall, London.Google Scholar
Halmos, P. R. (1982). A Hilbert Space Problem Book, 2nd edn. Springer, New York.Google Scholar
Kleptsyna, M. L. and Le Breton, A. (2002). Extension of the Kalman–Bucy filter to elementary linear systems with fractional Brownian noises. Statist. Infer. Stoch. Process. 5 249-271.Google Scholar
Le Breton, A. and Musiela, M. (1993). A generalization of the Kalman filter to models with infinite variance. Stoch. Process. Appl. 47 75-94.Google Scholar
Meyer-Brandis, T. and Proske, F. (2004). Explicit solution of a non-linear filtering problem for Lévy processes with application to finance. Appl. Math. Optimization 50 119-134.Google Scholar
Øksendal, B. (2003). Stochastic Differential Equations, 6th edn. Springer, Berlin.Google Scholar
Patel, A. and Kosko, B. (2008). Stochastic resonance in continuous and spiking neuron models with Lévy noise. IEEE Trans. Neural Networks 19 1993-2008.Google Scholar
Popa, S. and Sritharan, S. S. (2009). Nonlinear filtering of Itô–Lévy stochastic differential equations with continuous obervations. Commun. Stoch. Anal. 3 313-330.Google Scholar
Sornette, D. and Ide, K. (2001). The Kalman–Lévy filter. Physica D 151 142-174.Google Scholar
Xiong, J. (2008). An Introduction to Stochastic Filtering Theory (Oxford Graduate Texts Math. 18). Oxford University Press.Google Scholar