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Nonlinear filtering of stochastic dynamical systems with Lévy noises

Part of: Stochastic processes Stochastic analysis

Published online by Cambridge University Press:  21 March 2016

Huijie Qiao  and
Jinqiao Duan
Huijie Qiao*
Affiliation:
Southeast University
Jinqiao Duan*
Affiliation:
Illinois Institute of Technology
*
∗ Postal address: Department of Mathematics, Southeast University, Nanjing, Jiangsu, 211189, China. Email address: hjqiaogean@seu.edu.cn
∗∗ Postal address: Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA. Email address: duan@iit.edu
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Abstract

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Nonlinear filtering is investigated in a system where both the signal system and the observation system are under non-Gaussian Lévy fluctuations. Firstly, the Zakai equation is derived, and it is further used to derive the Kushner-Stratonovich equation. Secondly, by a filtered martingale problem, uniqueness for strong solutions of the Kushner-Stratonovich equation and the Zakai equation is proved. Thirdly, under some extra regularity conditions, the Zakai equation for the unnormalized density is also derived in the case of α-stable Lévy noise.

Keywords

Nonlinear filteringLévy processstochastic partial differential equations with jumpsZakai equation

MSC classification

Primary: 60G52: Stable processes 60G35: Signal detection and filtering 60H15: Stochastic partial differential equations
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 47 , Issue 3 , September 2015 , pp. 902 - 918
Copyright
Copyright © Applied Probability Trust 2015 

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