Hostname: page-component-76c49bb84f-lxspr Total loading time: 0 Render date: 2025-07-10T22:59:19.127Z Has data issue: false hasContentIssue false
  • English
  • Français

Uniqueness for measure-valued equations of nonlinear filtering for stochastic dynamical systems with Lévy noise

Part of: Stochastic processes Stochastic analysis

Published online by Cambridge University Press:  26 July 2018

Huijie Qiao
Huijie Qiao*
Affiliation:
Southeast University
*
* Postal address: Department of Mathematics, Southeast University, Nanjing, Jiangsu 211189, China. Email address: hjqiaogean@seu.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

In the paper we study the Zakai and Kushner–Stratonovich equations of the nonlinear filtering problem for a non-Gaussian signal-observation system. Moreover, we prove that under some general assumption, the Zakai equation has pathwise uniqueness and uniqueness in joint law, and the Kushner–Stratonovich equation is unique in joint law.

Keywords

Zakai equationKushner–Stratonovich equationuniqueness in joint lawpathwise uniqueness

MSC classification

Primary: 60G35: Signal detection and filtering 60G51: Processes with independent increments; L'evy processes 60H10: Stochastic ordinary differential equations

Information

Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 2 , June 2018 , pp. 396 - 413
Copyright
Copyright © Applied Probability Trust 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1]Ceci, C. and Colaneri, K. (2012). Nonlinear filtering for jump diffusion observations. Adv. Appl. Prob. 44, 678–701. Google Scholar
[2]Ceci, C. and Colaneri, K. (2014). The Zakai equation of nonlinear filtering for jump-diffusion observations: existence and uniqueness. Appl. Math. Optimization 69, 47–82. Google Scholar
[3]Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. John Wiley, New York. Google Scholar
[4]Heunis, A. J. and Lucic, V. M. (2008). On the innovations conjecture of nonlinear filtering with dependent data. Electron. J. Prob. 13, 2190–2216. Google Scholar
[5]Kurtz, T. G. and Ocone, D. L. (1988). Unique characterization of conditional distributions in nonlinear filtering. Ann. Prob. 16, 80–107. Google Scholar
[6]Lucic, V. M. and Heunis, A. J. (2001). On uniqueness of solutions for the stochastic differential equations of nonlinear filtering. Ann. Appl. Prob. 11, 182–209. Google Scholar
[7]Protter, P. and Shimbo, K. (2008). No arbitrage and general semimartingales. In Markov Processes and Related Topics, Institute of Mathematical Statistics, Beachwood, OH, pp. 267–283. Google Scholar
[8]Qiao, H. and Duan, J. (2015). Nonlinear filtering of stochastic dynamical systems with Lévy noises. Adv. Appl. Prob. 47, 902–918. Google Scholar
[9]Qiao, H. and Zhang, X. (2008). Homeomorphism flows for non-Lipschitz stochastic differential equations with jumps. Stoch. Process. Appl. 118, 2254–2268. Google Scholar
[10]Rogers, L. C. G. and Williams, D. (1987). Diffusions, Markov Processes, and Martingales, Vol. 2, Itô Calculus. John Wiley, New York. Google Scholar
[11]Situ, R. (2005). Theory of stochastic differential equations with jumps and applications. Springer, New York. Google Scholar
[12]Szpirglas, J. (1978). Sur l'équivalence d'équations différentielles stochastiques à valeurs mesures intervenant dans le filtrage markovien non linéaire. Ann. Inst. H. Poincaré B 14, 33–59. Google Scholar