Electronic Communications in Probability

Standardness and nonstandardness of next-jump time filtrations

Stéphane Laurent

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Abstract

The value of the next-jump time process at each time is the date of its the next jump. We characterize the standardness of the filtration generated by this process in terms of the asymptotic behavior at $n=-\infty$ of the probability that the process jumps at time $n$. In the case when the filtration is not standard we characterize the standardness of its extracted filtrations.

Article information

Source
Electron. Commun. Probab., Volume 18 (2013), paper no. 56, 11 pp.

Dates
Accepted: 7 July 2013
First available in Project Euclid: 7 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465315595

Digital Object Identifier
doi:10.1214/ECP.v18-2766

Mathematical Reviews number (MathSciNet)
MR3078019

Zentralblatt MATH identifier
1300.60041

Subjects
Primary: 60G05: Foundations of stochastic processes
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Keywords
Standard filtration Cosy filtration

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Laurent, Stéphane. Standardness and nonstandardness of next-jump time filtrations. Electron. Commun. Probab. 18 (2013), paper no. 56, 11 pp. doi:10.1214/ECP.v18-2766. https://projecteuclid.org/euclid.ecp/1465315595


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References

  • Ceillier, G., Leuridan, C.. Filtrations at the threshold of standardness. arXiv:1208.0110. To appear in: Probability Theory and Related Fields.
  • Émery, M.; Schachermayer, W. Brownian filtrations are not stable under equivalent time-changes. Séminaire de Probabilités XXXIII, 267–276, Lecture Notes in Math., 1709, Springer, Berlin, 1999.
  • Émery, M.; Schachermayer, W. On Vershik's standardness criterion and Tsirelson's notion of cosiness. Séminaire de Probabilités XXXV, 265–305, Lecture Notes in Math., 1755, Springer, Berlin, 2001.
  • Laurent, Stéphane. On standardness and I-cosiness. Séminaire de Probabilités XLIII, 127–186, Lecture Notes in Math., 2006, Springer, Berlin, 2011.
  • Laurent, S. On Vershikian and $I$-cosy random variables and filtrations. Teor. Veroyatn. Primen. 55 (2010), no. 1, 104–132; translation in Theory Probab. Appl. 55 (2011), no. 1, 54–76
  • Laurent, Stéphane. Further comments on the representation problem for stationary processes. Statist. Probab. Lett. 80 (2010), no. 7-8, 592–596.
  • Vershik, A.M.. Approximation in measure theory (in Russian). PhD Dissertation, Leningrad University (1973).
  • Vershik, A. M. Theory of decreasing sequences of measurable partitions. (Russian) Algebra i Analiz 6 (1994), no. 4, 1–68; translation in St. Petersburg Math. J. 6 (1995), no. 4, 705–761