Standardness and nonstandardness of next-jump time filtrations

Stéphane Laurent (Currently none)

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.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-11

Publication Date: July 7, 2013

DOI: 10.1214/ECP.v18-2766

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. MR1768000
  • 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. MR1837293
  • Laurent, Stéphane. On standardness and I-cosiness. Séminaire de Probabilités XLIII, 127--186, Lecture Notes in Math., 2006, Springer, Berlin, 2011. MR2790371
  • 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 MR2768520
  • Laurent, Stéphane. Further comments on the representation problem for stationary processes. Statist. Probab. Lett. 80 (2010), no. 7-8, 592--596. MR2595135
  • 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 MR1304093
Bookmark and Share


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