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Properties of a Special Type of Filtration and its Martingale Criteria

  • Assylliya K. Zhunussova
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Abstract

This article investigates the structural properties of stochastic processes relative to a generalized single jump filtration, extending the framework introduced by A.A. Gushchin (2020) to the case of a non-trivial initial σ-algebra ℋ. By leveraging the general theory of processes and optional projection techniques, we establish fundamental measurability criteria for random variables and a complete characterization of stopping times and adapted processes. Furthermore, we derive comprehensive martingale and local martingale criteria, providing necessary and sufficient conditions for the preservation of the martingale property in this extended setting.

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Cite

BibTeXRIS
BibTeX
@misc{zhunussova2026properties,
  title = {{Properties of a Special Type of Filtration and its Martingale Criteria}},
  author = {Assylliya K. Zhunussova},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2606.04245},
  eprint = {2606.04245},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.04245},
}
RIS
TY  - UNPB
TI  - Properties of a Special Type of Filtration and its Martingale Criteria
AU  - Assylliya K. Zhunussova
PY  - 2026
DA  - 2026/10/11
PB  - arXiv
DO  - 10.48550/arxiv.2606.04245
UR  - https://arxiv.org/abs/2606.04245
AB  - This article investigates the structural properties of stochastic processes relative to a generalized single jump filtration, extending the framework introduced by A.A. Gushchin (2020) to the case of a non-trivial initial σ-algebra ℋ. By leveraging the general theory of processes and optional projection techniques, we establish fundamental measurability criteria for random variables and a complete characterization of stopping times and adapted processes. Furthermore, we derive comprehensive martingale and local martingale criteria, providing necessary and sufficient conditions for the preservation of the martingale property in this extended setting.
ER  -