On the path structure of a semimartingale arising from monotone probability theory

Alexander C. R. Belton

Annales de l'I.H.P. Probabilités et statistiques (2008)

  • Volume: 44, Issue: 2, page 258-279
  • ISSN: 0246-0203

Abstract

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Let X be the unique normal martingale such that X0=0 and d[X]t=(1−t−Xt−) dXt+dt and let Yt:=Xt+t for all t≥0; the semimartingale Y arises in quantum probability, where it is the monotone-independent analogue of the Poisson process. The trajectories of Y are examined and various probabilistic properties are derived; in particular, the level set {t≥0: Yt=1} is shown to be non-empty, compact, perfect and of zero Lebesgue measure. The local times of Y are found to be trivial except for that at level 1; consequently, the jumps of Y are not locally summable.

How to cite

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Belton, Alexander C. R.. "On the path structure of a semimartingale arising from monotone probability theory." Annales de l'I.H.P. Probabilités et statistiques 44.2 (2008): 258-279. <http://eudml.org/doc/77969>.

@article{Belton2008,
abstract = {Let X be the unique normal martingale such that X0=0 and d[X]t=(1−t−Xt−) dXt+dt and let Yt:=Xt+t for all t≥0; the semimartingale Y arises in quantum probability, where it is the monotone-independent analogue of the Poisson process. The trajectories of Y are examined and various probabilistic properties are derived; in particular, the level set \{t≥0: Yt=1\} is shown to be non-empty, compact, perfect and of zero Lebesgue measure. The local times of Y are found to be trivial except for that at level 1; consequently, the jumps of Y are not locally summable.},
author = {Belton, Alexander C. R.},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {monotone independence; monotone Poisson process; non-commutative probability; quantum probability},
language = {eng},
number = {2},
pages = {258-279},
publisher = {Gauthier-Villars},
title = {On the path structure of a semimartingale arising from monotone probability theory},
url = {http://eudml.org/doc/77969},
volume = {44},
year = {2008},
}

TY - JOUR
AU - Belton, Alexander C. R.
TI - On the path structure of a semimartingale arising from monotone probability theory
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2008
PB - Gauthier-Villars
VL - 44
IS - 2
SP - 258
EP - 279
AB - Let X be the unique normal martingale such that X0=0 and d[X]t=(1−t−Xt−) dXt+dt and let Yt:=Xt+t for all t≥0; the semimartingale Y arises in quantum probability, where it is the monotone-independent analogue of the Poisson process. The trajectories of Y are examined and various probabilistic properties are derived; in particular, the level set {t≥0: Yt=1} is shown to be non-empty, compact, perfect and of zero Lebesgue measure. The local times of Y are found to be trivial except for that at level 1; consequently, the jumps of Y are not locally summable.
LA - eng
KW - monotone independence; monotone Poisson process; non-commutative probability; quantum probability
UR - http://eudml.org/doc/77969
ER -

References

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