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

Belton, Alexander C. R.

Ann. Inst. H. Poincaré Probab. Statist., Tome 44 (2008) no. 2, p. 258-279 / Harvested from Project Euclid

Résumé

Let X be the unique normal martingale such that X₀=0 and ¶ d[X]_t=(1−t−X_t−) dX_t+dt ¶ and let Y_t:=X_t+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: Y_t=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.

Publié le : 2008-04-15
Classification: Monotone independence, Monotone Poisson process, Non-commutative probability, Quantum probability, 60G44

@article{1207948219,
     author = {Belton, Alexander C. R.},
     title = {On the path structure of a semimartingale arising from monotone probability theory},
     journal = {Ann. Inst. H. Poincar\'e Probab. Statist.},
     volume = {44},
     number = {2},
     year = {2008},
     pages = { 258-279},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1207948219}
}

Belton, Alexander C. R. On the path structure of a semimartingale arising from monotone probability theory. Ann. Inst. H. Poincaré Probab. Statist., Tome 44 (2008) no. 2, pp.  258-279. http://gdmltest.u-ga.fr/item/1207948219/