Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
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TY - JOUR
T1 - On the path structure of a semimartingale arising from monotone probability theory
AU - Belton, Alexander C. R.
N1 - To appear in Annales de l'Institut Henri Poincare (B) Probability and Statistics; accepted in final form: 5th November 2006. Posted on the arXiv: 24th September 2007. RAE_import_type : Internet publication RAE_uoa_type : Pure Mathematics
PY - 2008
Y1 - 2008
N2 - Let X be the unique normal martingale such that X_0 = 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.
AB - Let X be the unique normal martingale such that X_0 = 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.
U2 - 10.1214/07-AIHP116
DO - 10.1214/07-AIHP116
M3 - Journal article
VL - 44
SP - 258
EP - 279
JO - Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques
JF - Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques
SN - 0246-0203
IS - 2
ER -