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

top
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

top

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

top
  1. [1] S. Attal. The structure of the quantum semimartingale algebras. J. Operator Theory 46 (2001) 391–410. Zbl0999.81035MR1870414
  2. [2] S. Attal and A. C. R. Belton. The chaotic-representation property for a class of normal martingales. Probab. Theory Related Fields 139 (2007) 543–562. Zbl1130.60049MR2322707
  3. [3] J. Azéma. Sur les fermés aléatoires. Séminaire de Probabilités XIX 397–495. J. Azéma and M. Yor (Eds). Lecture Notes in Math. 1123. Spring- er, Berlin, 1985. Zbl0563.60038MR889496
  4. [4] J. Azéma and M. Yor. Étude d’une martingale remarquable. Séminaire de Probabilités XXIII 88–130. J. Azéma, P.-A. Meyer and M. Yor (Eds). Lecture Notes in Math. 1372. Springer, Berlin, 1989. Zbl0743.60045MR1022900
  5. [5] A. C. R. Belton. An isomorphism of quantum semimartingale algebras. Q. J. Math. 55 (2004) 135–165. Zbl1059.81101MR2068315
  6. [6] A. C. R. Belton. A note on vacuum-adapted semimartingales and monotone independence. In Quantum Probability and Infinite Dimensional Analysis XVIII. From Foundations to Applications, 105–114. M. Schürmann and U. Franz (Eds), World Scientific, Singapore, 2005. MR2211883
  7. [7] A. C. R. Belton. The monotone Poisson process. In Quantum Probability 99–115. M. Bożejko, W. Młotkowski and J. Wysoczański (Eds). Banach Center Publications 73, Polish Academy of Sciences, Warsaw, 2006. Zbl1109.46052MR2423119
  8. [8] P. Billingsley. Probability and Measure, 3rd edition. Wiley, New York, 1995. Zbl0822.60002MR1324786
  9. [9] C. S. Chou. Caractérisation d’une classe de semimartingales. Séminaire de Probabilités XIII 250–252. C. Dellacherie, P.-A. Meyer and M. Weil (Eds). Lecture Notes in Math. 721. Springer, Berlin, 1979. Zbl0409.60045MR544798
  10. [10] R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey and D. E. Knuth. On the Lambert W function. Adv. Comput. Math. 5 (1996) 329–359. Zbl0863.65008MR1414285
  11. [11] F. Delbaen and W. Schachermayer. The fundamental theorem of asset pricing for unbounded stochastic processes. Math. Ann. 312 (1998) 215–250. Zbl0917.60048MR1671792
  12. [12] M. Émery. Compensation de processus à variation finie non localement intégrables. Séminaire de Probabilités XIV 152–160. J. Azéma and M. Yor (Eds). Lecture Notes in Math. 784. Springer, Berlin, 1980. Zbl0428.60054
  13. [13] M. Émery. On the Azéma martingales. Séminaire de Probabilités XXIII 66–87. J. Azéma, P.-A. Meyer and M. Yor (Eds). Lecture Notes in Math. 1372. Springer, Berlin, 1989. Zbl0753.60045
  14. [14] M. Émery. Personal communication, 2006. 
  15. [15] R. L. Graham, D. E. Knuth and O. Patashnik. Concrete Mathematics, 2nd edition. Addison-Wesley, Reading, MA, 1994. Zbl0668.00003MR1397498
  16. [16] N. Muraki. Monotonic independence, monotonic central limit theorem and monotonic law of small numbers. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 4 (2001) 39–58. Zbl1046.46049MR1824472
  17. [17] P. Protter. Stochastic Integration and Differential Equations. A New Approach. Springer, Berlin, 1990. Zbl0694.60047MR1037262
  18. [18] L. C. G. Rogers and D. Williams. Diffusions, Markov Processes and Martingales. Volume 1: Foundations, 2nd edition. Cambridge University Press, Cambridge, 2000. Zbl0949.60003MR1796539
  19. [19] W. Rudin. Real and Complex Analysis, 3rd edition. McGraw-Hill, New York, 1987. Zbl0925.00005MR924157
  20. [20] R. Speicher. A new example of “independence” and “white noise”. Probab. Theory Related Fields 84 (1990) 141–159. Zbl0671.60109MR1030725
  21. [21] C. Stricker. Représentation prévisible et changement de temps. Ann. Probab. 14 (1986) 1070–1074. Zbl0603.60038MR841606
  22. [22] C. Stricker and M. Yor. Calcul stochastique dépendant d’un paramètre. Z. Wahrsch. Verw. Gebiete 45 (1978) 109–133. Zbl0388.60056MR510530
  23. [23] G. Taviot. Martingales et équations de structure: étude géométrique. Thèse, Université Louis Pasteur Strasbourg 1, 1999. Zbl0953.60022MR1736397
  24. [24] S. J. Taylor. The α-dimensional measure of the graph and set of zeros of a Brownian path. Proc. Cambridge Philos. Soc. 51 (1955) 265–274. Zbl0064.05201MR74494

NotesEmbed ?

top

You must be logged in to post comments.