On the path structure of a semimartingale arising from monotone probability theory
Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 2, p. 258-279
Soit X l’unique martingale normale telle que X 0 =0 et dX t =(1-t-X t- )dX t +dt et soit Y t :=X t +t pour tout t0; la semimartingale Y se manifeste dans la théorie des probabilités quantiques, où c’est analogue du processus de Poisson pour l’indépendance monotone. Les trajectoires de Y sont examinées et diverses propriétés probabilistes sont déduites; en particulier, l’ensemble de niveau t0:Y t =1 est montré être non vide, compact, parfait et de mesure de Lebesgue nulle. Les temps locaux de Y sont trouvés être triviaux sauf celui au niveau 1; par conséquent les sauts de Y ne sont pas localements sommables.
Let X be the unique normal martingale such that X 0 =0 and dX t =(1-t-X t- )dX t +dt and let Y t :=X t +t for all t0; 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 t0: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.
@article{AIHPB_2008__44_2_258_0,
     author = {Belton, Alexander C. R.},
     title = {On the path structure of a semimartingale arising from monotone probability theory},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {44},
     number = {2},
     year = {2008},
     pages = {258-279},
     doi = {10.1214/07-AIHP116},
     zbl = {1180.60037},
     mrnumber = {2446323},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2008__44_2_258_0}
}
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, Tome 44 (2008) no. 2, pp. 258-279. doi : 10.1214/07-AIHP116. http://www.numdam.org/item/AIHPB_2008__44_2_258_0/

[1] S. Attal. The structure of the quantum semimartingale algebras. J. Operator Theory 46 (2001) 391-410. | MR 1870414 | Zbl 0999.81035

[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. | MR 2322707 | Zbl 1130.60049

[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. | Numdam | MR 889496 | Zbl 0563.60038

[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. | Numdam | MR 1022900 | Zbl 0743.60045

[5] A. C. R. Belton. An isomorphism of quantum semimartingale algebras. Q. J. Math. 55 (2004) 135-165. | MR 2068315 | Zbl 1059.81101

[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. | MR 2211883

[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. | MR 2423119 | Zbl 1109.46052

[8] P. Billingsley. Probability and Measure, 3rd edition. Wiley, New York, 1995. | MR 1324786 | Zbl 0822.60002

[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. | Numdam | MR 544798 | Zbl 0409.60045

[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. | MR 1414285 | Zbl 0863.65008

[11] F. Delbaen and W. Schachermayer. The fundamental theorem of asset pricing for unbounded stochastic processes. Math. Ann. 312 (1998) 215-250. | MR 1671792 | Zbl 0917.60048

[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. | Numdam | Zbl 0428.60054

[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. | Numdam | Zbl 0753.60045

[14] M. Émery. Personal communication, 2006.

[15] R. L. Graham, D. E. Knuth and O. Patashnik. Concrete Mathematics, 2nd edition. Addison-Wesley, Reading, MA, 1994. | MR 1397498 | Zbl 0668.00003

[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. | MR 1824472 | Zbl 1046.46049

[17] P. Protter. Stochastic Integration and Differential Equations. A New Approach. Springer, Berlin, 1990. | MR 1037262 | Zbl 0694.60047

[18] L. C. G. Rogers and D. Williams. Diffusions, Markov Processes and Martingales. Volume 1: Foundations, 2nd edition. Cambridge University Press, Cambridge, 2000. | MR 1796539 | Zbl 0949.60003

[19] W. Rudin. Real and Complex Analysis, 3rd edition. McGraw-Hill, New York, 1987. | MR 924157 | Zbl 0925.00005

[20] R. Speicher. A new example of “independence” and “white noise”. Probab. Theory Related Fields 84 (1990) 141-159. | MR 1030725 | Zbl 0671.60109

[21] C. Stricker. Représentation prévisible et changement de temps. Ann. Probab. 14 (1986) 1070-1074. | MR 841606 | Zbl 0603.60038

[22] C. Stricker and M. Yor. Calcul stochastique dépendant d'un paramètre. Z. Wahrsch. Verw. Gebiete 45 (1978) 109-133. | MR 510530 | Zbl 0388.60056

[23] G. Taviot. Martingales et équations de structure: étude géométrique. Thèse, Université Louis Pasteur Strasbourg 1, 1999. | MR 1736397

[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. | MR 74494 | Zbl 0064.05201