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.

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.

DOI : 10.1214/07-AIHP116
Classification : 60G44
Mots clés : monotone independence, monotone Poisson process, non-commutative probability, quantum probability
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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, Tome 44 (2008) no. 2, pp. 258-279. doi : 10.1214/07-AIHP116. http://www.numdam.org/articles/10.1214/07-AIHP116/

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

[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

[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

[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 | Zbl

[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 | Zbl

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

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

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

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

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

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

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

[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 | Zbl

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