×

On the path structure of a semimartingale arising from monotone probability theory. (English) Zbl 1180.60037

The authors prove the following result. 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 \(t \geq 0\); 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 \(\{t \geq 0: Y_t=1 \}\) 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.

MSC:

60G44 Martingales with continuous parameter
PDFBibTeX XMLCite
Full Text: DOI arXiv EuDML

References:

[1] S. Attal. The structure of the quantum semimartingale algebras. J. Operator Theory 46 (2001) 391-410. · 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. · Zbl 1130.60049 · doi:10.1007/s00440-006-0052-z
[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. · 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. · Zbl 0743.60045
[5] A. C. R. Belton. An isomorphism of quantum semimartingale algebras. Q. J. Math. 55 (2004) 135-165. · Zbl 1059.81101 · doi:10.1093/qmath/hag052
[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.
[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. · Zbl 1109.46052
[8] P. Billingsley. Probability and Measure , 3rd edition. Wiley, New York, 1995. · 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. · 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. · Zbl 0863.65008 · doi:10.1007/BF02124750
[11] F. Delbaen and W. Schachermayer. The fundamental theorem of asset pricing for unbounded stochastic processes. Math. Ann. 312 (1998) 215-250. · Zbl 0917.60048 · doi:10.1007/s002080050220
[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. · 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. · 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. · Zbl 0836.00001
[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. · Zbl 1046.46049 · doi:10.1142/S0219025701000334
[17] P. Protter. Stochastic Integration and Differential Equations. A New Approach . Springer, Berlin, 1990. · 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. · Zbl 0977.60005
[19] W. Rudin. Real and Complex Analysis , 3rd edition. McGraw-Hill, New York, 1987. · Zbl 0925.00005
[20] R. Speicher. A new example of “independence” and “white noise”. Probab. Theory Related Fields 84 (1990) 141-159. · Zbl 0671.60109 · doi:10.1007/BF01197843
[21] C. Stricker. Représentation prévisible et changement de temps. Ann. Probab. 14 (1986) 1070-1074. · Zbl 0603.60038 · doi:10.1214/aop/1176992460
[22] C. Stricker and M. Yor. Calcul stochastique dépendant d’un paramètre. Z. Wahrsch. Verw. Gebiete 45 (1978) 109-133. · Zbl 0388.60056 · doi:10.1007/BF00715187
[23] G. Taviot. Martingales et équations de structure: étude géométrique. Thèse, Université Louis Pasteur Strasbourg 1, 1999. · Zbl 0953.60022
[24] S. J. Taylor. The \alpha -dimensional measure of the graph and set of zeros of a Brownian path. Proc. Cambridge Philos. Soc. 51 (1955) 265-274. · Zbl 0064.05201 · doi:10.1017/S030500410003019X
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.