Hiding a constant drift
Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 2, pp. 498-514.

La question suivante a été posée par Marc Yor: Soit B un mouvement Brownien et St=t+Bt. Peut-on définir un processus H qui est -prévisible tel que l'intégrale stochastique (HS) soit un mouvement Brownien (sans drift) pour sa propre filtration ? Dans cet article nous fournissons une réponse affirmative en relâchant la condition que H soit -prévisible. Autrement dit, nous montrons qu'il existe une solution faible pour cette question de Yor. La question originale (c'est à dire, l'existence d'une solution forte) reste ouverte.

The following question is due to Marc Yor: Let B be a brownian motion and St=t+Bt. Can we define an -predictable process H such that the resulting stochastic integral (HS) is a brownian motion (without drift) in its own filtration, i.e. an -brownian motion? In this paper we show that by dropping the requirement of -predictability of H we can give a positive answer to this question. In other words, we are able to show that there is a weak solution to Yor's question. The original question, i.e., existence of a strong solution, remains open.

DOI : 10.1214/10-AIHP363
Classification : 60H05, 60G44, 60J65, 60G05, 60H10
Mots clés : brownian motion with drift, stochastic integral, enlargement of filtration
@article{AIHPB_2011__47_2_498_0,
     author = {Prokaj, Vilmos and R\'asonyi, Mikl\'os and Schachermayer, Walter},
     title = {Hiding a constant drift},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {498--514},
     publisher = {Gauthier-Villars},
     volume = {47},
     number = {2},
     year = {2011},
     doi = {10.1214/10-AIHP363},
     mrnumber = {2814420},
     zbl = {1216.60048},
     language = {en},
     url = {http://www.numdam.org/articles/10.1214/10-AIHP363/}
}
TY  - JOUR
AU  - Prokaj, Vilmos
AU  - Rásonyi, Miklós
AU  - Schachermayer, Walter
TI  - Hiding a constant drift
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2011
SP  - 498
EP  - 514
VL  - 47
IS  - 2
PB  - Gauthier-Villars
UR  - http://www.numdam.org/articles/10.1214/10-AIHP363/
DO  - 10.1214/10-AIHP363
LA  - en
ID  - AIHPB_2011__47_2_498_0
ER  - 
%0 Journal Article
%A Prokaj, Vilmos
%A Rásonyi, Miklós
%A Schachermayer, Walter
%T Hiding a constant drift
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2011
%P 498-514
%V 47
%N 2
%I Gauthier-Villars
%U http://www.numdam.org/articles/10.1214/10-AIHP363/
%R 10.1214/10-AIHP363
%G en
%F AIHPB_2011__47_2_498_0
Prokaj, Vilmos; Rásonyi, Miklós; Schachermayer, Walter. Hiding a constant drift. Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 2, pp. 498-514. doi : 10.1214/10-AIHP363. http://www.numdam.org/articles/10.1214/10-AIHP363/

[1] M. T. Barlow and M. Yor. Sur la construction d'une martingale continue, de valeur absolue donnée. In Seminar on Probability, XIV (Paris, 1978/1979) (French). Lecture Notes in Math. 784 62-75. Springer, Berlin, 1980. | Numdam | MR | Zbl

[2] R. F. Bass and K. Burdzy. Stochastic bifurcation models. Ann. Probab. 27 (1999) 50-108. | MR | Zbl

[3] M. Émery and W. Schachermayer. A remark on Tsirelson's stochastic differential equation. In Séminaire de Probabilités, XXXIII. Lecture Notes in Math. 1709 291-303. Springer, Berlin, 1999. | Numdam | MR | Zbl

[4] K. Itô and H. P. Mckean, Jr. Diffusion Processes and Their Sample Paths. Die Grundlehren der Mathematischen Wissenschaften 125. Academic Press, New York, 1965. | MR | Zbl

[5] R. Mansuy and M. Yor. Random Times and Enlargements of Filtrations in a Brownian Setting. Lecture Notes in Mathematics 1873. Springer, Berlin, 2006. | MR | Zbl

[6] H. P. Mckean, Jr. Stochastic Integrals. Probability and Mathematical Statistics 5. Academic Press, New York, 1969. | MR | Zbl

[7] V. Prokaj. Unfolding the Skorohod reflection of a semimartingale. Statist. Probab. Lett. 79 (2009) 534-536. | MR | Zbl

[8] P. E. Protter. Stochastic Integration and Differential Equations, 2nd edition. Applications of Mathematics (New York) 21. Springer, Berlin, 2004. | MR | Zbl

[9] M. Rásonyi, W. Schachermayer and R. Warnung. Hiding the drift. Ann. Probab. 37 (2009) 2459-2470. Available at http://arxiv.org/abs/0802.1152. | MR

[10] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion. Grundlehren der Mathematischen Wissenschaften 293. Springer, Berlin, 1991. | MR | Zbl

[11] L. Serlet. Creation or deletion of a drift on a Brownian trajectory. In Séminaire de Probabilités XLI. Lecture Notes in Math. 1934 215-232. Springer, Berlin, 2008. | MR | Zbl

[12] B. S. Tsirelson. An example of a stochastic differential equation that has no strong solution. Teor. Verojatnost. i Primenen. 20 (1975) 427-430. | MR | Zbl

Cité par Sources :