Hiding a constant drift

Vilmos Prokaj; Miklós Rásonyi; Walter Schachermayer

Annales de l'I.H.P. Probabilités et statistiques (2011)

  • Volume: 47, Issue: 2, page 498-514
  • ISSN: 0246-0203

Abstract

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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 (H⋅S) 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.

How to cite

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Prokaj, Vilmos, Rásonyi, Miklós, and Schachermayer, Walter. "Hiding a constant drift." Annales de l'I.H.P. Probabilités et statistiques 47.2 (2011): 498-514. <http://eudml.org/doc/240013>.

@article{Prokaj2011,
abstract = {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 (H⋅S) 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.},
author = {Prokaj, Vilmos, Rásonyi, Miklós, Schachermayer, Walter},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {brownian motion with drift; stochastic integral; enlargement of filtration; Brownian motion with drift},
language = {eng},
number = {2},
pages = {498-514},
publisher = {Gauthier-Villars},
title = {Hiding a constant drift},
url = {http://eudml.org/doc/240013},
volume = {47},
year = {2011},
}

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
PB - Gauthier-Villars
VL - 47
IS - 2
SP - 498
EP - 514
AB - 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 (H⋅S) 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.
LA - eng
KW - brownian motion with drift; stochastic integral; enlargement of filtration; Brownian motion with drift
UR - http://eudml.org/doc/240013
ER -

References

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  3. [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. Zbl0957.60064MR1768002
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  5. [5] R. Mansuy and M. Yor. Random Times and Enlargements of Filtrations in a Brownian Setting. Lecture Notes in Mathematics 1873. Springer, Berlin, 2006. Zbl1103.60003MR2200733
  6. [6] H. P. McKean, Jr.Stochastic Integrals. Probability and Mathematical Statistics 5. Academic Press, New York, 1969. Zbl0191.46603MR247684
  7. [7] V. Prokaj. Unfolding the Skorohod reflection of a semimartingale. Statist. Probab. Lett. 79 (2009) 534–536. Zbl1172.60012MR2494646
  8. [8] P. E. Protter. Stochastic Integration and Differential Equations, 2nd edition. Applications of Mathematics (New York) 21. Springer, Berlin, 2004. Zbl1041.60005MR2020294
  9. [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. Zbl1193.60073MR2573564
  10. [10] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion. Grundlehren der Mathematischen Wissenschaften 293. Springer, Berlin, 1991. Zbl0731.60002MR1083357
  11. [11] L. Serlet. Creation or deletion of a drift on a Brownian trajectory. In Séminaire de Probabilités XLI. Lecture Notes in Math. 1934215–232. Springer, Berlin, 2008. Zbl1157.60077MR2483734
  12. [12] B. S. Tsirelson. An example of a stochastic differential equation that has no strong solution. Teor. Verojatnost. i Primenen. 20 (1975) 427–430. Zbl0353.60061MR375461

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