Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Hiding a constant drift

Vilmos Prokaj, Miklós Rásonyi, and Walter Schachermayer

Full-text: Open access

Abstract

The following question is due to Marc Yor: Let B be a Brownian motion and St=t+Bt. Can we define an $\mathcal{F}^{B}$-predictable process H such that the resulting stochastic integral (HS) is a Brownian motion (without drift) in its own filtration, i.e. an $\mathcal{F}^{(H\cdot S)}$-Brownian motion?

In this paper we show that by dropping the requirement of $\mathcal{F}^{B}$-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.

Résumé

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 $\mathcal{F}^{B}$-prévisible tel que l’intégrale stochastique (HS) soit un mouvement Brownien (sans drift) pour sa propre filtration $\mathcal{F}^{(H\cdot S)}$?

Dans cet article nous fournissons une réponse affirmative en relâchant la condition que H soit $\mathcal{F}^{B}$-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.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist. Volume 47, Number 2 (2011), 498-514.

Dates
First available in Project Euclid: 23 March 2011

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1300887279

Digital Object Identifier
doi:10.1214/10-AIHP363

Mathematical Reviews number (MathSciNet)
MR2814420

Zentralblatt MATH identifier
1216.60048

Subjects
Primary: 60H05: Stochastic integrals 60G44: Martingales with continuous parameter 60J65: Brownian motion [See also 58J65]
Secondary: 60G05: Foundations of stochastic processes 60H10: Stochastic ordinary differential equations [See also 34F05]

Keywords
Brownian motion with drift Stochastic integral Enlargement of filtration

Citation

Prokaj, Vilmos; Rásonyi, Miklós; Schachermayer, Walter. Hiding a constant drift. Ann. Inst. H. Poincaré Probab. Statist. 47 (2011), no. 2, 498--514. doi:10.1214/10-AIHP363. https://projecteuclid.org/euclid.aihp/1300887279


Export citation

References

  • [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.
  • [2] R. F. Bass and K. Burdzy. Stochastic bifurcation models. Ann. Probab. 27 (1999) 50–108.
  • [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.
  • [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.
  • [5] R. Mansuy and M. Yor. Random Times and Enlargements of Filtrations in a Brownian Setting. Lecture Notes in Mathematics 1873. Springer, Berlin, 2006.
  • [6] H. P. McKean, Jr. Stochastic Integrals. Probability and Mathematical Statistics 5. Academic Press, New York, 1969.
  • [7] V. Prokaj. Unfolding the Skorohod reflection of a semimartingale. Statist. Probab. Lett. 79 (2009) 534–536.
  • [8] P. E. Protter. Stochastic Integration and Differential Equations, 2nd edition. Applications of Mathematics (New York) 21. Springer, Berlin, 2004.
  • [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.
  • [10] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion. Grundlehren der Mathematischen Wissenschaften 293. Springer, Berlin, 1991.
  • [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.
  • [12] B. S. Tsirelson. An example of a stochastic differential equation that has no strong solution. Teor. Verojatnost. i Primenen. 20 (1975) 427–430.