Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Hiding a constant drift

Vilmos Prokaj, Miklós Rásonyi, and Walter Schachermayer

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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 $\mathcal{F}^{B}$-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.


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

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

First available in Project Euclid: 23 March 2011

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Zentralblatt MATH identifier

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]

Brownian motion with drift Stochastic integral Enlargement of filtration


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.

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