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

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.

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.

DOI : https://doi.org/10.1214/10-AIHP363
Classification:  60H05,  60G44,  60J65,  60G05,  60H10
Keywords: 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},
     publisher = {Gauthier-Villars},
     volume = {47},
     number = {2},
     year = {2011},
     pages = {498-514},
     doi = {10.1214/10-AIHP363},
     zbl = {1216.60048},
     mrnumber = {2814420},
     language = {en},
     url = {http://www.numdam.org/item/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, Volume 47 (2011) no. 2, pp. 498-514. doi : 10.1214/10-AIHP363. http://www.numdam.org/item/AIHPB_2011__47_2_498_0/

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