Abstract
A filtration on a probability space is said to be Brownian when it is generated by some Brownian motion started from 0. Recognising whether a given filtration F = (F t ) t≥0 is Brownian may be a difficult problem; but when F is Brownian after zero, a necessary and sufficient condition for F to be Brownian is available, namely, the self-coupling property (ii) of Theorem 1 of [4]. (‘Brownian after zero’ means that for each ɛ > 0, the shifted filtration F ɛ = (F ɛ+t ) t≥0 is generated by its initial σ-field F ɛ and by some F ɛ-Brownian motion.) In all concrete examples where this self-coupling criterion has been used to establish Brownianity, another, more constructive proof was also available. The situation presented below is different. We are interested in a certain process, introduced in 1991 by Beneš, Karatzas and Rishel; the natural filtration of this process turns out to be also generated by some Brownian motion, but we have not been able to exhibit such a generating Brownian motion; the general, non constructive criterion is the only proof we know that this filtration is indeed Brownian.
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References
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Émery, M. (2009). Recognising Whether a Filtration is Brownian: a Case Study. In: Donati-Martin, C., Émery, M., Rouault, A., Stricker, C. (eds) Séminaire de Probabilités XLII. Lecture Notes in Mathematics(), vol 1979. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01763-6_14
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DOI: https://doi.org/10.1007/978-3-642-01763-6_14
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