The non-closeness and non-denseness of \(H^{\infty}\) in BMO. (English) Zbl 0605.60048
We construct a reference system \((\Omega^ 0,F^ 0,(F_ t^ 0)_{t\geq 0},P^ 0)\), on which the martingale space \(H^{\infty}\) is not closed and not dense in the martingale space BMO.
To find a suitable closed sub-space or a suitable dense sub-space for the BMO, especially the latter is valuable. It has been proved by C. Dellacherie, P. A. Meyer and M. Yor [Lect. Notes Math. 649, 98-113 (1978; Zbl 0392.60009), theorem 10] that, if \(L^{\infty}\neq BMO\), then \(L^{\infty}\) is not closed and not dense in BMO. Hence \(L^{\infty}\) is not a suitable closed sub-space or a suitable dense sub-space of BMO. As for \(H^{\infty}\), there is a conjecture in the quoted paper: ”... et il est possible qu’il \((H^{\infty})\) soit toujours dense dans BMO”. But our example gives a negative answer to it.
To find a suitable closed sub-space or a suitable dense sub-space for the BMO, especially the latter is valuable. It has been proved by C. Dellacherie, P. A. Meyer and M. Yor [Lect. Notes Math. 649, 98-113 (1978; Zbl 0392.60009), theorem 10] that, if \(L^{\infty}\neq BMO\), then \(L^{\infty}\) is not closed and not dense in BMO. Hence \(L^{\infty}\) is not a suitable closed sub-space or a suitable dense sub-space of BMO. As for \(H^{\infty}\), there is a conjecture in the quoted paper: ”... et il est possible qu’il \((H^{\infty})\) soit toujours dense dans BMO”. But our example gives a negative answer to it.
MSC:
| 60G44 | Martingales with continuous parameter |
| 60G46 | Martingales and classical analysis |
| 46J15 | Banach algebras of differentiable or analytic functions, \(H^p\)-spaces |
Citations:
Zbl 0392.60009References:
| [1] | Dellacherie, C., Meyer, P. A., Yor, M., Sur certaines propriétés des espaces de BanachH 1 et O.Sém. Probab. XII,Lect. Notes in Math., 649, 1978. |
| [2] | Yan Jia-an, An Introduction to the Theory of Martingales and Stochastic Integrals,Shanghai Science and Technical Press, 1981. (in Chinese) |
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