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A note on the Doob-Meyer-decomposition of Lp-valued submartingales

Published online by Cambridge University Press:  17 April 2009

Bernhard Burgstaller
Bernhard Burgstaller
Affiliation:
Department of Financial Mathematics, Institute of Analysis, University of Linz, Altenberger Strasse 69, A-4040 Linz, Austria e-mail: bernhardburgstaller@yahoo.de
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Let p > 1 real. We Doob-Meyer-decompose Lp(ℙ)-valued positive submartingales such that the martingale and predictable parts are also in Lp(ℙ). We give two variants of such a decomposition. The first one handles also not necessarily right continuous submartingales, since its proof is as discrete in its nature as Doob's archaically decomposition. The second decomposition acts in Lp (ℝ × Ω ℬ ⊗ ℱ, μ ⊗ ℙ) for some finite measure μ on ℝ.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 69 , Issue 2 , April 2004 , pp. 227 - 235
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Burgstaller, B., ‘Doob-Meyer-decomposition of Hilbert space valued functions’, J. Theoret. Probab. 16 (2003).CrossRefGoogle Scholar
[2]Doléans, C., ‘Existence du processus croissant naturel associé à un potential de la classe (D)’, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 9 (1968), 309–314.CrossRefGoogle Scholar
[3]Doob, J.L.. Stochastic processes (J. Wiley & Son, New York, 1953).Google Scholar
[4]Garsia, A.M., Martingale inequalities: Seminar notes on recent progress (W.A. Benjamin, Reading, MA, 1973).Google Scholar
[5]Kallenberg, O.Foundations of modern probability (Springer-Verlag, Berlin, Heidelberg, New York, 1997).Google Scholar
[6]Meyer, P.A., ‘A decomposition theorem for supermartingales’, Illinois J. Math. 6 (1962), 193–205.CrossRefGoogle Scholar
[7]Neveu, J., Discrete parameter martingales, (English translation) (North-Holland, Amsterdam, 1972–1975).Google Scholar
[8]Rao, K.M., ‘On decomposition theorems of Meyer’, Math. Scand. 24 (1969), 66–78.CrossRefGoogle Scholar