F. Delbaen, W. Schachermayer
Proceedings of the Seminar of Stochastic Analysis, Random Fields and Applications, Progress in Probability, Vol. 45 (1999), pp. 137-173.
For $\Cal H^1$ bounded sequences, we introduce a technique, related to the Kadec-Pelczynski-decomposition for $L^1$ sequences, that allows us to prove compactness theorems. Roughly speaking, a bounded sequence in $\Cal H^1$ can be split into two sequences, one of which is weakly compact, the other forms the singular part. If the martingales are continuous then the singular part tends to zero in the semi-martingale topology. In the general case the singular parts give rise to a process of bounded variation. The technique allows to give a new proof of the optional decomposition theorem in Mathematical Finance.
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