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Föllmer-Schweizer decomposition and closedness of \(G_ T(\Theta)\). (Décomposition de Föllmer-Schweizer et fermeture de \(G_ T(\Theta)\).) (French) Zbl 0797.60040

Summary: Let \((X_ t)_{0\leq t\leq T}\) be a special semimartingale on a filtered probability space and \(\Theta\) the space of all predictable, \(X\)- integrable processes \(\theta\) such that \(\int \theta dX\) is in the space \(S^ 2\) of semimartingales. If \(H\in {\mathcal L}^ 2\), we prove, under additional assumptions on \(X\), that \(H\) admits a unique Föllmer- Schweizer decomposition. Moreover, we prove that this decomposition is continuous. Finally, we deduce from this property the closedness of the subspaces of \({\mathcal L}^ 2\), \(G_ T(\Theta):= \Bigl\{\int^ T_ 0 \theta_ s dX_ s\mid \theta\in \Theta\Bigr\}\) and \(\{c+ G_ T(\Theta)\mid c\in\mathbb{R}\}\).

MSC:

60G44 Martingales with continuous parameter
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