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We prove results on the existence of Doléans-Dade measures and of the Doob-Meyer decomposition for supermartingales indexed by a general index set.We prove, in Theorem 1, a necessary and sufficient condition, the class D 0 property, for the existence of a Doléans-Dade measure associated to a supermartingale. Based on this, we establish, in Theorem 2, a sufficient condition, the class D * property, for the existence of a Doob Meyer decomposition. In Corollary 1 we consider supermartingales of uniformly integrable variation. The two key properties involved in our results are the possibility of extending the original supermartingale to a larger filtration and some form of the optional sampling theorem. From a mathematical perspective we exploit extensively results from the theory of finitely additive measures.

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Cited by 2 publications

References 5 publications

The set-indexed Ito integral

The set-indexed Ito integral

Supermartingale decomposition with a general index set

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