Supermartingale decomposition with a general index set

Abstract

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.

Sure wins, separating probabilities and the representation of linear functionals

“…is coherent in the sense of Corollary 1 and if (L τ : τ ∈ T ) is increasing in τ then necessarily φ υ |L τ ≥ φ τ whenever τ, υ ∈ T and τ ≤ υ. This conclusion has a direct application to the theory of finitely additive supermartingales, treated in [6].…”