Supermartingales as Radon-Nikodym densities and related measure extensions
The authors compare two approaches of associating countably and finitely additive measures to supermartingales. In the countably additive case, they prove the existence of a probability measure such that a given supermartingale can be interpreted as Radon-Nikodym density of that measure. In particular, it is shown that this probability measure can be constructed on the canonical space itself and not only on the product space, even if the given supermartingale is not a local martingale. Moreover, precise necessary and sufficient conditions are provided for the uniqueness of such a probability measure associated to the given supermartingale. In the finitely additive case, the existence of a finitely additive measure associated to the given semimartingale is shown as long as the filtering probability space is sufficiently rich, supporting a Brownian motion. It is also shown that such a finitely additive measure is never unique. The argument for the existence of such finitely additive measures also yields the existence of uniformly integrable martingales that Fatou converge to a given supermartingale.
scientific article; zbMATH DE number 6541354
scientific article