Supermartingale Decomposition with a General Index Set

By Doob’s theorem, supermartingales indexed by the natural numbers decompose into the difference of a uniformly integrable martingale and an increasing process. The relative ease of working with increasing processes rather than supermartingales explains the prominent role of this result in stochastic analysis and in the theory of stochastic integration. Meyer [19] then proved that, under the usual conditions, Doob’s decomposition exists for right continuous supermartingales indexed by the positive reals if and only if the class D property is satisfied. Doléans-Dade [8] was the first to represent supermartingales as measures over predictable rectangles and to prove that a supermartingale is of class D if and only if its Doléans-Dade measure is countably additive. This line of approach has then become dominant in the work of authors such as Föllmer [12] and Metivier and Pellaumail [18]. In this paper we cosider the case of processes indexed by general index sets, illustrated in the following