Predictability and stopping on lattices of sets

Abstract

As a first step in the development of a general theory of set-indexed martingales, we define predictability on a general space with respect to a filtration indexed by a lattice of sets. We prove a characterization of the predictable σ-algebra in terms of adapted and “left-continuous” processes without any form of topology for the index set. We then define a stopping set and show that it is a natural generalization of the stopping time; in particular, the predictable σ-algebra can be characterized by various stochastic intervals generated by stopping sets.