A Notion of Stopping Line for Set-Indexed Processes

Abstract

In this paper, we introduce the class of $$\mathcal {A}$$ -stopping lines which generalize the planar stopping lines in Merzbach [(1980), Stochastic Process. Appl. 10, 49–63] by replacing the positive quadrant of the plane by a collection $$\mathcal {A}$$ of compact subsets of a fixed topological space. Our notion of stopping line also compliments and generalizes the stopping sets defined in Ivanoff and Merzbach [(1995), Stochastic Process. Appl. 57, 83–98].