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An introduction to the general theory of set-indexed martingales. (English) Zbl 1026.60066

Merzbach, Ely (ed.), Topics in spatial stochastic processes. Lectures given at the C.I.M.E. summer school, Martina Franca, Italy, July 1-8, 2001. Berlin: Springer. Lect. Notes Math. 1802, 41-84 (2003).
This work contains the general first steps towards the theory of martingales indexed by a class of sets. The index set used throughout the paper is a class of compact connected subsets of a locally compact topological space. The paper is organized as follows. Section 2 contains the framework and definitions. Section 3 is devoted to one of the key tools which will be used subsequently: the concept of stopping. In Section 4, the fundamental concept of predictability is studied, and two different predictable \(\sigma\)-algebras are given which depend on the definition of the “history” of a set. The special Section 5 is devoted to the notion of announcability. In contrast to the classical one-parameter case, in the general context predictability and announcability are not equivalent. Martingales are defined and studied in Section 6. There are three different types of martingales: strong, weak and simple martingales. Section 7 is devoted to stopping theorems. The different results correspond to the different types of martingales and the role of stopping sets becomes clear. Section 8 deals with local martingales. Various concepts of set-indexed quasimartingales are defined in Section 9. In Section 10, the compensator of a submartingale and the quadratic variation of a square-integrable martingale are introduced. Sections 11 and 12 are devoted to Doob-Meyer decomposition for strong submartingales and submartingales. The existence of a quadratic variation process for both martingales and strong martingales is proved.
For the entire collection see [Zbl 1005.00040].

MSC:

60G60 Random fields
60G40 Stopping times; optimal stopping problems; gambling theory
60G48 Generalizations of martingales
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