Set-Indexed Stochastic Processes and Predictability

Abstract

The objective of this work is to propose a beginning of a general theory for stochastic processes indexed by a family of subsets of a topological space. The following concepts will be defined and intensively studied: Random sets, stopping sets, announceable stopping sets, and $\sigma $-fields associated with stopping sets. These notions lead to the study of the predictable $\sigma $-field and its different characterizations. Different kinds of martingales will be defined, as well as some extensions (quasi-martingales, local martingales) and the optional sampling theorems will be discussed. A Doob–Meyer decomposition will be given. The notions of stochastic integral and local time will be introduced. The Markov properties for set-indexed processes will be discussed in our context such as the germ-field property and the strong Markov property. We will study convergence theorems for sequences of set-indexed processes. Finally, the results obtained will be applied to Gaussian fields and to jump processes.