Geodesic
Set-Indexed Stochastic Processes and Predictability
Teoriâ veroâtnostej i ee primeneniâ, Tome 37 (1992) no. 1, pp. 57-63 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
@article{TVP_1992_37_1_a8,
     author = {B. G. Ivanova and E. Merzach},
     title = {Set-Indexed {Stochastic} {Processes} and {Predictability}},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {57--63},
     year = {1992},
     volume = {37},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1992_37_1_a8/}
}
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B. G. Ivanova; E. Merzach. Set-Indexed Stochastic Processes and Predictability. Teoriâ veroâtnostej i ee primeneniâ, Tome 37 (1992) no. 1, pp. 57-63. http://geodesic.mathdoc.fr/item/TVP_1992_37_1_a8/