|
Teoriya Veroyatnostei i ee Primeneniya, 1992, Volume 37, Issue 1, Pages 57–63
(Mi tvp4124)
|
|
|
|
Set-Indexed Stochastic Processes and Predictability
B. G. Ivanova, E. Merzach University of Ottawa
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.
Citation:
B. G. Ivanova, E. Merzach, “Set-Indexed Stochastic Processes and Predictability”, Teor. Veroyatnost. i Primenen., 37:1 (1992), 57–63; Theory Probab. Appl., 37:1 (1993), 61–66
Linking options:
https://www.mathnet.ru/eng/tvp4124 https://www.mathnet.ru/eng/tvp/v37/i1/p57
|
Statistics & downloads: |
Abstract page: | 155 | Full-text PDF : | 50 | First page: | 5 |
|