A nonstandard approach to stochastic processes under probability bounding

Matthias C. M. Troffaes
Proceedings of the Thirteenth International Symposium on Imprecise Probability: Theories and Applications, PMLR 215:450-460, 2023.

Abstract

This paper studies stochastic processes under probability bounding, using nonstandard conditional lower previsions within the framework of internal set theory. Following Nelson’s approach to stochastic processes, we introduce elementary processes which are defined over a finite number of time points and that serve to approximate any standard process, including processes over continuous time. We show that every standard process can be represented by an elementary process, and that the shadow of every elementary process constitutes again a standard process. We then move to demonstrate how elementary processes can be used to define imprecise Markov chains both in discrete and continuous time. To demonstrate the benefits and downsides of this approach, we show how to recover some basic results for continuous time Markov chains through analysis of a nonstandard elementary process.

Cite this Paper

BibTeX
@InProceedings{pmlr-v215-troffaes23a, title = {A nonstandard approach to stochastic processes under probability bounding}, author = {Troffaes, Matthias C. M.}, booktitle = {Proceedings of the Thirteenth International Symposium on Imprecise Probability: Theories and Applications}, pages = {450--460}, year = {2023}, editor = {Miranda, Enrique and Montes, Ignacio and Quaeghebeur, Erik and Vantaggi, Barbara}, volume = {215}, series = {Proceedings of Machine Learning Research}, month = {11--14 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v215/troffaes23a/troffaes23a.pdf}, url = {https://proceedings.mlr.press/v215/troffaes23a.html}, abstract = {This paper studies stochastic processes under probability bounding, using nonstandard conditional lower previsions within the framework of internal set theory. Following Nelson’s approach to stochastic processes, we introduce elementary processes which are defined over a finite number of time points and that serve to approximate any standard process, including processes over continuous time. We show that every standard process can be represented by an elementary process, and that the shadow of every elementary process constitutes again a standard process. We then move to demonstrate how elementary processes can be used to define imprecise Markov chains both in discrete and continuous time. To demonstrate the benefits and downsides of this approach, we show how to recover some basic results for continuous time Markov chains through analysis of a nonstandard elementary process.} }
Endnote
%0 Conference Paper %T A nonstandard approach to stochastic processes under probability bounding %A Matthias C. M. Troffaes %B Proceedings of the Thirteenth International Symposium on Imprecise Probability: Theories and Applications %C Proceedings of Machine Learning Research %D 2023 %E Enrique Miranda %E Ignacio Montes %E Erik Quaeghebeur %E Barbara Vantaggi %F pmlr-v215-troffaes23a %I PMLR %P 450--460 %U https://proceedings.mlr.press/v215/troffaes23a.html %V 215 %X This paper studies stochastic processes under probability bounding, using nonstandard conditional lower previsions within the framework of internal set theory. Following Nelson’s approach to stochastic processes, we introduce elementary processes which are defined over a finite number of time points and that serve to approximate any standard process, including processes over continuous time. We show that every standard process can be represented by an elementary process, and that the shadow of every elementary process constitutes again a standard process. We then move to demonstrate how elementary processes can be used to define imprecise Markov chains both in discrete and continuous time. To demonstrate the benefits and downsides of this approach, we show how to recover some basic results for continuous time Markov chains through analysis of a nonstandard elementary process.
APA
Troffaes, M.C.M.. (2023). A nonstandard approach to stochastic processes under probability bounding. Proceedings of the Thirteenth International Symposium on Imprecise Probability: Theories and Applications, in Proceedings of Machine Learning Research 215:450-460 Available from https://proceedings.mlr.press/v215/troffaes23a.html.

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