Countable-state stochastic processes with c\`adl\`ag sample paths
Abstract
The Daniell-Kolmogorov Extension Theorem is a fundamental result in the theory of stochastic processes, as it allows one to construct a stochastic process with prescribed finite-dimensional distributions. However, it is well-known that the domain of the constructed probability measure - the product sigma-algebra in the set of all paths - is not sufficiently rich. This problem is usually dealt with through a modification of the stochastic process, essentially changing the sample paths so that they become c\`adl\`ag. Assuming a countable state space, we provide an alternative version of the Daniell-Kolmogorov Extension Theorem that does not suffer from this problem, in that the domain is sufficiently rich and we do not need a subsequent modification step: we assume a rather weak regularity condition on the finite-dimensional distributions, and directly obtain a probability measure on the product sigma-algebra in the set of all c\`adl\`ag paths.
Keywords
Cite
@article{arxiv.2301.07992,
title = {Countable-state stochastic processes with c\`adl\`ag sample paths},
author = {Alexander Erreygers and Jasper De Bock},
journal= {arXiv preprint arXiv:2301.07992},
year = {2023}
}