2023/01/19 by Alexander Erreygers, Erreygers, Alexander, Jasper De Bock +1
Economics, Econometrics and Finance · Physics and Astronomy · #60G05 #60G17 #60G30 #60J27 #60J75 #FOS: Mathematics #Probability (math.PR) #Quantum Mechanics and Applications #Statistical Mechanics and Entropy #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2301.07992
openalex publication_date 2023/01/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 càdlàg. 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 càdlàg paths.