Alexander Erreygers

Countable-state stochastic processes with càdlàg sample paths

Alexander Erreygers & Jasper De Bock

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.