Kolmogorov type and general extension results for nonlinear expectations

TL;DR

This paper develops extension methods for nonlinear expectations, including a nonlinear Kolmogorov extension theorem, enabling the construction of nonlinear Markov processes from given transition kernels.

Contribution

It introduces new extension procedures for convex nonlinear expectations, including a nonlinear Daniell-Stone type theorem and a robust Kolmogorov extension theorem.

Findings

Established a maximal extension for convex expectations with finitely additive measures.

Provided a continuous-from-above extension for convex expectations with countably additive measures.

Constructed nonlinear Markov processes from nonlinear transition kernels.

Abstract

We provide extension procedures for nonlinear expectations to the space of all bounded measurable functions. We first discuss a maximal extension for convex expectations which have a representation in terms of finitely additive measures. One of the main results of this paper is an extension procedure for convex expectations which are continuous from above and therefore admit a representation in terms of countably additive measures. This can be seen as a nonlinear version of the Daniell-Stone theorem. From this, we deduce a robust Kolmogorov extension theorem which is then used to extend nonlinear kernels to an infinite dimensional path space. We then apply this theorem to construct nonlinear Markov processes with a given family of nonlinear transition kernels.