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arXiv · 1511.08726

Kolmogorov type and general extension results for nonlinear expectations

Robert Denk·Michael Kupper·Max Nendel
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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.

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Robert Denk, Michael Kupper, Max Nendel. 2017-06-20. Kolmogorov type and general extension results for nonlinear expectations. https://doi.org/10.1215/17358787-2017-0024

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