Banach Journal of Mathematical Analysis

Kolmogorov-type and general extension results for nonlinear expectations

Robert Denk, Michael Kupper, and Max Nendel

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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 article 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.

Article information

Banach J. Math. Anal., Volume 12, Number 3 (2018), 515-540.

Received: 21 February 2017
Accepted: 19 June 2017
First available in Project Euclid: 17 November 2017

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Zentralblatt MATH identifier

Primary: 28C05: Integration theory via linear functionals (Radon measures, Daniell integrals, etc.), representing set functions and measures
Secondary: 47H07: Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces 46A55: Convex sets in topological linear spaces; Choquet theory [See also 52A07] 46A20: Duality theory 28A12: Contents, measures, outer measures, capacities

nonlinear expectations extension results Kolmogorov’s extension theorem nonlinear kernels


Denk, Robert; Kupper, Michael; Nendel, Max. Kolmogorov-type and general extension results for nonlinear expectations. Banach J. Math. Anal. 12 (2018), no. 3, 515--540. doi:10.1215/17358787-2017-0024.

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