## Banach Journal of Mathematical Analysis

### Kolmogorov-type and general extension results for nonlinear expectations

#### 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 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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1510909221

Digital Object Identifier
doi:10.1215/17358787-2017-0024

Mathematical Reviews number (MathSciNet)
MR3824739

Zentralblatt MATH identifier
06946069

#### Citation

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. https://projecteuclid.org/euclid.bjma/1510909221

#### References

• [1] P. Artzner, F. Delbaen, J.-M. Eber, and D. Heath, Coherent measures of risk, Math. Finance 9 (1999), no. 3, 203–228.
• [2] D. Bartl, Pointwise dual representation of dynamic convex expectations, preprint, arXiv:1612.09103v1 [math.PR].
• [3] D. Bierlein, Über die Fortsetzung von Wahrscheinlichkeitsfeldern, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 1 (1962/1963), 28–46.
• [4] V. I. Bogachev, Measure Theory, Vol. I, Springer, Berlin, 2007; Vol. 2, Springer, Berlin, 2007.
• [5] S. Cerreia-Vioglio, F. Maccheroni, M. Marinacci, and A. Rustichini, Niveloids and their extensions: Risk measures on small domains, J. Math. Anal. Appl. 413 (2014), no. 1, 343–360.
• [6] P. Cheridito, F. Delbaen, and M. Kupper, Dynamic monetary risk measures for bounded discrete-time processes, Electron. J. Probab. 11 (2006), no. 3, 57–106.
• [7] P. Cheridito, M. Kupper, and L. Tangpi, Representation of increasing convex functionals with countably additive measures, preprint, arXiv:1502.05763v1 [math.FA].
• [8] P. Cheridito, M. Kupper, and L. Tangpi, Duality formulas for robust pricing and hedging in discrete time, SIAM J. Financial Math. 8 (2017), no. 1, 738–765.
• [9] P. Cheridito, H. M. Soner, N. Touzi, and N. Victoir, Second-order backward stochastic differential equations and fully nonlinear parabolic PDEs, Comm. Pure Appl. Math. 60 (2007), no. 7, 1081–1110.
• [10] G. Choquet, Forme abstraite du théorème de capacitabilité, Ann. Inst. Fourier (Grenoble) 9 (1959), 83–89.
• [11] F. Coquet, Y. Hu, J. Mémin, and S. Peng, Filtration-consistent nonlinear expectations and related $g$-expectations, Probab. Theory Related Fields 123 (2002), no. 1, 1–27.
• [12] F. Delbaen, Coherent Risk Measures, Cattedra Galileiana, Scuola Normale Superiore, Classe di Scienze, Pisa, 2000.
• [13] F. Delbaen, “Coherent risk measures on general probability spaces” in Advances in Finance and Stochastics, Springer, Berlin, 2002, 1–37.
• [14] F. Delbaen, “The structure of m-stable sets and in particular of the set of risk neutral measures” in In Memoriam Paul-André Meyer: Séminaire de Probabilités XXXIX, Lecture Notes in Math. 1874, Springer, Berlin, 2006, 215–258.
• [15] F. Delbaen, S. Peng, and E. Rosazza Gianin, Representation of the penalty term of dynamic concave utilities, Finance Stoch. 14 (2010), no. 3, 449–472.
• [16] C. Dellacherie and P.-A. Meyer, Probabilities and Potential, North-Holland Math. Stud. 29, North-Holland, Amsterdam, 1978.
• [17] L. Denis, M. Hu, and S. Peng, Function spaces and capacity related to a sublinear expectation: Application to $G$-Brownian motion paths, Potential Anal. 34 (2011), no. 2, 139–161.
• [18] Y. Dolinsky, M. Nutz, and H. M. Soner, Weak approximation of $G$-expectations, Stochastic Process. Appl. 122 (2012), no. 2, 664–675.
• [19] N. Dunford and J. T. Schwartz, Linear Operators, I: General Theory, with the assistance of W. G. Bade and R. G. Bartle, reprint of the 1958 original, Wiley Classics Library, Wiley, New York, 1988.
• [20] K. Fan, Minimax theorems, Proc. Natl. Acad. Sci. USA 39 (1953), 42–47.
• [21] W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, 2nd ed., Stoch. Model. Appl. Probab. 25, Springer, New York, 2006.
• [22] H. Föllmer and C. Klüppelberg, “Spatial risk measures: Local specification and boundary risk” in Stochastic Analysis and Applications 2014, Springer Proc. Math. Stat. 100, Springer, Cham, 2014, 307–326.
• [23] H. Föllmer and I. Penner, Convex risk measures and the dynamics of their penalty functions, Statist. Decisions 24 (2006), no. 1, 61–96.
• [24] H. Föllmer and A. Schied, Stochastic Finance: An Introduction in Discrete Time, 3rd revised and extended ed., de Gruyter, Berlin, 2011.
• [25] D. J. Hartfiel, Markov Set-Chains, Lecture Notes in Math. 1695, Springer, Berlin, 1998.
• [26] S. Peng, Nonlinear expectations and nonlinear Markov chains, Chinese Ann. Math. Ser. B 26 (2005), no. 2, 159–184.
• [27] S. Peng, “$G$-expectation, $G$-Brownian motion and related stochastic calculus of Itô type” in Stochastic Analysis and Applications, Abel Symp 2, Springer, Berlin, 2007, 541–567.
• [28] S. Peng, Multi-dimensional $G$-Brownian motion and related stochastic calculus under $G$-expectation, Stochastic Process. Appl. 118 (2008), no. 12, 2223–2253.
• [29] H. M. Soner, N. Touzi, and J. Zhang, Martingale representation theorem for the $G$-expectation, Stochastic Process. Appl. 121 (2011), no. 2, 265–287.
• [30] H. M. Soner, N. Touzi, and J. Zhang, Quasi-sure stochastic analysis through aggregation, Electron. J. Probab. 16 (2011), no. 67, 1844–1879.
• [31] B. Z. Vulikh, Introduction to the Theory of Partially Ordered Spaces, Wolters-Noordhoff Sci., Groningen, 1967.
• [32] J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Appl. Math. 43, Springer, New York, 1999.