Quasi-sure analysis, aggregation and dual representations of sublinear expectations in general spaces

Samuel Cohen (University of Oxford)

Abstract


We consider coherent sublinear expectations on a measurable space, without assuming the existence of a dominating probability measure. By considering a decomposition of the space in terms of the supports of the measures representing our sublinear expectation, we give a simple construction, in a quasi-sure sense, of the (linear) conditional expectations, and hence give a representation for the conditional sublinear expectation. We also show an aggregation property holds, and give an equivalence between consistency and a pasting property of measures.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-15

Publication Date: August 6, 2012

DOI: 10.1214/EJP.v17-2224

References

  • Bion-Nadal, Jocelyne. A free Girsanov property for free Brownian motions. J. Operator Theory 55 (2006), no. 2, 373--392. MR2242856
  • Bion-Nadal, Jocelyne; Kervarec, Magali. Risk measuring under model uncertainty. Ann. Appl. Probab. 22 (2012), no. 1, 213--238. MR2932546
  • Samuel N. Cohen, Shaolin Ji, and Shige Peng, Nonlinear expectations and martingales in discrete time, arXiv:1104.5390v1, 2011.
  • Delbaen, Freddy. Coherent risk measures on general probability spaces. Advances in finance and stochastics, 1--37, Springer, Berlin, 2002. MR1929369
  • Delbaen, Freddy. The structure of m-stable sets and in particular of the set of risk neutral measures. In memoriam Paul-André Meyer: Séminaire de Probabilités XXXIX, 215--258, Lecture Notes in Math., 1874, Springer, Berlin, 2006. MR2276899
  • Denis, Laurent; Hu, Mingshang; Peng, Shige. Function spaces and capacity related to a sublinear expectation: application to $G$-Brownian motion paths. Potential Anal. 34 (2011), no. 2, 139--161. MR2754968
  • Detlefsen, Kai; Scandolo, Giacomo. Conditional and dynamic convex risk measures. Finance Stoch. 9 (2005), no. 4, 539--561. MR2212894
  • Föllmer, Hans; Schied, Alexander. Stochastic finance. An introduction in discrete time. de Gruyter Studies in Mathematics, 27. Walter de Gruyter & Co., Berlin, 2002. x+422 pp. ISBN: 3-11-017119-8 MR1925197
  • Jacod, Jean; Shiryaev, Albert N. Limit theorems for stochastic processes. Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 288. Springer-Verlag, Berlin, 2003. xx+661 pp. ISBN: 3-540-43932-3 MR1943877
  • Karandikar, Rajeeva L. On pathwise stochastic integration. Stochastic Process. Appl. 57 (1995), no. 1, 11--18. MR1327950
  • Frank Knight, Risk, uncertainty and profit, Houghton Mifflin, 1921.
  • Terry J. Lyons, Uncertain volatility and the risk-free synthesis of derivatives, Applied Mathematical Finance 2 (1995), no. 2, 117--133.
  • Marcel Nutz, Pathwise construction of stochastic integrals, arXiv:1108.2981v1, 2011.
  • Marcel Nutz and H. Mete Soner, Superhedging and dynamic risk measures under volatility uncertainty, arxiv:1011.2958v1, 2010.
  • Peng, Shige. Backward stochastic differential equation, nonlinear expectation and their applications. Proceedings of the International Congress of Mathematicians. Volume I, 393--432, Hindustan Book Agency, New Delhi, 2010. MR2827899
  • Soner, H. Mete; Touzi, Nizar; Zhang, Jianfeng. Martingale representation theorem for the $G$-expectation. Stochastic Process. Appl. 121 (2011), no. 2, 265--287. MR2746175
  • Soner, H. Mete; Touzi, Nizar; Zhang, Jianfeng. Quasi-sure stochastic analysis through aggregation. Electron. J. Probab. 16 (2011), no. 67, 1844--1879. MR2842089
Bookmark and Share


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.