Nonlinear Expectations and g-Expectations

  • Łukasz Delong
Part of the EAA Series book series (EAAS)


We investigate nonlinear expectations. We briefly discuss Choquet expectations and we focus on g-expectations defined by BSDEs. The connection between filtration-consistent nonlinear expectations and g-expectations is presented. We study the properties of translation invariance, positive homogeneity, convexity and sub-linearity of g-expectations and we show that these properties are determined by the generator of the BSDE defining the g-expectation.




  1. Chen, Z., Kulperger, R.: Minimax pricing and Choquet expectations. Insur. Math. Econ. 38, 518–528 (2006) MathSciNetMATHCrossRefGoogle Scholar
  2. Chen, Z., Chen, T., Davison, M.: Choquet expectation and Peng’s g-expectations. Ann. Probab. 33, 1179–1199 (2005) MathSciNetMATHCrossRefGoogle Scholar
  3. Choquet, G.: Theory of capacities. Ann. Inst. Fourier 5, 131–195 (1953) MathSciNetCrossRefGoogle Scholar
  4. Cohen, S.N.: Representing filtration consistent non-linear expectations as g-expectations in general probability spaces. Preprint (2011) Google Scholar
  5. Coquet, F., Hu, Y., Mémin, J., Peng, S.: Filtration-consistent non-linear expectations and related g-expectations. Probab. Theory Relat. Fields 123, 1–27 (2002) MATHCrossRefGoogle Scholar
  6. Jiang, L.: Convexity, translation invariance and subadditivity for g-expectations and related risk measures. Ann. Appl. Probab. 18, 245–258 (2008) MathSciNetMATHCrossRefGoogle Scholar
  7. Nguyen, H., Pham, U., Tran, H.: On some claims related to Choquet integral risk measures. Ann. Oper. Res. 195, 5–31 (2012) MathSciNetMATHCrossRefGoogle Scholar
  8. Peng, S.: Backward SDE and related g-expectations. In: El Karoui, N., Mazliak, L. (eds.) Backward Stochastic Differential Equations, Pitman Research Notes, pp. 141–161. Pitman, London (1997) Google Scholar
  9. Rosazza Gianin, E.: Risk measures via g-expectations. Insur. Math. Econ. 39, 19–34 (2006) MathSciNetMATHCrossRefGoogle Scholar
  10. Royer, M.: Backward stochastic differential equations with jumps and related non-linear expectations. Stoch. Process. Appl. 116, 1358–1376 (2006) MathSciNetMATHCrossRefGoogle Scholar
  11. Wang, S.: A class of distortion operators for pricing financial and insurance risks. J. Risk Insur. 1, 15–36 (2000) Google Scholar

Copyright information

© Springer-Verlag London 2013

Authors and Affiliations

  • Łukasz Delong
    • 1
  1. 1.Institute of Econometrics, Division of Probabilistic MethodsWarsaw School of EconomicsWarsawPoland

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