Skip to main content
SpringerLink
Log in

The minimal sublinear expectations and their related properties

  • Published:
Science in China Series A: Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we prove that for a sublinear expectation ɛ[·] defined on L 2(Ω,\( \mathcal{F} \)), the following statements are equivalent:

  1. (i)

    ɛ is a minimal member of the set of all sublinear expectations defined on L 2(Ω,\( \mathcal{F} \))

  2. (ii)

    ɛ is linear

  3. (iii)

    the two-dimensional Jensen’s inequality for ɛ holds.

Furthermore, we prove a sandwich theorem for subadditive expectation and superadditive expectation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Coquet F, Hu Y, Mémin J, et al. Filtration-consistent nonlinear expectations and related g-expectations. Probab Theory Related Fields, 123(1): 1–27 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  2. Peng S. BSDE and related g-expectation. In: El Karoui N, Mazliak L, eds. Pitman Research Notes in Mathematics Series, No 364, “Backward Stochastic Differential Equations”. 1997, 141–159

  3. Denneberg D. Non-additive Measure and Integration. Dodrecht: Kluwer, 1994

    Google Scholar 

  4. Artzner P, Delbaen F, Eber J, et al. Coherent measures of risk. Math Finance, 9(3): 203–228 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  5. Peng S. Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation. Stochastic Processes Appl, 118(12): 2223–2253 (2008)

    Article  MATH  Google Scholar 

  6. Peng S. Nonlinear expectations, nonlinear evaluations and risk measures. In: Stochastic methods in finance. Lecture Notes in Math, 1856. Berlin: Springer-Verlag, 2004, 165–253

    Google Scholar 

  7. Song Y, Yan J. The representations of two types of functionals on L ∞(Ω,\( \mathcal{F} \)) and L ∞(Ω,\( \mathcal{F} \)). Sci China Ser A, 49(10): 1376–1382 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  8. Yosida K. Functional Analysis. Berlin, Heidelberg, New York: Spriger-Verlag, 1980

    MATH  Google Scholar 

  9. Fuchsteiner B, König H. New versions of the Hahn-Banach theorem. In: Beckenbah E, ed. General Inequalities 2. Basel: Birkhäuser Verlag, 1978, 255–266

    Google Scholar 

  10. Hu Y. On Jensen’s inequality for g-expectation and for nonlinear expectation. Arch Math (Basel), 85: 572–580 (2005)

    MATH  MathSciNet  Google Scholar 

  11. Jia G. On n-dimensional Jensen inequalities for g-expectation and nonlinear expectation, and their Applications. Preprint, 2007

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, Shandong University, Jinan, 250100, China

    GuangYan Jia

Authors
  1. GuangYan Jia
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to GuangYan Jia.

Additional information

This work was supported by National Basic Research Program of China (973 Program) (Grant No. 2007CB814901) (Financial Risk) and National Natural Science Foundation of China (Grant No. 10671111)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jia, G. The minimal sublinear expectations and their related properties. Sci. China Ser. A-Math. 52, 785–793 (2009). https://doi.org/10.1007/s11425-008-0164-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-008-0164-2

Keywords

MSC(2000)

Search

Navigation