Skip to main content
SpringerLink
Log in

A simple characterization of tightness for convex solid sets of positive random variables

  • Published:
Positivity Aims and scope Submit manuscript

Abstract

We show that for a convex solid set of positive random variables to be tight, or equivalently bounded in probability, it is necessary and sufficient to be is radially bounded, i.e. that every ray passing through one of its elements eventually leaves the set. The result is motivated by problems arising in mathematical finance.

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

Notes

  1. We thank an anonymous referee for pointing this out.

References

  1. Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis: A Hitchhiker’s Guide, 3rd edn. Springer, Berlin (2006)

    MATH  Google Scholar 

  2. Bogachev, V.I.: Measure Theory, vol. 1. Springer, Berlin (2007)

    Book  MATH  Google Scholar 

  3. Brannath, W., Schachermayer, W.: A bipolar theorem for \(L_+^0(\Omega ,\cal{F},{\mathbb{P}})\). In: Séminaire de Probabilités XXXIII, pp. 349–354. Springer, Berlin (1999)

  4. Delbaen, F., Schachermayer, W.: A general version of the fundamental theorem of asset pricing. Mathematische Annalen. 300, 463–520 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  5. Föllmer, H., Schied, A.: Stochastic Finance: An Introduction in Discrete Time, 3rd edn. Walter de Gruyter, Berlin (2011)

    Book  MATH  Google Scholar 

  6. Kardaras, C.: A structural characterization of numeraires of convex sets of nonnegative random variables. Positivity 16(2), 245–253 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Kardaras, C.: Uniform integrability and local convexity in \(L^0\). J. Funct. Anal. 266, 1913–1927 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Kardaras, C.: Maximality and numeraires in convex sets of nonnegative random variables. J. Funct. Anal. 268, 3219–3231 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kardaras, C., Žitković, G.: Forward-convex convergence of sequences of nonnegative random variables. Proc. Am. Math. Soc. 141, 919–929 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kramkov, D., Schachermayer, W.: The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Annals of Applied Probability. 904–950 (1999)

  11. Kupper, M., Svindland, G.: Dual representation of monotone convex functions on \(L^0\). Proc. Am. Math. Soc. 139(11), 4073–4086 (2011)

    Article  MATH  Google Scholar 

  12. Žitković, G.: Convex-compactness and its applications. Math. Financ. Econ. 3(1), 1–12 (2009)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Center for Finance and Insurance, University of Zurich, Zurich, Switzerland

    Pablo Koch-Medina, Cosimo Munari & Mario Šikić

Authors
  1. Pablo Koch-Medina
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Cosimo Munari
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mario Šikić
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Cosimo Munari.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Koch-Medina, P., Munari, C. & Šikić, M. A simple characterization of tightness for convex solid sets of positive random variables. Positivity 22, 1015–1022 (2018). https://doi.org/10.1007/s11117-018-0556-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11117-018-0556-7

Keywords

Mathematics Subject Classification

Search

Navigation