Doklady Mathematics

, Volume 95, Issue 2, pp 164–167 | Cite as

Banach geometry of financial market models

Mathematics
  • 32 Downloads

Abstract

Banach geometric objects imitating a phenomenon of the type of the absence of arbitrage in financial markets models are analyzed. The role played in this field by reflexive subspaces (which replace classically considered finite-dimensional subspaces) and by plasterable cones is revealed. A series of new geometric criteria for the absence of arbitrage are proved. An alternative description of the existence of a martingale measure is given, which does not use dual objects.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    F. Delbaen and W. Schachermayer, The Mathematics of Arbitrage (Springer, New York, 2006).MATHGoogle Scholar
  2. 2.
    W. Schachermayer, in Encyclopedia of Quantitative Finance (Wiley, New York, 2010), Vol. 2, pp. 792–801.Google Scholar
  3. 3.
    H. Follmer and A. Schied, Stochastic Finance: An Introduction in Discrete Time, 2nd ed. (Walter de Gruyter, Berlin, 2004).CrossRefMATHGoogle Scholar
  4. 4.
    M. A. Krasnosel’skii, Positive Solutions of Operator Equations (Nauka, Moscow, 1962) [in Russian].Google Scholar
  5. 5.
    M. A. Krasnosel’skii, E. A. Lifshits, and A.V. Sobolev, Positive Linear Systems: The Method of Positive Operators (Nauka, Moscow, 1985) [in Russian].MATHGoogle Scholar
  6. 6.
    N. Dunford and J. T. Schwartz, Linear Operators, Vol. 1: General Theory (Interscience, New York, 1958; Inostrannaya Literatura, Moscow, 1962).MATHGoogle Scholar
  7. 7.
    A. V. Lebedev and P. P. Zabreiko, Arbitrage Free Markets Geometry. http://arxiv.org/pdf/1410.4807v1.pdf. Submitted October 17, 2014.Google Scholar
  8. 8.
    F. Albiac and N. J. Kalton, Topics in Banach Space Theory (Springer, New York, 2006).MATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Mechanics and Mathematics FacultyBelarussian State UniversityMinskBelarus
  2. 2.University of BialystokBialystokPoland

Personalised recommendations