Journal of Theoretical Probability

, Volume 15, Issue 1, pp 41–61 | Cite as

A Filtered Version of the Bipolar Theorem of Brannath and Schachermayer

  • Gordan Žitković


We extend the Bipolar Theorem of Kramkov and Schachermayer(12) to the space of nonnegative càdlàg supermartingales on a filtered probability space. We formulate the notion of fork-convexity as an analogue to convexity in this setting. As an intermediate step in the proof of our main result we establish a conditional version of the Bipolar theorem. In an application to mathematical finance we describe the structure of the set of dual processes of the utility maximization problem of Kramkov and Schachermayer(12) and give a budget-constraint characterization of admissible consumption processes in an incomplete semimartingale market.

bipolar theorem stochastic processes positive supermartingales duality mathematical finance 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Brannath, W., and Schachermayer, W. (1999). A bipolar theorem for subsets of L +0(Ω,ℱ, ℘). Séminaire de Probabilités XXXIII, 349-354.Google Scholar
  2. 2.
    Delbaen, F., and Schachermayer, W. (1998). The fundamental theorem of asset pricing for unbounded stochastic processes. Mathematische Annalen 312, 215-250.Google Scholar
  3. 3.
    Delbean, F., and Schachermayer, W. (1993). A general version of the fundamental theorem of asset pricing. Mathematische Annalen 300, 463-520.Google Scholar
  4. 4.
    Delbean, F., and Schachermayer, W. (1999). A compactness principle for bounded sequences of martingales with applications. In Proceedings of the Seminar of Stochastic Analysis, Random Fields and Applications.Google Scholar
  5. 5.
    El Karoui, N., and Quenez, M. C. (1995). Dynamic programming and pricing of contingent claims in an incomplete market. SIAM Journal on Control and Optimization 33/1, 29-66.Google Scholar
  6. 6.
    Föllmer, H., and Kabanov, Y. M. (1998). Optional decomposition and lagrange multipliers. Finance and Stochastics 2, 69-81.Google Scholar
  7. 7.
    Föllmer, H., and Kramkov, D. (1997). Optional decomposition under constraints. Probability Theory and Related Fields 109, 1-25.Google Scholar
  8. 8.
    Kalton, S., Peck, N. T., and Roberts, J. W. (1984). An F-space sampler. London Math. Soc. Lecture Notes 89.Google Scholar
  9. 9.
    Karatzas, I., and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus. Springer-Verlag, NewYork, 2nd edn..Google Scholar
  10. 10.
    Karatzas, I., and Shreve, S. E. (1998). Methods of Mathematical Finance. Springer, NewYork.Google Scholar
  11. 11.
    Kramkov, D. (1996). Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets. Probability Theory and Related Fields 105, 459-479.Google Scholar
  12. 12.
    Kramkov, D., and Schachermayer, W. (1999). A condition on the asymptotic elasticity of utility functions and optimal investment in incomplete markets. Annals of Applied Probability 93, 904-950.Google Scholar
  13. 13.
    Revuz, D., and Yor, M. (1991). Continuous Martingales and Brownian Motion. Springer-Verlag, NewYork.Google Scholar
  14. 14.
    Schwartz, M. (1986). New proofs of a theorem of Komlós. Acta Math. Hung. 47, 181-185.Google Scholar
  15. 15.
    Žitković G. (1999). Maximization of utility of consumption in incomplete semimartingale markets. working paper, Tehnische Universität Wien.Google Scholar

Copyright information

© Plenum Publishing Corporation 2002

Authors and Affiliations

  • Gordan Žitković
    • 1
  1. 1.Department of StatisticsColumbia UniversityNew York

Personalised recommendations