Convergence in the Semimartingale Topology and Constrained Portfolios

  • Christoph Czichowsky
  • Nicholas Westray
  • Harry ZhengEmail author
Part of the Lecture Notes in Mathematics book series (LNM, volume 2006)


Consider an \({\mathbb{R}}^{d}\)-valued semimartingale S and a sequence of \({\mathbb{R}}^{d}\)-valued S-integrable predictable processes H n valued in some closed convex set \(\mathcal{K}\subset {\mathbb{R}}^{d}\), containing the origin. Suppose that the real-valued sequence H n S converges to X in the semimartingale topology. We would like to know whether we may write X = H 0S for some \({\mathbb{R}}^{d}\)-valued, S-integrable process H 0 valued in \(\mathcal{K}\)? This question is of crucial importance when looking at superreplication under constraints. The paper considers a generalization of the above problem to \(\mathcal{K} = \mathcal{K}(\omega,t)\) possibly time dependent and random.

Stochastic integrals Semimartingale topology Integrands in polyhedral and continuous convex constraints Optimization under constraints Mathematical finance 


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We thank Freddy Delbaen, Dmitry Kramkov, Aleksandar Mijatović and Martin Schweizer for helpful advice and valuable discussions on the content of this paper.

Nicholas Westray was supported via an EPSRC grant and Christoph Czichowsky acknowledges financial support from the National Centre of Competence in Research “Financial Valuation and Risk Management” (NCCR FINRISK), Project D1 (Mathematical Methods in Financial Risk Management).


  1. 1.
    Émery, M.: Une topologie sur l’espace des semimartingales. In: Séminaire de Probabilités, XIII (Univ. Strasbourg, Strasbourg, 1977/1978). Lecture Notes in Mathematics, vol. 721, pp. 260–280. Springer, Berlin (1979)Google Scholar
  2. 2.
    Föllmer, H., Kramkov, D.: Optional decompositions under constraints. Probab. Theory Relat. Field. 109(1), 1–25 (1997)zbMATHCrossRefGoogle Scholar
  3. 3.
    Gale, D., Klee, V.: Continuous convex sets. Math. Scand. 7, 379–391 (1959)MathSciNetGoogle Scholar
  4. 4.
    Goldman, A.J., Tucker, A.W.: Polyhedral convex cones. In: Linear Equalities and Related Systems. Annals of Mathematics Studies, vol. 38, pp. 19–40. Princeton University Press, Princeton, NJ (1956)Google Scholar
  5. 5.
    Jacod, J., Shiryaev A.N.: Limit theorems for stochastic processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 288, 2nd edn. Springer, Berlin (2003)Google Scholar
  6. 6.
    Karatzas, I., Kardaras, C.: The numéraire portfolio in semimartingale financial models. Finance Stochast. 11(4), 447–493 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Karatzas, I., Žitković, G.: Optimal consumption from investment and random endowment in incomplete semimartingale markets. Ann. Probab. 31(4), 1821–1858 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Klee, V.: Maximal separation theorems for convex sets. Trans. Am. Math. Soc. 134, 133–147 (1968)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Kuratowski, K., Ryll-Nardzewski, C.: A general theorem on selectors. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 13, 397–403 (1965)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Mémin, J.: Espaces de semi martingales et changement de probabilité. Z. Wahrsch. Verw. Gebiete. 52(1), 9–39 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Mnif, M., Pham, H.: Stochastic optimization under constraints. Stochast. Process. Appl. 93(1), 149–180 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Pham, H.: Minimizing shortfall risk and applications to finance and insurance problems. Ann. Appl. Probab. 12(1), 143–172 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Rockafellar, R.T.: Measurable dependence of convex sets and functions on parameters. J. Math. Anal. Appl. 28, 4–25 (1969)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Rockafellar, R.T.: Convex Analysis. Princeton Mathematical Series, vol. 28. Princeton University Press, Princeton, NJ (1970)Google Scholar
  15. 15.
    Rockafellar, R.T.: Integral functionals, normal integrands and measurable selections. In: Nonlinear Operators and the Calculus of Variations (Summer School. Univ. Libre Bruxelles, Brussels, 1975). Lecture Notes in Mathematics, vol. 543, pp. 157–207. Springer, Berlin (1976)Google Scholar
  16. 16.
    Wagner, D.H.: Survey of measurable selection theorems. SIAM J. Contr. Optim. 15(5), 859–903 (1977)zbMATHCrossRefGoogle Scholar
  17. 17.
    Westray, N., Zheng, H.: Constrained non-smooth utility maximization without quadratic inf convolution. Stochast. Process. Appl. 119(5), 1561–1579 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Yosida, K.: Functional Analysis. Grundlehren der Mathematischen Wissenschaften, vol. 123, 5th edn. Springer, Berlin (1978)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Christoph Czichowsky
    • 1
  • Nicholas Westray
    • 2
  • Harry Zheng
    • 3
    Email author
  1. 1.Department of MathematicsETH ZurichZurichSwitzerland
  2. 2.Department of MathematicsHumboldt Universität BerlinBerlinGermany
  3. 3.Department of MathematicsImperial CollegeLondonUK

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