Closedness in the Semimartingale Topology for Spaces of Stochastic Integrals with Constrained Integrands

  • Christoph Czichowsky
  • Martin SchweizerEmail author
Part of the Lecture Notes in Mathematics book series (LNM, volume 2006)


Let S be an d -valued semimartingale and (ψ n ) a sequence of C-valued integrands, i.e. predictable, S-integrable processes taking values in some given closed set C(ω, t) ⊆ d which may depend on the state ω and time t in a predictable way. Suppose that the stochastic integrals (ψ n S) converge to X in the semimartingale topology. When can X be represented as a stochastic integral with respect to S of some C-valued integrand? We answer this with a necessary and sufficient condition (on S and C), and explain the relation to the sufficient conditions introduced earlier in (Czichowsky, Westray, Zheng, Convergence in the semimartingale topology and constrained portfolios, 2010; Mnif and Pham, Stochastic Process Appl 93:149–180, 2001; Pham, Ann Appl Probab 12:143–172, 2002). The existence of such representations is equivalent to the closedness (in the semimartingale topology) of the space of all stochastic integrals of C-valued integrands, which is crucial in mathematical finance for the existence of solutions to most optimisation problems under trading constraints. Moreover, we show that a predictably convex space of stochastic integrals is closed in the semimartingale topology if and only if it is a space of stochastic integrals of C-valued integrands, where each , t is convex.

Stochastic integrals Constrained strategies Semimartingale topology Closedness Predictably convex Projection on predictable range Predictable correspondence Optimisation under constraints Mathematical finance 


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We thank an anonymous referee for careful reading and helpful suggestions. Financial support by the National Centre of Competence in Research “Financial Valuation and Risk Management” (NCCR FINRISK), Project D1 (Mathematical Methods in Financial Risk Management) is gratefully acknowledged. The NCCR FINRISK is a research instrument of the Swiss National Science Foundation.


  1. 1.
    Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis, 3rd edn. Springer, Berlin (2006)zbMATHGoogle Scholar
  2. 2.
    Artstein, Z.: Set-valued measures. Trans. Am. Math. Soc. 165, 103–125 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Cuoco, D.: Optimal consumption and equilibrium prices with portfolio constraints and stochastic income. J. Econ. Theory 72, 33–73 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Cvitanić, J., Karatzas, I.: Convex duality in constrained portfolio optimization. Ann. Appl. Probab. 2, 767–818 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Czichowsky, C., Schweizer, M.: On the Markowitz problem with cone constraints. Working paper, ETH Zurich (2010, in preparation),
  6. 6.
    Czichowsky, C., Westray, N., Zheng, H.: Convergence in the semimartingale topology and constrained portfolios. In: Séminaire de Probabilités XLIII (2010, this volume)Google Scholar
  7. 7.
    Debreu, G., Schmeidler, D.: The Radon–Nikodým derivative of a correspondence. In: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, vol. II: Probability Theory, pp. 41–56. University of California Press, Berkeley, CA (1972)Google Scholar
  8. 8.
    Delbaen, F.: The structure of m-stable sets and in particular of the set of risk neutral measures. In: In memoriam Paul-André Meyer: Séminaire de Probabilités XXXIX. Lecture Notes in Mathematics, vol. 1874, pp. 215–258. Springer, Berlin (2006)Google Scholar
  9. 9.
    Delbaen, F., Schachermayer, W.: The Mathematics of Arbitrage. Springer Finance. Springer, Berlin (2006)zbMATHGoogle Scholar
  10. 10.
    Emery, M.: Une topologie sur l’espace des semimartingales. In: Séminaire de Probabilités XIII. Lecture Notes in Mathematics, vol. 721, pp. 260–280. Springer, Berlin (1979)Google Scholar
  11. 11.
    Föllmer, H., Kramkov, D.: Optional decompositions under constraints. Probab. Theory Relat. Field. 109, 1–25 (1997)zbMATHCrossRefGoogle Scholar
  12. 12.
    Föllmer, H., Schied, A.: Stochastic Finance. An Introduction in Discrete Time. de Gruyter Studies in Mathematics, vol. 27. Walter de Gruyter & Co., Berlin, second revised and extended edition (2004)Google Scholar
  13. 13.
    Gale, D., Klee, V.: Continuous convex sets. Math. Scand. 7, 379–391 (1959)MathSciNetGoogle Scholar
  14. 14.
    Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes. Grundlehren der Mathematischen Wissenschaften, vol. 288, 2nd edn. Springer, Berlin (2003)Google Scholar
  15. 15.
    Jin, H., Zhou, X.Y.: Continuous-time Markowitz’s problems in an incomplete market, with no-shorting portfolios. In: Benth, F.E., et al. (eds.) Stochastic Analysis and Applications. Proceedings of the Second Abel Symposium, Oslo, 2005, pp. 435–459. Springer, Berlin (2007)Google Scholar
  16. 16.
    Karatzas, I., Kardaras, C.: The numéraire portfolio in semimartingale financial models. Finance Stochast. 11, 447–493 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Karatzas, I., Žitković, G.: Optimal consumption from investment and random endowment in incomplete semimartingale markets. Ann. Probab. 31, 1821–1858 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Klee, V.: Some characterizations of convex polyhedra. Acta Math. 102, 79–107 (1959)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Labbé, C., Heunis, A.J.: Convex duality in constrained mean-variance portfolio optimization. Adv. Appl. Probab. 39, 77–104 (2007)zbMATHCrossRefGoogle Scholar
  20. 20.
    Mémin, J.: Espaces de semi martingales et changement de probabilité. Z. Wahrsch. verw. Gebiete 52, 9–39 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Mnif, M., Pham, H.: Stochastic optimization under constraints. Stochastic Process. Appl. 93, 149–180 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Pham, H.: Minimizing shortfall risk and applications to finance and insurance problems. Ann. Appl. Probab. 12, 143–172 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Rockafellar, R.T.: Integral functionals, normal integrands and measurable selections. In: Nonlinear Operators and the Calculus of Variations. Lecture Notes in Mathematics, vol. 543, pp. 157–207. Springer, Berlin (1976)Google Scholar
  24. 24.
    Schachermayer, W.: A Hilbert space proof of the fundamental theorem of asset pricing in finite discrete time. Insur. Math. Econ. 11, 249–257 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Westray, N., Zheng, H.: Constrained nonsmooth utility maximization without quadratic inf convolution. Stochastic Process. Appl. 119, 1561–1579 (2009)MathSciNetzbMATHCrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Department of MathematicsETH ZurichZurichSwitzerland

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