Advances in Applied Probability

Convex duality in mean-variance hedging under convex trading constraints

Christoph Czichowsky and Martin Schweizer

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We study mean-variance hedging under portfolio constraints in a general semimartingale model. The constraints are formulated via predictable correspondences, meaning that the trading strategy is restricted to lie in a closed convex set which may depend on the state and time in a predictable way. To obtain the existence of a solution, we first establish the closedness in L2 of the space of all gains from trade (i.e. the terminal values of stochastic integrals with respect to the price process of the underlying assets). This is a first main contribution which enables us to tackle the problem in a systematic and unified way. In addition, using the closedness allows us to explain and generalise in a systematic way the convex duality results obtained previously by other authors via ad-hoc methods in specific frameworks.

Article information

Adv. in Appl. Probab., Volume 44, Number 4 (2012), 1084-1112.

First available in Project Euclid: 5 December 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G48: Generalizations of martingales 91G10: Portfolio theory 93E20: Optimal stochastic control 49N10: Linear-quadratic problems
Secondary: 60H05: Stochastic integrals

Mean-variance hedging constraints stochastic integral convex duality


Czichowsky, Christoph; Schweizer, Martin. Convex duality in mean-variance hedging under convex trading constraints. Adv. in Appl. Probab. 44 (2012), no. 4, 1084--1112. doi:10.1239/aap/1354716590.

Export citation


  • Aliprantis, C. D. and Border, K. C. (2006). Infinite Dimensional Analysis, 3rd edn. Springer, Berlin.
  • Aubin, J.-P. (2000). Applied Functional Analysis, 2nd edn. Wiley-Interscience, New York.
  • Bielecki, T. R., Jin, H., Pliska, S. R. and Zhou, X. Y. (2005). Continuous-time mean-variance portfolio selection with bankruptcy prohibition. Math. Finance 15, 213–244.
  • Choulli, T., Krawczyk, L. and Stricker, C. (1998). $\mathcal{E}$-martingales and their applications in mathematical finance. Ann. Prob. 26, 853–876.
  • Cvitanić, J. and Karatzas, I. (1992). Convex duality in constrained portfolio optimization. Ann. Appl. Prob. 2, 767–818.
  • Czichowsky, C. and Schweizer, M. (2011). Closedness in the semimartingale topology for spaces of stochastic integrals with constrained integrands. In Séminaire de Probabilités XLIII (Lecture Notes Math. 2006), Springer, Berlin, pp. 413–436
  • Czichowsky, C. and Schweizer, M. (2012). Cone-constrained continuous-time Markowitz problems. To appear in Ann. Appl. Prob.
  • Delbaen, F. (2006). 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 Math. 1874), Springer, Berlin, pp. 215–258.
  • Delbaen, F. \et (1997). Weighted norm inequalities and hedging in incomplete markets. Finance Stoch. 1, 181–227.
  • Ekeland, I. and Temam, R. (1976). Convex Analysis and Variational Problems. North-Holland, Amsterdam.
  • Föllmer, H. and Kramkov, D. (1997). Optional decompositions under constraints. Prob. Theory Relat. Fields 109, 1–25.
  • Hou, C. and Karatzas, I. (2004). Least-squares approximation of random variables by stochastic integrals. In Stochastic Analysis and Related Topics in Kyoto (Adv. Stud. Pure Math. 41), Mathematical Society, Japan, Tokyo, pp. 141–166.
  • Hu, Y. and Zhou, X. Y. (2005). Constrained stochastic LQ control with random coefficients, and application to portfolio selection. SIAM J. Control Optimization 44, 444–466.
  • Jin, H. and Zhou, X. Y. (2007). Continuous-time Markowitz's problems in an incomplete market, with no-shorting portfolios. In Stochastic Analysis and Applications (Abel Symp. 2), Springer, Berlin, pp. 435–459.
  • Karatzas, I. and Kardaras, C. (2007). The numéraire portfolio in semimartingale financial models. Finance Stoch. 11, 447–493.
  • Karatzas, I. and Shreve, S. E. (1998). Methods of Mathematical Finance (Appl. Math. 39). Springer, New York.
  • Karatzas, I. and Žitković, G. (2003). Optimal consumption from investment and random endowment in incomplete semimartingale markets. Ann. Prob. 31, 1821–1858.
  • Kramkov, D. and Schachermayer, W. (1999). The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Ann. Appl. Prob. 9, 904–950.
  • Labbé, C. and Heunis, A. J. (2007). Convex duality in constrained mean-variance portfolio optimization. Adv. Appl. Prob. 39, 77–104.
  • Mémin, J. (1980). Espaces de semi martingales et changement de probabilité. Z. Wahrscheinlichkeitsth. 52, 9–39.
  • Mnif, M. and Pham, H. (2001). Stochastic optimization under constraints. Stoch. Process. Appl. 93, 149–180.
  • Pham, H. (2000). Dynamic $L\sp p$-hedging in discrete time under cone constraints. SIAM J. Control Optimization 38, 665–682.
  • Pham, H. (2002). Minimizing shortfall risk and applications to finance and insurance problems. Ann. Appl. Prob. 12, 143–172.
  • Protter, P. E. (2005). Stochastic Integration and Differential Equations (Stoch. Modelling Appl. Prob. 21). Springer, Berlin.
  • Rockafellar, R. T. (1970). Convex Analysis. (Princeton Math. Ser. 28). Princeton University Press, Princeton, NJ.
  • Rockafellar, R. T. (1976). Integral functionals, normal integrands and measurable selections. In Nonlinear Operators and the Calculus of Variations (Lecture Notes Math. 543), Springer, Berlin, pp. 157–207
  • Schweizer, M. (2001). A guided tour through quadratic hedging approaches. In Option Pricing, Interest Rates and Risk Management. Cambridge University Press, pp. 538–574.
  • Schweizer, M. (2010). Mean-variance hedging. In Encyclopedia of Quantitative Finance, ed. R. Cont, John Wiley, pp. 1177–1181.
  • Sun, W. G. and Wang, C. F. (2006). The mean-variance investment problem in a constrained financial market. J. Math. Econom. 42, 885–895.