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Convergence in the Semimartingale Topology and Constrained Portfolios

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

Abstract

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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Notes

Écknowledgements

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).

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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

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

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