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

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Séminaire de Probabilités XLIII

Part of the book series: Lecture Notes in Mathematics ((SEMPROBAB,volume 2006))

Abstract

Let S be an ℝ d-valued semimartingale and (ψn) a sequence of C-valued integrands, i.e. predictable, S-integrable processes taking values in some given closed set C(ω, t) ⊆ ℝ d which may depend on the state ω and time t in a predictable way. Suppose that the stochastic integrals (ψn ⋅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 Cω, t is convex.

MSC 2000 Classification Numbers: 60G07, 60H05, 28B20, 91B28.

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Acknowledgements

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.

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Authors and Affiliations

  1. Department of Mathematics, ETH Zurich, Rämistrasse 101, CH-8092, Zurich, Switzerland

    Christoph Czichowsky & Martin Schweizer

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  1. Christoph Czichowsky
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  2. Martin Schweizer
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Correspondence to Martin Schweizer .

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Editors and Affiliations

  1. Lab. de Probabilités et Modèles Aléatoir, Université Pierre et Marie Curie, place Jussieu 4, Paris Cedex 05, 75252, France

    Catherine Donati-Martin

  2. IECN, Campus scientifique, BP 239, Nancy-Université, INRIA, Vandoeuvre-lès-Nancy, 54506, France

    Antoine Lejay

  3. Laboratoire de Mathématiques, Université de Versailles-St-Quentin, avenue des Etats-Unis 45, Versailles, 78035, France

    Alain Rouault

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Czichowsky, C., Schweizer, M. (2011). Closedness in the Semimartingale Topology for Spaces of Stochastic Integrals with Constrained Integrands. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) Séminaire de Probabilités XLIII. Lecture Notes in Mathematics(), vol 2006. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15217-7_18

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