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

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

Abstract

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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Notes

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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© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Department of MathematicsETH ZurichZurichSwitzerland

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