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Chaotic expansion of powers and martingale representation (v1.5)

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  • Farshid Jamshidian

    (Univ. of Twente, NIBCapital)

Abstract

This paper extends a recent martingale representation result of [N-S] for a Levy process to filtrations generated by a rather large class of semimartingales. As in [N-S], we assume the underlying processes have moments of all orders, but here we allow angle brackets to be stochastic. Following their approach, including a chaotic expansion, and incorporating an idea of strong orthogonalization from [D], we show that the stable subspace generated by Teugels martingales is dense in the space of square-integrable martingales, yielding the representation. While discontinuities are of primary interest here, the special case of a (possibly infinite-dimensional) Brownian filtration is an easy consequence.

Suggested Citation

  • Farshid Jamshidian, 2005. "Chaotic expansion of powers and martingale representation (v1.5)," GE, Growth, Math methods 0507009, University Library of Munich, Germany.
    • Handle: RePEc:wpa:wuwpge:0507009
    Note: Type of Document - pdf; pages: 22. Martingale representation results for filtration generated by a large class of processes, including Levy processes. (Minor improvements to version 1.4)
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    File URL: https://econwpa.ub.uni-muenchen.de/econ-wp/ge/papers/0507/0507009.pdf
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    Cited by:

    1. Ariel Neufeld & Philipp Schmocker, 2022. "Chaotic Hedging with Iterated Integrals and Neural Networks," Papers 2209.10166, arXiv.org, revised Feb 2023.

    More about this item

    Keywords

    Martingale representation; stochastic integration; stable subspaces; power brackets; Teugels martingales; polynomial; chaos; Hilbert space direct sum decomposition; Levy processes; finite moements semimartingales; dense.;
    All these keywords.

    JEL classification:

    • C - Mathematical and Quantitative Methods
    • G - Financial Economics

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