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Polynomial processes and their applications to mathematical finance

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Abstract

We introduce a class of Markov processes, called m-polynomial, for which the calculation of (mixed) moments up to order m only requires the computation of matrix exponentials. This class contains affine processes, processes with quadratic diffusion coefficients, as well as Lévy-driven SDEs with affine vector fields. Thus, many popular models such as exponential Lévy models or affine models are covered by this setting. The applications range from statistical GMM estimation procedures to new techniques for option pricing and hedging. For instance, the efficient and easy computation of moments can be used for variance reduction techniques in Monte Carlo methods.

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Notes

  1. All statements concerning the characteristics are meant up to an evanescent set.

  2. We thank Martin Schweizer for pointing out this result to us.

  3. We write here μ 0 for the constant part of the jump measure in contrast to [7], where it is denoted by m.

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Acknowledgements

All authors gratefully acknowledge the support from the FWF grant Y 328 (START prize from the Austrian Science Fund) and from ETH Foundation. Furthermore the authors are grateful for many comments and valuable suggestions by Chris Rogers, which improved our paper a lot.

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Authors and Affiliations

  1. Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, 1090, Wien, Austria

    Christa Cuchiero

  2. Fakultät II, Institut für Mathematik, TU Berlin, Strasse des 17. Juni 136, 10623, Berlin, Germany

    Martin Keller-Ressel

  3. D-MATH, ETH Zürich, Rämistrasse 101, 8092, Zürich, Switzerland

    Josef Teichmann

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  1. Christa Cuchiero
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  2. Martin Keller-Ressel
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  3. Josef Teichmann
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Correspondence to Josef Teichmann.

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Cuchiero, C., Keller-Ressel, M. & Teichmann, J. Polynomial processes and their applications to mathematical finance. Finance Stoch 16, 711–740 (2012). https://doi.org/10.1007/s00780-012-0188-x

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