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Linear hyperfinite Lévy integrals

Author

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  • Herzberg, Frederik

    (Center for Mathematical Economics, Bielefeld University)

Abstract

This article shows that the nonstandard approach to stochastic integration with respect to (C^2 functions of) Lévy processes is consistent with the classical theory of pathwise stochastic integration with respect to (C^2 functions of) jump-diffusions with finite-variation jump part. It is proven that internal stochastic integrals with respect to hyperfinite Lévy processes possess right standard parts, and that these standard parts coincide with the classical pathwise stochastic integrals, provided the integrator's jump part is of finite variation. If the integrator's Lévy measure is bounded from below, one can obtain a similar result for stochastic integrals with respect to C^2 functions of Lévy processes. As a by-product, this yields a short, direct nonstandard proof of the generalized Itô formula for stochastic differentials of smooth functions of Lévy processes.

Suggested Citation

  • Herzberg, Frederik, 2011. "Linear hyperfinite Lévy integrals," Center for Mathematical Economics Working Papers 404, Center for Mathematical Economics, Bielefeld University.
    • Handle: RePEc:bie:wpaper:404
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    File URL: https://pub.uni-bielefeld.de/download/2316147/2319851
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    Cited by:

    1. Frederik Herzberg, 2013. "First steps towards an equilibrium theory for Lévy financial markets," Annals of Finance, Springer, vol. 9(3), pages 543-572, August.
    2. Herzberg, Frederik, 2011. "On the foundations of Lévy finance. Equilibrium for a single-agent financial market with jumps," Center for Mathematical Economics Working Papers 406, Center for Mathematical Economics, Bielefeld University.

    More about this item

    Keywords

    Itô formula; Nonstandard analysis; Lévy processes; Stochastic integration;
    All these keywords.

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