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The Smirnov Property for Weighted Lebesgue Spaces

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  • Eberhard Mayerhofer

    (Department of Mathematics and Statistics, University of Limerick, V94 T9PX Limerick, Ireland)

Abstract

We establish lower norm bounds for multivariate functions within weighted Lebesgue spaces, characterised by a summation of functions whose components solve a system of nonlinear integral equations. This problem originates in portfolio selection theory, where these equations allow one to identify mean-variance optimal portfolios, composed of standard European options on several underlying assets. We elaborate on the Smirnov property—an integrability condition for the weights that guarantees the uniqueness of solutions to the system. Sufficient conditions on weights to satisfy this property are provided, and counterexamples are constructed, where either the Smirnov property does not hold or the uniqueness of solutions fails.

Suggested Citation

  • Eberhard Mayerhofer, 2024. "The Smirnov Property for Weighted Lebesgue Spaces," Mathematics, MDPI, vol. 12(19), pages 1-16, October.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:19:p:3135-:d:1493459
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    References listed on IDEAS

    as
    1. Semyon MALAMUD, 2014. "Portfolio Selection with Options and Transaction Costs," Swiss Finance Institute Research Paper Series 14-08, Swiss Finance Institute.
    2. Paolo Guasoni & Eberhard Mayerhofer, 2020. "Technical Note—Options Portfolio Selection," Operations Research, INFORMS, vol. 68(3), pages 733-740, May.
    3. Mathias Beiglbock & Benjamin Jourdain & William Margheriti & Gudmund Pammer, 2021. "Stability of the Weak Martingale Optimal Transport Problem," Papers 2109.06322, arXiv.org, revised Apr 2022.
    4. Breeden, Douglas T & Litzenberger, Robert H, 1978. "Prices of State-contingent Claims Implicit in Option Prices," The Journal of Business, University of Chicago Press, vol. 51(4), pages 621-651, October.
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