The Smirnov Property for Weighted Lebesgue Spaces

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

Mathematics, 2024, vol. 12, issue 19, 1-16

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.

Keywords: weighted Lebesgue spaces; multivariate distributions; estimates; integral equations (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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