The Smirnov Property for weighted Lebesgue spaces

TL;DR

This paper investigates the Smirnov property in weighted Lebesgue spaces, providing conditions for uniqueness of solutions to nonlinear integral systems relevant in portfolio optimization.

Contribution

It establishes sufficient conditions for the Smirnov property in weighted Lebesgue spaces and analyzes their implications for solution uniqueness in portfolio models.

Findings

Sufficient conditions for the Smirnov property are identified.

Counterexamples demonstrate failure cases of the property.

Implications for mean-variance portfolio optimization are discussed.

Abstract

We establish lower norm bounds for multivariate functions within weighted Lebesgue spaces, characterized 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 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.