Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Smirnov Property for weighted Lebesgue spaces

Eberhard Mayerhofer

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.

The Smirnov Property for weighted Lebesgue spaces

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.

Paper Structure

This paper contains 14 sections, 7 theorems, 67 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Program of Paper
  3. Mathematical Framework
  4. Notation
  5. Heuristic Derivation of Integral Equations
  6. Orthogonality in weighted $\mathcal{L}^p$-spaces
  7. The Smirnov property
  8. Sufficient Conditions
  9. A Counterexample
  10. Main Results
  11. Theorem and Proof
  12. A Counterexample concerning Uniqueness
  13. Mixture models
  14. Conclusion

Key Result

Lemma 2.1

Let $1<p<\infty$, $g\in\mathcal{L}^p(w)$, and $Y$ be a closed subspace of $\mathcal{L}^p(w)$. The following are equivalent:

Figures (1)

  • Figure 1: This 2D Contour plot depicts the fit of a Gaussian mixture model to a sample of size $1000$ from a correlated bivariate normal distribution with zero means, unit variances and correlation $\rho=0.5$. The fitted normal mixture is comprised of two normal densities, each with zero correlation. Their estimated weights $\alpha_i$ are essentially the same, with the first one $\alpha_1=0.0.498582$. The other parameter estimates are $\mu^1=(-0.64,-0.62)^\top$, $\mu^2=(0.62,0,62)$, $\Sigma^1=\text{diag}(0.62, 0.62)$ and $\Sigma^1=\text{diag}(0.61, 0.64)$.

Theorems & Definitions (16)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • Definition 2.4
  • Corollary 2.5
  • Proposition 2.6
  • proof
  • Remark 2.7
  • Remark 2.8
  • ...and 6 more