Paper: arXiv 2405.09766

Authors: Niushan Gao, Foivos Xanthos

Abstract

In this short note, we show that every convex, order bounded above functional on a Frechet lattice is automatically norm continuous. This improves a result in \cite{RS06} and applies to many deviation and variability measures. We also show that an order-continuous, law-invariant functional on an Orlicz space is strongly consistent everywhere, extending a result in \cite{KSZ14}.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 1.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper is a purely theoretical note in functional analysis and risk measure theory, establishing automatic continuity on Fréchet lattices and asymptotic consistency in Orlicz spaces. While mathematically rigorous and novel in extending established results by Ruszczyński-Shapiro and Krätschmer et al., it contains no empirical data, backtests, or computational experiments.

Research Flowchart

  flowchart TD
    A[Goal: Analyze continuity & consistency of risk/variability measures] --> B[Input: Convex functionals on Fréchet Lattices & Orlicz Spaces]
    B --> C[Method 1: Prove automatic norm continuity for convex, order bounded above functionals]
    B --> D[Method 2: Extend strong asymptotic consistency to law-invariant, order-continuous functionals]
    C --> E[Outcome 1: Improved RS06 result for deviation & variability measures]
    D --> F[Outcome 2: Extended KSZ14 result for Orlicz spaces everywhere]

Cited References (27)

  1. Stopping Times and Directed Processes (1992) — link
  2. Principles of Real Analysis (2018) — link
  3. Topics in Optimal Transportation (2003) — link
  4. Estimation of Distortion Risk Measures (2013) — link
  5. Consistency of Sample Estimates of Risk Averse Stochastic Programs (2013) — link
  6. Coherent Measures of Risk (1999) — link
  7. Haezendonck–Goovaerts risk measures and Orlicz quantiles (2012) — link
  8. Comparative and qualitative robustness for law-invariant risk measures (2014) — link
and 19 more
  1. Spectral measures of risk: A coherent representation of subjective risk aversion (2002) — link
  2. INDIFFERENCE PRICE WITH GENERAL SEMIMARTINGALES (2010) — link
  3. Domains of weak continuity of statistical functionals with a view toward robust statistics (2017) — link
  4. Optimization of Convex Risk Functions (2006) — link
  5. PROBABILITY AND MATHEMATICAL STATISTICS (1999) — link
  6. Fatou property, representations, and extensions of law-invariant risk measures on general Orlicz spaces (2018) — link
  7. Closedness of convex sets in Orlicz spaces with applications to dual representation of risk measures (2019) — link
  8. On the C‐property and ‐representations of risk measures (2017) — link
  9. Dynamic robust Orlicz premia and Haezendonck–Goovaerts risk measures (2019) — link
  10. Stability properties of Haezendonck–Goovaerts premium principles (2020) — link
  11. Beyond cash-additive risk measures: when changing the numéraire fails (2013) — link
  12. Generalized deviations in risk analysis (2005) — link
  13. Law invariant risk measures have the Fatou property (2007) — link
  14. Parametric measures of variability induced by risk measures (2022) — link
  15. Automatic Fatou property of law-invariant risk measures (2022) — link
  16. Stochastic Finance (2011) — link
  17. Optimality and Risk - Modern Trends in Mathematical Finance (2009) — link
  18. Positive Operators (2006) — link
  19. Distortion risk measures: Prudence, coherence, and the expected shortfall (2024) — link

0 of 27 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.