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<p>We establish a profound connection between coherent risk measures, a prominent object in quantitative finance, and uniform integrability, a fundamental concept in probability theory. Instead of working with absolute values of random variables, which is convenient in studying integrability, we work directly with random losses and gains, which have a clear financial interpretation. We introduce a technical tool called the folding score of distortion risk measures. The analysis of the folding score allows us to convert some conditions on absolute values to those on losses and gains. As our main results, we obtain three sets of equivalent conditions for uniform integrability. In particular, a set is uniformly integrable if and only if one can find a coherent distortion risk measure that is bounded on the set, but not finite on <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>1</mn> </msup> </math></EquationSource> <EquationSource Format="TEX">$L^{1}$</EquationSource> </InlineEquation>.</p>

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Coherent risk measures and uniform integrability

  • 期刊论文

  • 发表日期：

    2026年2月9日

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  • Muqiao Huang,
  • Ruodu Wang

摘要

We establish a profound connection between coherent risk measures, a prominent object in quantitative finance, and uniform integrability, a fundamental concept in probability theory. Instead of working with absolute values of random variables, which is convenient in studying integrability, we work directly with random losses and gains, which have a clear financial interpretation. We introduce a technical tool called the folding score of distortion risk measures. The analysis of the folding score allows us to convert some conditions on absolute values to those on losses and gains. As our main results, we obtain three sets of equivalent conditions for uniform integrability. In particular, a set is uniformly integrable if and only if one can find a coherent distortion risk measure that is bounded on the set, but not finite on L 1 $L^{1}$ .