We discuss when law-invariant convex functionals “collapse to the mean”. More precisely, we show that, in a large class of spaces of random variables and under mild semicontinuity assumptions, the expectation functional is, up to an affine transformation, the only law-invariant convex functional that is linear along the direction of a nonconstant random variable with nonzero expectation. This extends results obtained in the literature in a bounded setting and under additional assumptions on the functionals. We illustrate the implications of our general results for pricing rules and risk measures.

中文翻译：

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律律不变的功能崩溃到平均水平

我们讨论何时定律不变的凸泛函“崩溃到均值”。更准确地说，我们表明，在一大类随机变量空间中，在温和的半连续假设下，期望函数是直到仿射变换为止，唯一沿着非恒定随机方向线性的定律不变凸函数。具有非零期望值的变量。这扩展了在有限范围内以及在功能的其他假设下在文献中获得的结果。我们说明了总体结果对定价规则和风险衡量的影响。