SIAM Journal on Financial Mathematics, Volume 12, Issue 1, Page 318-341, January 2021.
We establish general versions of a variety of results for quasiconvex, lower-semicontinuous, and law-invariant functionals. Our results extend well-known results from the literature to a large class of spaces of random variables. We sometimes obtain sharper versions, even for the well-studied case of bounded random variables. Our approach builds on two fundamental structural results for law-invariant functionals: the equivalence of law invariance and Schur convexity, i.e., monotonicity with respect to the convex stochastic order, and the fact that a law-invariant functional is fully determined by its behavior on bounded random variables. We show how to apply these results to provide a unifying perspective on the literature on law-invariant functionals, with special emphasis on quantile-based representations, including Kusuoka representations, dilatation monotonicity, and infimal convolutions.

中文翻译：

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随机变量一般空间上的律不变泛函

SIAM 金融数学杂志，第 12 卷，第 1 期，第 318-341 页，2021 年 1 月。
我们为拟凸函数、下半连续函数和律不变函数建立了各种结果的一般版本。我们的结果将文献中众所周知的结果扩展到一大类随机变量空间。我们有时会获得更清晰的版本，即使对于有界随机变量的研究充分的情况也是如此。我们的方法建立在规律不变泛函的两个基本结构结果上：规律不变性和 Schur 凸性的等价性，即关于凸随机阶的单调性，以及规律不变泛函完全由其在有界随机变量。我们展示了如何应用这些结果为关于律不变泛函的文献提供一个统一的视角，特别强调基于分位数的表示，包括 Kusuoka 表示，