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Law-Invariant Functionals on General Spaces of Random Variables


Bellini, Fabio; Koch-Medina, Pablo; Munari, Cosimo; Svindland, Gregor (2021). Law-Invariant Functionals on General Spaces of Random Variables. SIAM Journal on Financial Mathematics, 12(1):318-341.

Abstract

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.

Abstract

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.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:03 Faculty of Economics > Department of Banking and Finance
Dewey Decimal Classification:330 Economics
Scopus Subject Areas:Physical Sciences > Numerical Analysis
Social Sciences & Humanities > Finance
Physical Sciences > Applied Mathematics
Uncontrolled Keywords:Law invariance, Schur convexity, Dilation monotonicity, Extension results, Quantile representations, Kusuoka representations, Infimal convolutions
Scope:Discipline-based scholarship (basic research)
Language:English
Date:1 January 2021
Deposited On:11 Jan 2024 14:00
Last Modified:06 Mar 2024 14:41
Publisher:Society for Industrial and Applied Mathematics
ISSN:1945-497X
OA Status:Green
Publisher DOI:https://doi.org/10.1137/20m1341258
Other Identification Number:merlin-id:24262
  • Content: Published Version
  • Language: English