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Minkowski deviation measures

Author

Listed:
  • Moresco Marlon

    (Mathematics and Statistics, Concordia University, 455 Boul. de Maisonneuve Ouest, Montréal, QC H3G 1M8, Canada)

  • Brutti Righi Marcelo

    (Universidade Federal do Rio Grande do Sul, Escola de Administração, Rua Washington Luiz, 855 Centro Histórico, Porto Alegre, RS 90010-460, Brazil)

  • Horta Eduardo

    (Universidade Federal do Rio Grande do Sul, Campus do Vale - Prédio A1 (43111) Sala A106 Av Bento Gonçalves, 9500 - Porto Alegre, RS 91509-900, Brazil)

Abstract

We propose to derive deviation measures through the Minkowski gauge of a given set of acceptable positions. We show that, given a suitable acceptance set, any positive homogeneous deviation measure can be accommodated in our framework. In doing so, we provide a new interpretation for such measures, namely, that they quantify how much one must shrink or deleverage a financial position for it to become acceptable. In particular, the Minkowski Deviation of a set which is convex, translation insensitive, and radially bounded at non-constants, is a generalized deviation measure in the sense of [R. T. Rockafellar, S. Uryasev and M. Zabarankin, Generalized deviations in risk analysis, Finance Stoch. 10 2006, 1, 51–74]. Furthermore, we explore the converse relations from properties of a Minkowski Deviation to its sub-level sets, introducing the notion of acceptance sets for deviations. Hence, we fill a gap existing in the literature, namely the lack of a well-defined concept of acceptance sets for deviation measures. Dual characterizations in terms of polar sets and support functionals are provided.

Suggested Citation

  • Moresco Marlon & Brutti Righi Marcelo & Horta Eduardo, 2023. "Minkowski deviation measures," Statistics & Risk Modeling, De Gruyter, vol. 40(1-2), pages 1-19, January.
    • Handle: RePEc:bpj:strimo:v:40:y:2023:i:1-2:p:1-19:n:2
    DOI: 10.1515/strm-2021-0033
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