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Natural risk measures

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Abstract

A coherent risk measure with a proper continuity condition cannot be defined on a large set of random variables. However, if one relaxes the sub-additivity condition and replaces it with co-monotone sub-additivity, the proper domain of risk measures can contain the set of all random variables. In this study, by replacing the sub-additivity axiom of law invariant coherent risk measures with co-monotone sub-additivity, we introduce the class of natural risk measures on the space of all bounded-below random variables. We characterize the class of natural risk measures by providing a dual representation of its members.

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Notes

  1. This convexification can only be regarded as a technical extension since in [25] the authors have a different objective: to compare their co-monotone additive premium function in a competitive market with an arbitrage-free pricing rule, where additivity holds for all risks.

  2. Unlike in financial mathematics literature, which considers a profit variable, we found the loss variable more convenient to deal with.

  3. Càdlàg is a French acronym that translates into English as “right continuous and left limited”.

  4. The definition of a natural risk measure is motivated by the definition of a natural risk statistics introduced on \(\mathbb {R}^n\) in [17].

  5. Robust optimization is an approach to model uncertainty when the uncertain parameters are known to be within certain bounds. For more reading on the robust analysis approach, see [7, 20, 26, 27].

  6. In general, for any two topological vector spaces \(V,V'\) with bilinear dual relation \((v.v')\), \(\sigma (V,V')\) denotes the smallest topology on V under which all members of \(V'\) are continuous.

  7. \(\mathrm {supp}\left( \lambda \right) \) stands for support of \(\lambda \).

  8. \(\vert \mu \vert =\mu _{+}+\mu _{-}\) is the absolute value of \(\mu \).

  9. One would wonder why we use the term ‘statistics’ instead of ‘statistic.’ Actually, there is no reason except that it is the exact term that has been used in the literature; see, e.g. [1].

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Authors and Affiliations

  1. Institute for Financial and Actuarial Mathematics, University of Liverpool, Liverpool, UK

    Hirbod Assa

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  1. Hirbod Assa
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Correspondence to Hirbod Assa.

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The author would like to sincerely thank the reviewers and in particular the associate editor, Alexander Zimper, for their valuable comments, which considerably improved the quality of this paper.

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Assa, H. Natural risk measures. Math Finan Econ 10, 441–456 (2016). https://doi.org/10.1007/s11579-016-0165-9

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