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Multivariate coherent risk measures induced by multivariate convex risk measures

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Abstract

In this paper, we study the close relationship between multivariate coherent and convex risk measures. Namely, starting from a multivariate convex risk measure, we propose a family of multivariate coherent risk measures induced by it. In return, the convex risk measure can be represented by its induced coherent risk measures. The representation result for the induced coherent risk measures is given in terms of the minimal penalty function of the convex risk measure. Finally, an example is also given.

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Acknowledgements

The authors are very grateful to the Editors and the anonymous referees for their valuable and constructive comments and suggestions which led to the present greatly improved version of the manuscript.

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Authors and Affiliations

  1. College of Finance and Statistics, Hunan University, Changsha, 410082, Hunan, People’s Republic of China

    Yanhong Chen

  2. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, Hubei, People’s Republic of China

    Yijun Hu

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  1. Yanhong Chen
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  2. Yijun Hu
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Correspondence to Yanhong Chen.

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Supported by the Fundamental Research Funds for the Central Universities (No. 531107051210) and the National Natural Science Foundation of China (Nos. 11771343, 11901184).

Appendix

Appendix

The following Lemma A.1 is an immediate corollary of Hiriart-Urruty and Lemaréchal [19, Proposition 2.2.1].

Lemma A.1

Let \(\rho : {\mathscr {X}}^N \rightarrow (-\infty , +\infty ]\) be a convex function, then \(F: {\mathscr {X}}^N \times {\mathbf {R}}_{++} \rightarrow (-\infty , +\infty ]\) with \(F({\mathbf {M}}, t) := t \rho (t^{-1}{\mathbf {M}})\) for any \({\mathbf {M}} \in {\mathscr {X}}^N\) and \(t \in {\mathbf {R}}_{++}\) is jointly convex in \({\mathbf {M}}\) and t.

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Chen, Y., Hu, Y. Multivariate coherent risk measures induced by multivariate convex risk measures. Positivity 24, 711–727 (2020). https://doi.org/10.1007/s11117-019-00703-2

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