Multivariate shortfall risk statistics with scenario analysis

Abstract

In the present paper, we construct a new class of multivariate risk statistics, which are multivariate shortfall risk statistics. First, we propose the multivariate shortfall risk statistics. Then their basic properties are investigated, and the dual representation results are studied. Furthermore, their coherency is also characterized by means of the corresponding loss function. Finally, examples are given to illustrate the proposed multivariate shortfall risk statistics.

KEYWORDS:

• (Multivariate) risk measures
• multivariate shortfall risk statistics
• convex risk statistics
• loss function

Mathematics Subject Classification (2010):

• 91B30; 91B32; 91B70

Acknowledgments

The authors are very grateful to the Editor-in-Chief Professor Narayanaswamy Balakrishnan and the anonymous referees for their constructive comments and suggestions, which led to the present greatly improved version of the manuscript.

Additional information

Funding

Supported by the National Natural Science Foundation of China (Nos. 11771343, 11901184) and the Natural Science Basic Research Plan in Shaanxi Province of China (No. 2014JQ2-1003).