Since the money is of time value, we will study a new class of risk statistics, named cash sub-additive risk statistics in this paper. This new class of risk statistics can be considered as a kind of risk extension of risk statistics introduced by Kou, Peng and Heyde (2013), and also data-based versions of cash sub-additive risk measures introduced by El Karoui and Ravanelli (2009) and Sun and Hu (2019).
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