Discrete-time Risk Measures with Time Consistency

According to the properties of general probability space, we propose the conception of acceptance set and capital requirement in the discrete-time risk measures framework. Related propositions are put forward and proved in the second part. Then we mainly focus on the time consistency properties shown during the course of discrete-time risk measures. Time consistency has been certified as one of the most important properties for discrete-time risk measures. In particular, the poor, middle and strong consistency based on the original recursiveness and consistency are proposed in the third section. These time consistency properties provide the mathematical description of dynamic risk measures on the general probability space. Finally, the validation of time consistency of existing methods including VaR, CVaR and ES is provided