A Unified Approach to Time Consistency of Dynamic Risk Measures and Dynamic Performance Measures in Discrete Time

In this paper, we provide a flexible framework allowing for a unified study of time consistency of risk measures and performance measures (also known as acceptability indices). The proposed framework not only integrates existing forms of time consistency, but also provides a comprehensive toolbox for analysis and synthesis of the concept of time consistency in decision making. In particular, it allows for in-depth comparative analysis of (most of) the existing types of time consistency—a feat that has not been possible before and, which is done in the companion paper by the authors. In our approach, the time consistency is studied for a large class of maps that are postulated to satisfy only two properties—monotonicity and locality. We call these maps LM-measures. The time consistency is defined in terms of an update rule. The form of the update rule introduced here is novel, and is perfectly suited for developing the unifying framework that is worked out in this paper. As an illustration of the applicability of our approach, we show how to recover almost all concepts of weak time consistency by means of constructing appropriate update rules.