A unified approach to time consistency of dynamic risk measures and dynamic performance measures in discrete time

TL;DR

This paper introduces a flexible, unified framework for analyzing the time consistency of dynamic risk and performance measures, enabling comprehensive comparison and synthesis of existing concepts through a novel update rule approach.

Contribution

It develops a general, adaptable framework that unifies various forms of time consistency in dynamic measures using minimal assumptions and a new update rule methodology.

Findings

Framework unifies multiple types of time consistency

Novel update rule effectively captures weak time consistency

Allows for in-depth comparative analysis of dynamic measures

Abstract

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 be possible before and which is done in the companion paper [BCP2016] to this one. 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. 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.