What risk measures are time consistent for all filtrations?

TL;DR

This paper characterizes the types of coherent risk measures that remain time-consistent across all filtrations, revealing that only four main types satisfy this property and identifying conditions under which they are linear or supremum-based.

Contribution

It provides a complete characterization of coherent risk measures that are universally time-consistent across all filtrations, including conditions for linearity and supremum forms.

Findings

A coherent risk measure is time-consistent for all filtrations iff it belongs to four main types.

Strictly monotone risk measures are linear.

On non-atomic probability spaces, such measures are either linear or an essential supremum.

Abstract

We study coherent risk measures which are time-consistent for multiple filtrations. We show that a coherent risk measure is time-consistent for every filtration if and only if it is one of four main types. Furthermore, if the risk measure is strictly monotone it is linear, and if the reference probability space is not atomic then it is either linear or an essential supremum.