Time-consistency of cash-subadditive risk measures

TL;DR

This paper investigates the conditions under which cash-subadditive convex dynamic risk measures are time-consistent, providing a dual representation and a sufficient condition for strong time-consistency based on a generalized cocycle condition.

Contribution

It extends the understanding of time-consistency for cash-subadditive convex risk measures by offering a dual representation and a new sufficient condition for strong time-consistency.

Findings

Provides a dual representation of cash-subadditive convex risk measures.

Identifies a sufficient condition for strong time-consistency via a generalized cocycle condition.

Extends previous results to the cash-subadditive case, though not fully characterizing strong time-consistency.

Abstract

The main goal of this paper is to investigate under which conditions cash-subadditive convex dynamic risk measures are time-consistent. Proceeding as in Detlefsen and Scandolo \cite{detlef-scandolo} and inspired by their result, we give a dual representation of dynamic cash-subadditive convex risk measures (that can also be seen as particular case of the dual quasiconvex representation). The main result of the paper consists in providing, in the cash-subadditive case, a sufficient condition for strong time-consistency (or recursivity) in terms of a generalized cocycle condition. On one hand, our result can be seen as an extension to cash-subadditive convex dynamic risk measures of Theorem 2.5 in Bion-Nadal \cite{bion-nadal-FS}; on the other hand, it is weaker since strong time-consistency is not fully characterized. Finally, we exploit the relation between different notions of time-consistency.