Time consistency of nonconvex risk measures

Although most of the theory development concerning risk measures has concentrated on convex or even coherent risk measures, nonconvex risk measures are used in practice, the prime example of course being Value-at-Risk. The purpose of this paper is to investigate the relations between various notions of time consistency for nonconvex risk measures. We focus on three notions in particular which, as we show, all satisfy a certain compatibility property that other notions of consistency do not always have.One of these notions is the strong consistency (also called dynamic consistency) that has received most attention in the literature. Despite the fact that the other two notions are weaker, we show that they are still strong enough to make consistent updates uniqueif they exist at all. We give a number of su±cient and necessary conditions that an aggregate risk measure has to satisfy for consistent updating to be possible.