Weakly time consistent concave valuations and their dual representations

Abstract

We derive dual characterizations of two notions of weak time consistency for concave valuations, which are convex risk measures under a positive sign convention. Combined with a suitable risk aversion property, these notions are shown to amount to three simple rules for not necessarily minimal representations, describing precisely which features of a valuation determine its unique consistent update. A compatibility result shows that for a time-indexed sequence of valuations, it is sufficient to verify these rules only pairwise with respect to the initial valuation, or in discrete time, only stepwise. We conclude by describing classes of consistently risk averse dynamic valuations with prescribed static properties per time step. This gives rise to a new formalism for recursive valuation. Keywords

Time-consistency of cash-subadditive risk measures

“…While strong time-consistency guarantees that the riskiness of a position at time s can be equivalently calculated in two ways (that is, directly at time s or in two steps -from time u to time t and then to time s), weak and weak* time-consistency imply that if a position is riskier than (or as risky as) another at time t then the same holds at any time s ≤ t. Further notions of timeconsistency can be also found in the recent paper of Roorda and Schumacher [28].…”

“…While time-consistency of dynamic coherent risk measures is strongly related to m-stability (or rectangularity) of the set of probability measures appearing in the dual representation of such risk measures (see Delbaen [13]), in the dynamic convex case strong time-consistency has been characterized by means of a decomposition property on acceptance sets (see Cheridito et al [9]) and in terms of a property (called cocycle) on the minimal penalty term (see Bion-Nadal [4] in continuous time and Föllmer and Penner [20] in discrete time). Further studies on time-consistency of risk measures can be found in Acciaio and Penner [1], Cheridito and Kupper [10], Delbaen et al [14], Detlefsen and Scandolo [15], Drapeau et al [17], Klöppel and Schweizer [26], Riedel [27], Roorda and Schumacher [28] and Rosazza Gianin [29], among others.…”

Robust optimal control using conditional risk mappings in infinite horizon

“…Namely, a multistage stochastic decision problem is time-consistent, if resolving the problem at later stages (i.e., after observing some random outcomes), the original solutions remain optimal for the later stages. We refer the reader to [9,10,15,38,22] for further elaboration and examples on this type of inconsistency. Hence, optimal control problems on multi-period setting using risk measures on bounded and unbounded costs are not vast, but still, some works in this direction are [11,12,14,13].…”

Mean‐variance Policy for Discrete‐time Cone‐constrained Markets: Time Consistency in Efficiency and the Minimum‐variance Signed Supermartingale Measure

“…As the time consistency requirement may result in a very conservative risk measure (see Roorda and Schumacher ), there have been numerous proposals to relax it to weak time consistency (or sequential consistency), which represents the idea that a monetary position that is surely (un)acceptable at some future date should also be (un)acceptable now. We refer to Riedel (), Föllmer and Penner (, ), Tutsch (, ), and Roorda and Schumacher (, ) for detailed discussions of weak time consistency.…”