Building up time-consistency for risk measures and dynamic optimization

Michel de Lara (1) , Vincent Leclère (1)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique

In stochastic optimal control, one deals with sequential decision-making under uncertainty; with dynamic risk measures, one assesses stochastic processes (costs) as time goes on and information accumulates. Under the same vocable of time-consistency (or dynamic-consistency), both theories coin two different notions: the latter is consistency between successive evaluations of a stochas-tic processes by a dynamic risk measure (a form of monotonicity); the former is consistency between solutions to intertemporal stochastic optimization problems. Interestingly, both notions meet in their use of dynamic programming, or nested, equations. We provide a theoretical framework that offers i) basic ingredients to jointly define dynamic risk measures and corresponding intertemporal stochastic optimization problems ii) common sets of assumptions that lead to time-consistency for both. We highlight the role of time and risk preferences — materialized in one-step aggregators — in time-consistency. Depending on how one moves from one-step time and risk preferences to intertemporal time and risk preferences, and depending on their compatibility (commutation), one will or will not observe time-consistency. We also shed light on the relevance of information structure by giving an explicit role to a state control dynamical system, with a state that parameterizes risk measures and is the input to optimal policies.

Keywords: dynamic programming; time-consistency; dynamic risk measures

Domains: Optimization and Control [math.OC]

Michel de Lara, Vincent Leclère. Building up time-consistency for risk measures and dynamic optimization. European Journal of Operational Research, 2015, ⟨10.1016/j.ejor.2015.03.046⟩. ⟨hal-01208297⟩