Time-Consistency: from Optimization to Risk Measures

Stochastic optimal control is concerned with sequential decision-making under uncertainty. The theory of dynamic risk measures gives values to stochastic processes (costs) as time goes on and information accumulates. Both theories coin, under the same vocable of time-consistency (or dynamic-consistency), two different notions: the latter is consistency between successive evaluations of a stochastic processes by a dynamic risk measure as information accumulates (a form of monotonicity); the former is consistency between solutions to intertemporal stochastic optimization problems as information accumulates. Interestingly, time-consistency in stochastic optimal control and time-consistency for dynamic risk measures 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. Our theoretical framework highlights the role of time and risk preferences, materialized in one-step aggregators, in time-consistency. Depending on how you move from one-step time and risk preferences to intertemporal time and risk preferences, and depending on their compatibility (commutation), you 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.