Dynamic consistency for Stochastic Optimal Control problems

TL;DR

This paper explores the concept of dynamic consistency in stochastic optimal control, linking it to state variables in Markov Decision Processes, and identifies conditions under which problems are dynamically consistent over time.

Contribution

It establishes a connection between time consistency and state variables in MDPs, showing that appropriate state variable selection ensures dynamic consistency in stochastic control problems.

Findings

Dynamic consistency depends on the choice of state variables.

A significant class of problems are dynamically consistent with proper state variables.

The paper bridges concepts from economics and stochastic programming.

Abstract

For a sequence of dynamic optimization problems, we aim at discussing a notion of consistency over time. This notion can be informally introduced as follows. At the very first time step $t_0$, the decision maker formulates an optimization problem that yields optimal decision rules for all the forthcoming time step $t_0, t_1, ..., T$; at the next time step $t_1$, he is able to formulate a new optimization problem starting at time $t_1$ that yields a new sequence of optimal decision rules. This process can be continued until final time $T$ is reached. A family of optimization problems formulated in this way is said to be time consistent if the optimal strategies obtained when solving the original problem remain optimal for all subsequent problems. The notion of time consistency, well-known in the field of Economics, has been recently introduced in the context of risk measures, notably by Artzner et al. (2007) and studied in the Stochastic Programming framework by Shapiro (2009) and for Markov Decision Processes (MDP) by Ruszczynski (2009). We here link this notion with the concept of "state variable" in MDP, and show that a significant class of dynamic optimization problems are dynamically consistent, provided that an adequate state variable is chosen.