Dynamic consistency for stochastic optimal control problems

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 t₀, the decision maker formulates an optimization problem that yields optimal decision rules for all the forthcoming time steps t₀,t₁,...,T; at the next time step t₁, he is able to formulate a new optimization problem starting at time t₁ that yields a new sequence of optimal decision rules. This process can be continued until the final time T is reached. A family of optimization problems formulated in this way is said to be dynamically consistent if the optimal strategies obtained when solving the original problem remain optimal for all subsequent problems. The notion of dynamic consistency, well-known in the field of economics, has been recently introduced in the context of risk measures, notably by Artzner et al. (Ann. Oper. Res. 152(1):5-22, 2007) and studied in the stochastic programming framework by Shapiro (Oper. Res. Lett. 37(3):143-147, 2009) and for Markov Decision Processes (MDP) by Ruszczynski (Math. Program. 125(2):235-261, 2010). 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.