Pathwise Information States and Dynamic Programming Equations for Decentralized Stochastic Optimal Control

We extend the classical dynamic programming approach of stochastic optimal control to decentralized stochastic discrete-time nonlinear dynamical optimal control problems, in which the control strategies have access to different information structures, and aim to optimize a common pay-off. We derive a set of generalized dynamic programming (DP) equations, where the information states are pathwise conditional distributions that satisfy the Markov property, and play the role of the sufficient statistics for the different strategies. The main theorem states that each control strategy, which is obtained from the solutions of the DP equations is optimal, and a functional of its own pathwise information state.