The role of information in high dimensional stochastic optimal control

Palmer, Aaron Zeff

The stochastic optimal control of many agents minimizes a cost aggregated over the agents. We investigate the problem of partial information, where the state of each agent is not known and the control must be decided based on noisy observations. This results in a high dimensional controlled Markov process that is impractical to handle directly. In the limit as the number of agents approaches infinity, a finite dimensional mean field optimal control problem emerges, where the dependence on the available information vanishes. In this work, we calculate the Gaussian fluctuations about the mean field optimal control, which incorporates the available observations. The method we establish uses an approximate Kalman filter on the fluctuations about the mean field solution. It is straightforward to compute, even when the number of states is large. We consider an example of an epidemic model with observation of positive tests, as well as simple two state model that exhibits a phase transition at which point the fluctuations diverge.