Discrete-Time Control for Systems of Interacting Objects with Unknown Random Disturbance Distributions: A Mean Field Approach

We are concerned with stochastic control systems composed of a large number of N interacting objects sharing a common environment. The evolution of each object is determined by a stochastic difference equation where the random disturbance density $$\rho $$ź is unknown for the controller. We present the Markov control model (N-model) associated to the proportions of objects in each state, which is analyzed according to the mean field theory. Thus, combining convergence results as $$N\rightarrow \infty $$Nźź (the mean field limit) with a suitable statistical estimation method for $$\rho $$ź, we construct the so-named eventually asymptotically optimal policies for the N-model under a discounted optimality criterion. A consumption-investment problem is analyzed to illustrate our results.