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

• Published in: Applied mathematics & optimization 2016-08, Vol.74 (1), p.197-227
• Main Authors: Higuera-Chan, Carmen G., Jasso-Fuentes, Héctor, Minjárez-Sosa, J. Adolfo
• Format: Article
• Language: English
• Subjects: Applied mathematics; Asymptotic properties; Calculus of Variations and Optimal Control; Optimization; Consumption; Control; Disturbances; Field theory; Markov models; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematical models; Mathematics; Mathematics and Statistics; Numerical and Computational Physics; Optimization; Policies; Random variables; Simulation; Studies; Systems Theory; Theoretical
• ISSN: 0095-4616; 1432-0606
• DOI: 10.1007/s00245-015-9312-6

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 ρ 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 → ∞ (the mean field limit) with a suitable statistical estimation method for ρ , 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.