2010/04/14 by Nicolas Gast, Gast, Nicolas, Bruno Gaujal +3 · 1 citation
Business, Management and Accounting · Decision Sciences · #Advanced Queuing Theory Analysis #Artificial Intelligence (cs.AI) #FOS: Computer and information sciences #FOS: Electrical engineering #FOS: Mathematics #Game Theory and Applications #Optimization and Control (math.OC) #Performance (cs.PF) #Probability (math.PR) #Simulation Techniques and Applications #Systems and Control (eess.SY) #electronic engineering #information engineering
paper · doi:10.48550/arxiv.1004.2342
openalex publication_date 2010/04/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the convergence of Markov Decision Processes made of a large number of objects to optimization problems on ordinary differential equations (ODE). We show that the optimal reward of such a Markov Decision Process, satisfying a Bellman equation, converges to the solution of a continuous Hamilton-Jacobi-Bellman (HJB) equation based on the mean field approximation of the Markov Decision Process. We give bounds on the difference of the rewards, and a constructive algorithm for deriving an approximating solution to the Markov Decision Process from a solution of the HJB equations. We illustrate the method on three examples pertaining respectively to investment strategies, population dynamics control and scheduling in queues are developed. They are used to illustrate and justify the construction of the controlled ODE and to show the gain obtained by solving a continuous HJB equation rather than a large discrete Bellman equation.