2022/09/29 by Somnath Pradhan, Serdar Yüksel, Pradhan, Somnath +1 · 2 citations
Economics, Econometrics and Finance · Engineering · Physics and Astronomy · #60J60 #Advanced Thermodynamics and Statistical Mechanics #Economic theories and models #FOS: Mathematics #Optimization and Control (math.OC) #Primary 93E20 #Secondary 35Q93 #Smart Grid Energy Management
paper · pdf · doi:10.48550/arxiv.2209.14982
openalex publication_date 2022/09/29 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
We first show that the discounted cost, cost up to an exit time, and ergodic cost involving controlled non-degenerate diffusions are continuous on the space of stationary control policies when the policies are given a topology introduced by Borkar [V. S. Borkar, A topology for Markov controls, Applied Mathematics and Optimization 20 (1989), 55-62]. The same applies for finite horizon problems when the control policies are markov and the topology is revised to include time also as a parameter. We then establish that finite action/piecewise constant stationary policies are dense in the space of stationary Markov policies under this topology. Using the above mentioned continuity and denseness results we establish that finite action/piecewise constant policies approximate optimal stationary policies with arbitrary precision. This gives rise to the applicability of many numerical methods such as policy iteration and stochastic learning methods for discounted cost, cost up to an exit time, and ergodic cost optimal control problems in continuous-time. For the finite-horizon setup, we establish additionally near optimality of time-discretized policies by an analogous argument. We thus present a unified and concise approach for approximations directly applicable under several commonly adopted cost criteria.