Stochastic control via direct comparison

The standard approach to stochastic control is dynamic programming. In this paper, we introduce an alternative approach based on direct comparison of the performance of any two policies. This is achieved by modeling the state process as a continuous-time and continuous-state Markov process and applying the same ideas as for the discrete-time and discrete-state case. This approach is simple and intuitively clear; it applies to different problems with, finite and infinite horizons, discounted and long-run-average performance, continuous and jump diffusions, in the same way. Discounting is not needed when dealing with long-run average performance. The approach provides a unified framework for stochastic control and other optimization theory and methodologies, including Markov decision processes, perturbation analysis, and reinforcement learning.