Capacities, Measurable Selection and Dynamic Programming Part II: Application in Stochastic Control Problems

TL;DR

This paper discusses how measurable selection techniques can be used to establish the dynamic programming principle in stochastic control problems, including diffusion processes, and explores approximation and equivalence of different problem formulations.

Contribution

It provides a framework for applying measurable selection to derive dynamic programming principles and demonstrates the equivalence of various control problem formulations.

Findings

Measurable selection techniques enable derivation of dynamic programming principles.

Piecewise constant control approximations are effective for stochastic control problems.

Strong, weak, and relaxed formulations of controlled diffusion problems are shown to be equivalent.

Abstract

We provide an overview on how to use the measurable selection techniques to derive the dynamic programming principle for a general stochastic optimal control/stopping problem. By considering its martingale problem formulation on the canonical space of paths, one can check the required measurability conditions. This covers in particular the most classical controlled/stopped diffusion processes problems. Further, we study the approximation property of the optimal control problems by piecewise constant control problems. As a byproduct, we obtain an equivalence result of the strong, weak and relaxed formulations of the controlled/stopped diffusion processes problem.