A Framework for the Dynamic Programming Principle and Martingale-generated Control Correspondences

TL;DR

This paper develops an abstract framework to prove the dynamic programming principle (DPP) for various stochastic control problems, including controlled diffusions and singular control, facilitating analysis and solution characterization.

Contribution

It introduces a broad, unified framework for establishing the DPP in weak and martingale formulations, simplifying proofs and extending applicability to complex control problems.

Findings

Established DPP for controlled diffusions with viscosity solutions.

Applied framework to singular control, specifically the monotone-follower problem.

Demonstrated minimal conditions for DPP and solution characterization.

Abstract

We construct an abstract framework in which the dynamic programming principle (DPP) can be readily proven. It encompasses a broad range of common stochastic control problems in the weak formulation, and deals with problems in the "martingale formulation" with particular ease. We give two illustrations; first, we establish the DPP for general controlled diffusions and show that their value functions are viscosity solutions of the associated Hamilton-Jacobi-Bellman equations under minimal conditions. After that, we show how to treat singular control on the example of the classical monotone-follower problem.