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

Nicole El Karoui, Xiaolu Tan

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.

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

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.

Paper Structure

This paper contains 35 sections, 26 theorems, 214 equations.

Table of Contents

  1. Introduction and examples
  2. Introduction
  3. Examples: controlled/stopped diffusion processes problems
  4. A strong formulation of the optimal control/stopping problem
  5. A piecewise constant control problem
  6. A weak formulation of the optimal control/stopping problem
  7. A relaxed formulation of the optimal control/stopping problem
  8. Strong, weak and relaxed formulations on the canonical space
  9. Weak and relaxed formulations by martingale problem
  10. An overview on the dynamic programming principle
  11. Canonical space, measurable selection theorem
  12. Optimization and dynamic programming principle
  13. Some direct consequences of the DPP
  14. Dynamic programming principle of the optimal control and stopping problem
  15. Generators and a controlled/stopped martingale problem
  16. ...and 20 more sections

Key Result

Proposition 1.1

One has and

Theorems & Definitions (54)

  • Remark 1.1
  • Example 1.2: Nisio semi-group problem
  • Definition 1.3
  • Definition 1.4
  • Remark 1.5
  • Remark 1.6
  • Proposition 1.1
  • Remark 1.7
  • Theorem 2.1
  • Theorem 2.2
  • ...and 44 more