Capacities, Measurable Selection and Dynamic Programming Part I: Abstract Framework

TL;DR

This paper introduces a unified abstract framework for capacity theory, measurable selection, and dynamic programming, extending classical discrete methods to continuous time stochastic control problems.

Contribution

It provides a novel abstract framework connecting capacity theory with measurable selection and dynamic programming in both discrete and continuous time settings.

Findings

Derived a measurable selection theorem from capacity theory.

Presented a classical method for discrete time stochastic control using measurable selection.

Proposed an abstract framework for continuous time dynamic programming.

Abstract

We give a brief presentation of the capacity theory and show how it derives naturally a measurable selection theorem following the approach of Dellacherie (1972). Then we present the classical method to prove the dynamic programming of discrete time stochastic control problem, using measurable selection arguments. At last, we propose a continuous time extension, that is an abstract framework for the continuous time dynamic programming principle (DPP).