Dynamic Programming for controlled Markov families: abstractly and over Martingale Measures

TL;DR

This paper develops an abstract control-theoretic framework for dynamic programming in continuous time, applicable to financial mathematics problems involving sets of martingale measures, such as hedging and utility maximization.

Contribution

It introduces a novel abstract framework that verifies dynamic programming principles with minimal structural assumptions, specifically tailored for problems over martingale measures.

Findings

Established dynamic programming principle in continuous time for controlled Markov families.

Applied framework to lower-hedging and utility-maximization in finance.

Demonstrated the approach's effectiveness in complex financial models.

Abstract

We describe an abstract control-theoretic framework in which the validity of the dynamic programming principle can be established in continuous time by a verification of a small number of structural properties. As an application we treat several cases of interest, most notably the lower-hedging and utility-maximization problems of financial mathematics both of which are naturally posed over ``sets of martingale measures''.