Dynamic Programming Principle for Stochastic Control Problems driven by General L\'{e}vy Noise

TL;DR

This paper extends the dynamic programming principle to stochastic control problems driven by general Lévy noise, even when the state process lacks moments, broadening the applicability of the DPP in stochastic control theory.

Contribution

It provides a proof of the DPP for control problems with Lévy noise without requiring the state process to have moments, which was not previously established.

Findings

DPP holds for Lévy-driven control problems without moment conditions

The proof extends to more general stochastic processes

Broadens the theoretical foundation of stochastic control

Abstract

We extend the proof of the dynamic programming principle (DPP) for standard stochastic optimal control problems driven by general L\'{e}vy noises. Under appropriate assumptions, it is shown that the DPP still holds when the state process fails to have any moments at all.