Martingale approach to control for general jump processes

TL;DR

This paper develops verification theorems for infinite horizon stochastic control problems involving semimartingales, allowing control over drift, diffusion, and jump intensity, thus generalizing classical diffusion control frameworks.

Contribution

It introduces a martingale formulation for control problems that includes control over jump intensities, extending standard Itô and Lévy diffusion control models.

Findings

Generalizes control frameworks to include jump intensity control

Provides explicit solutions in illustrative examples

Establishes verification theorems for broad classes of semimartingale controls

Abstract

We provide verification theorems (at different levels of generality) for infinite horizon stochastic control problems in continuous time for semimartingales. The control framework is given as an abstract "martingale formulation", which encompasses a broad range of standard control problems. Under appropriate conditions we show that the set of admissible controls gives rise to a certain class of controlled special semimartingales. Our results generalise both the standard controlled It\^o- and L\'evy-diffusion settings as we allow ourselves to locally control not only the drift and diffusion coefficients, but also the jump intensity measure of the jumps. As an illustration, we present a few examples with explicit solutions.