Ergodic control of a class of jump diffusions with finite L\'evy measures and rough kernels

TL;DR

This paper investigates the ergodic control of jump diffusions with finite Levy measures and rough kernels, providing a weak formulation approach, deriving the Hamilton-Jacobi-Bellman equation, and characterizing optimality without strong assumptions.

Contribution

It introduces a novel weak formulation for ergodic control of jump diffusions with minimal assumptions and characterizes optimal controls comprehensively.

Findings

Derived the Hamilton-Jacobi-Bellman equation under minimal assumptions

Established verification of optimality using analytical methods

Provided regularity results for invariant measures

Abstract

We study the ergodic control problem for a class of jump diffusions in $\mathbb{R}^d$, which are controlled through the drift with bounded controls. The Levy measure is finite, but has no particular structure; it can be anisotropic and singular. Moreover, there is no blanket ergodicity assumption for the controlled process. Unstable behavior is `discouraged' by the running cost which satisfies a mild coercive hypothesis (i.e., is near-monotone). We first study the problem in its weak formulation as an optimization problem on the space of infinitesimal ergodic occupation measures, and derive the Hamilton-Jacobi-Bellman equation under minimal assumptions on the parameters, including verification of optimality results, using only analytical arguments. We also examine the regularity of invariant measures. Then, we address the jump diffusion model, and obtain a complete characterization of optimality.