# Ergodic control of diffusions with compound Poisson jumps under a general structural hypothesis

## TL;DR

This paper extends ergodic control theory for jump diffusions driven by compound Poisson processes, providing a full characterization of optimality, regularity of solutions, and practical methods for near-optimal control in complex network applications.

## Contribution

It generalizes previous results to non-near-monotone costs, characterizes optimal controls via HJB equations, and offers practical approaches for scheduling in large-scale networks.

## Key findings

- Full characterization of optimality via HJB equations.
- Regularity results for solutions under mild conditions.
- Method to construct near-optimal controls by solving HJB on large domains.

## Abstract

We study the ergodic control problem for a class of controlled jump diffusions driven by a compound Poisson process. This extends the results of [SIAM J. Control Optim. 57 (2019), no. 2, 1516-1540] to running costs that are not near-monotone. This generality is needed in applications such as optimal scheduling of large-scale parallel server networks. We provide a full characterization of optimality via the Hamilton-Jacobi-Bellman (HJB) equation, for which we additionally exhibit regularity of solutions under mild hypotheses. In addition, we show that optimal stationary Markov controls are a.s. pathwise optimal. Lastly, we show that one can fix a stable control outside a compact set and obtain near-optimal solutions by solving the HJB on a sufficiently large bounded domain. This is useful for constructing asymptotically optimal scheduling policies for multiclass parallel server networks.