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We study the ergodic control problem for a class of controlled jumpdiffusions 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 arenot near-monotone. This generality is needed in applications such as optimalscheduling of large-scale parallel server networks.  We provide a full characterization of optimality via theHamilton-Jacobi-Bellman (HJB) equation, for which we additionally exhibitregularity of solutions under mild hypotheses. In addition, we show thatoptimal stationary Markov controls are a.s. pathwise optimal. Lastly, we showthat one can fix a stable control outside a compact set and obtain near-optimalsolutions by solving the HJB on a sufficiently large bounded domain. This isuseful for constructing asymptotically optimal scheduling policies formulticlass parallel server networks.

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