Risk-sensitive control for a class of diffusions with jumps

We consider a class of diffusions controlled through the drift and jump size, driven by Lévy process nondegenerate Wiener process, we study infinite horizon (ergodic) risk-sensitive control problems for this model. start with Dirichlet eigenvalue problem in smooth bounded domains, which also allows us to generalize current results literature on exit rate problems. Then average minimization maximization whole domain. Under suitable hypotheses, establish existence uniqueness principal eigenfunction Hamilton–Jacobi–Bellman (HJB) operator space, fully characterize stationary Markov optimal controls as measurable selectors HJB equation.