A VARIATIONAL CHARACTERIZATION OF THE RISK-SENSITIVE AVERAGE REWARD FOR CONTROLLED DIFFUSIONS ON Rd

We address the variational formulation of the risk-sensitive reward problem for nondegenerate diffusions on R-d controlled through the drift. We establish a variational formula on the whole space and also show that the risk-sensitive value equals the generalized principal eigenvalue of the semilinear operator. This can be viewed as a controlled version of the variational formulas for principal eigenvalues of diffusion operators arising in large deviations. We also revisit the average risk-sensitive minimization problem, and by employing a gradient estimate developed in this paper, we extend earlier results to unbounded drifts and running costs.