  We address the variational formulation of the risk-sensitive reward problem
for non-degenerate diffusions on $\mathbb{R}^d$ 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.
