# A variational characterization of the risk-sensitive average reward for controlled diffusions on $\mathbb{R}^d$

## TL;DR

This paper develops a variational framework for the risk-sensitive reward problem in controlled diffusions on bd, linking it to eigenvalues of associated operators and extending results to unbounded drifts and costs.

## Contribution

It introduces a variational formula for the risk-sensitive value and connects it to the principal eigenvalue of a semilinear operator, extending previous results.

## Key findings

- Established a variational formula on bd for the risk-sensitive reward.
- Showed the risk-sensitive value equals the generalized principal eigenvalue.
- Extended results to unbounded drifts and costs using a new gradient estimate.

## Abstract

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.