SIAM Journal on Control and Optimization, Volume 58, Issue 6, Page 3785-3813, January 2020.
We address the variational formulation of the risk-sensitive reward problem for nondegenerate 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.

中文翻译：

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$ \ mathbb {R} ^ d $上受控扩散的风险敏感平均奖励的变化特征

SIAM杂志上控制与优化，第58卷，第6期，第3785-3813，2020年一月
我们应对风险敏感的奖励问题的变分列式上$ \ mathbb {R} ^ d $简并扩散通过漂移控制。我们在整个空间上建立了一个变分公式，并且表明风险敏感值等于半线性算子的广义本征值。这可以看作是偏差较大的扩散算子的主要特征值的变分公式的受控形式。我们还重新审视了平均风险敏感的最小化问题，并且通过采用本文中开发的梯度估计，我们将较早的结果扩展到无穷大的漂移和运营成本。