We consider the problems of estimation and optimization of utility-based shortfall risk (UBSR), which is a popular risk measure in finance. In the context of UBSR estimation, we derive a non-asymptotic bound on the mean-squared error of the classical sample average approximation (SAA) of UBSR. Next, in the context of UBSR optimization, we derive an expression for the UBSR gradient under a smooth parameterization. This expression is a ratio of expectations, both of which involve the UBSR. We use SAA for the numerator as well as denominator in the UBSR gradient expression to arrive at a biased gradient estimator. We derive non-asymptotic bounds on the estimation error, which show that our gradient estimator is asymptotically unbiased. We incorporate the aforementioned gradient estimator into a stochastic gradient (SG) algorithm for UBSR optimization. Finally, we derive non-asymptotic bounds that quantify the rate of convergence of our SG algorithm for UBSR optimization.

翻译：我们考虑了基于效用的短fall风险（UBSR）的估计与优化问题，这是金融领域一种常用的风险度量。在UBSR估计方面，我们推导了经典样本均值近似（SAA）的均方误差的非渐近界。接着，在UBSR优化方面，我们推导了光滑参数化下UBSR梯度的表达式，该表达式为两个期望的比值，且两者均涉及UBSR。我们采用SAA来近似UBSR梯度表达式中的分子与分母，从而得到有偏的梯度估计量。我们推导了估计误差的非渐近界，表明该梯度估计量是渐近无偏的。我们将上述梯度估计量融入随机梯度（SG）算法中，用于UBSR优化。最后，我们推导了量化SG算法在UBSR优化问题中收敛速度的非渐近界。