Utility-Based Shortfall Risk (UBSR) is a risk metric that is increasingly popular in financial applications, owing to certain desirable properties that it enjoys. We consider the problem of estimating UBSR in a recursive setting, where samples from the underlying loss distribution are available one-at-a-time. We cast the UBSR estimation problem as a root finding problem, and propose stochastic approximation-based estimations schemes. We derive non-asymptotic bounds on the estimation error in the number of samples. We also consider the problem of UBSR optimization within a parameterized class of random variables. We propose a stochastic gradient descent based algorithm for UBSR optimization, and derive non-asymptotic bounds on its convergence.

翻译：基于效用的短缺风险（UBSR）是一种因其具有某些理想性质而在金融应用中日益流行的风险度量指标。我们研究在递归场景下估计UBSR的问题，即底层损失分布的样本按序逐个获取。我们将UBSR估计问题转化为求根问题，并提出基于随机逼近的估计方案。我们推导了关于样本数量的估计误差的非渐近界。我们还研究了参数化随机变量类中UBSR的优化问题。针对UBSR优化，我们提出一种基于随机梯度下降的算法，并推导其收敛性的非渐近界。