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（Utility-Based Shortfall Risk, UBSR）是一种在金融应用中日益流行的风险度量指标，因其具备某些理想性质而受到青睐。我们探讨在递归设定下估计UBSR的问题，其中基础损失分布的样本依次可用。我们将UBSR估计问题转化为寻根问题，并提出基于随机逼近的估计方案。我们推导了样本数量下估计误差的非渐近界。同时，我们考虑了参数化随机变量类别内的UBSR优化问题。我们提出了一种基于随机梯度下降的UBSR优化算法，并推导了其收敛性的非渐近界。