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:实用-基于短期风险(UBSR)是一种风险衡量标准,在金融应用中越来越受欢迎,这是由于它享有某些可取的特性。我们考虑在循环环境中估算UBSR的问题,因为基础损失分布的样本是一次性的。我们把UBSR估算问题作为一个根本发现问题,并提出了基于随机估算的随机估算方法。我们从样本数量的估计误差中得出非抽取的近似估计方法。我们还考虑UBSR优化在随机变量参数分类中的问题。我们提出了基于UBSR优化的随机梯度梯度下降算法,并得出了非抽查的趋同界限。