Online Estimation and Optimization of Utility-Based Shortfall Risk

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, in which 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 estimation schemes. We derive nonasymptotic 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 nonasymptotic bounds on its convergence.