Online Estimation and Optimization of Utility-Based Shortfall Risk

TL;DR

This paper develops recursive estimation and optimization algorithms for Utility-Based Shortfall Risk (UBSR), providing theoretical bounds on their accuracy and convergence in a sequential sampling setting.

Contribution

It introduces stochastic approximation methods for UBSR estimation and a stochastic gradient algorithm for UBSR optimization, with non-asymptotic error bounds.

Findings

Derived non-asymptotic bounds on estimation error.

Established convergence bounds for the optimization algorithm.

Proposed recursive schemes suitable for sequential data.

Abstract

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.