Bounding Conditional Value-at-Risk via Auxiliary Distributions with Bounded Discrepancies

TL;DR

This paper introduces a theoretical method to bound the Conditional Value-at-Risk (CVaR) of a random variable by leveraging an auxiliary variable with bounded discrepancies, aiding in risk estimation when direct data is limited.

Contribution

It develops a novel framework for bounding CVaR using related variables with bounded discrepancies, enabling practical risk assessment with limited information.

Findings

Provides bounds for CVaR using auxiliary distributions

Offers concentration inequalities for tail risk control

Facilitates risk estimation with limited data

Abstract

In this paper, we develop a theoretical framework for bounding the CVaR of a random variable $X$ using another related random variable $Y$, under assumptions on their cumulative and density functions. Our results yield practical tools for approximating $\operatorname{CVaR}_\alpha(X)$ when direct information about $X$ is limited or sampling is computationally expensive, by exploiting a more tractable or observable random variable $Y$. Moreover, the derived bounds provide interpretable concentration inequalities that quantify how the tail risk of $X$ can be controlled via $Y$.