On Exponential Utility and Conditional Value-at-Risk as Risk-Averse Performance Criteria

Abstract

The standard approach to risk-averse control is to use the exponential utility (EU) functional, which has been studied for several decades. Like other risk-averse utility functionals, EU encodes risk aversion through an increasing convex mapping of objective costs to subjective costs. An objective cost is a realization y of a random variable Y. In contrast, a subjective cost is a realization (y)of a random variable (Y)that has been transformed to measure preferences about the outcomes. For EU, the transformation is (y) = exp ((-θ /2)y), and under certain conditions, the quantity -1 E(\varphi (Y)))can be approximated by a linear combination of the mean and variance of Y. More recently, there has been growing interest in risk-averse control using the conditional value-at-risk (CVaR) functional. In contrast to the EU functional, the CVaR of a random variable Yconcerns a fraction of its possible realizations. If Yis a continuous random variable with finite E(|Y|), then the CVaR of Yat level α is the expectation of Yin the α 100\%worst cases. Here, we study the applications of risk-averse functionals to controller synthesis and safety analysis through the development of numerical examples, with an emphasis on EU and CVaR. Our contribution is to examine the decision-theoretic, mathematical, and computational tradeoffs that arise when using EU and CVaR for optimal control and safety analysis. We are hopeful that this work will advance the interpretability of risk-averse control technology and elucidate its potential benefits.