On Exponential Utility and Conditional Value-at-Risk as Risk-Averse Performance Criteria
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 $\varphi$ 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 $\varphi(y)$ of a random variable $\varphi(Y)$ that has been transformed to measure preferences about the outcomes. For EU, the transformation is $\varphi(y) = \exp(({-\theta}/{2})y)$ , and under certain conditions, the quantity $\varphi^{-1}(E(\varphi(Y)))$ 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 $Y$ concerns a fraction of its possible realizations. If $Y$ is a continuous random variable with finite $E(|Y|)$ , then the CVaR of $Y$ at level $\alpha$ is the expectation of $Y$ in the $\alpha \cdot 100\%$ 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.