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 phi 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 phi(y) of a random variable phi(Y) that has been transformed to measure preferences about the outcomes. For EU, the transformation is phi(y) = exp((-theta/2) y), and under certain conditions, the quantity phi(-1)(E(phi(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(vertical bar Y vertical bar), then the CVaR of Y at level a is the expectation of Y in the alpha . 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.

Automated summary

The standard approach to risk-averse control is to use the exponential utility (EU) and more recently, there has been growing interest in using the conditional value-at-risk (CVaR) for risk-averse control. This study aims to examine the applications of these risk-averse functionals to controller synthesis and safety analysis through numerical examples.