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 t