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.