  Variance plays a crucial role in risk-sensitive reinforcement learning, and
most risk measures can be analyzed via variance. In this paper, we consider two
law-invariant risks as examples: mean-variance risk and exponential utility
risk. With the aid of the state-augmentation transformation (SAT), we show
that, the two risks can be estimated in Markov decision processes (MDPs) with a
stochastic transition-based reward and a randomized policy. To relieve the
enlarged state space, a novel definition of isotopic states is proposed for
state lumping, considering the special structure of the transformed transition
probability. In the numerical experiment, we illustrate state lumping in the
SAT, errors from a naive reward simplification, and the validity of the SAT for
the two risk estimations.
