  It is well-known from the work of Kupper and Schachermayer that most
law-invariant risk measures do not admit a time-consistent representation. In
this work we show that in a Brownian filtration the "Optimized Certainty
Equivalent" risk measures of Ben-Tal and Teboulle can be computed through PDE
techniques, i.e. dynamically. This can be seen as a substitute of sorts
whenever they lack time consistency, and covers the cases of conditional
value-at-risk and monotone mean-variance. Our method consists of focusing on
the convex dual representation, which suggests extending the state space. With
this we can obtain a dynamic programming principle and use stochastic control
techniques, along with the theory of viscosity solutions, which we must adapt
to cover the present singular situation.
