On the dynamic representation of some time-inconsistent risk measures in a Brownian filtration

It is well-known from the work of Kupper and Schachermayer that most law-invariant risk measures are not time-consistent, and thus do not admit dynamic representations as backward stochastic differential equations. 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 an expression of the risk measure as the value of a stochastic control problem on an extended 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.