  Optimized certainty equivalents (OCEs) is a family of risk measures widely
used by both practitioners and academics. This is mostly due to its
tractability and the fact that it encompasses important examples, including
entropic risk measures and average value at risk. In this work we consider
stochastic optimal control problems where the objective criterion is given by
an OCE risk measure, or put in other words, a risk minimization problem for
controlled diffusions. A major difficulty arises since OCEs are often time
inconsistent. Nevertheless, via an enlargement of state space we achieve a
substitute of sorts for time consistency in fair generality. This allows us to
derive a dynamic programming principle and thus recover central results of
(risk-neutral) stochastic control theory. In particular, we show that the value
of our risk minimization problem can be characterized via the viscosity
solution of a Hamilton--Jacobi--Bellman--Issacs equation. We further establish
the uniqueness of the latter under suitable technical conditions.
