  In this paper we present a dynamic programing approach to stochastic optimal
control problems with dynamic, time-consistent risk constraints. Constrained
stochastic optimal control problems, which naturally arise when one has to
consider multiple objectives, have been extensively investigated in the past 20
years, however, in most formulations, the constraints are formulated as either
risk-neutral (i.e., by considering an expected cost), or by applying static,
single-period risk metrics with limited attention to "time-consistency" (i.e.,
to whether such metrics ensure rational consistency of risk preferences across
multiple periods). Recently, significant strides have been made in the
development of a rigorous theory of dynamic, \emph{time-consistent} risk
metrics for multi-period (risk-sensitive) decision processes, however, their
integration within constrained stochastic optimal control problems has received
little attention. The goal of this paper is to bridge this gap. First, we
formulate the stochastic optimal control problem with dynamic, time-consistent
risk constraints and we characterize the tail subproblems (which requires the
addition of a Markovian structure to the risk metrics). Second, we develop a
dynamic programming approach for its solution, which allows to compute the
optimal costs by value iteration. Finally, we discuss both theoretical and
practical features of our approach, such as generalizations, construction of
optimal control policies, and computational aspects. A simple, two-state
example is given to illustrate the problem setup and the solution approach.
