  This paper addresses risk awareness of stochastic optimization problems.
Nested risk measures appear naturally in this context, as they allow beneficial
reformulations for algorithmic treatments. The reformulations presented extend
usual Hamilton-Jacobi-Bellman equations in dynamic optimization by involving
risk awareness in the problem formulation.
  Nested risk measures are built on risk measures, which originate by
conditioning on the history of a stochastic process. We derive martingale
properties of these risk measures and use them to prove continuity. It is
demonstrated that stochastic optimization problems, which incorporate risk
awareness via nesting risk measures, are continuous with respect to the natural
distance governing these optimization problems, the nested distance.
