A Time Consistent Formulation of Risk Constrained Stochastic Optimal Control

Time-consistency is an essential requirement in risk sensitive optimal control problems to make rational decisions. An optimization problem is time consistent if its solution policy does not depend on the time sequence of solving the optimization problem. On the other hand, a dynamic risk measure is time consistent if a certain outcome is considered less risky in the future implies this outcome is also less risky at current stage. In this paper, we study time-consistency of risk constrained problem where the risk metric is time consistent. From the Bellman optimality condition in [1], we establish an analytical "risk-to-go" that results in a time consistent optimal policy. Finally we demonstrate the effectiveness of the analytical solution by solving Haviv's counter-example [2] in time inconsistent planning.