Weak Dynamic Programming for Generalized State Constraints
  We provide a dynamic programming principle for stochastic optimal control problems with expectation constraints. A weak formulation, using test functions and a probabilistic relaxation of the constraint, avoids restrictions related to a measurable selection but still implies the Hamilton-Jacobi-Bellman equation in the viscosity sense. We treat open state constraints as a special case of expectation constraints and prove a comparison theorem to obtain the equation for closed state constraints. 
Bruno Bouchard, Marcel Nutz
math.OC cs.SY math.AP math.PR q-fin.RM