Weak Dynamic Programming for Generalized State Constraints
It provides a theoretical foundation for solving stochastic control problems with general state constraints, which is important for researchers in optimal control and stochastic analysis.
The paper develops a weak dynamic programming principle for stochastic optimal control problems with expectation constraints, enabling the derivation of the Hamilton-Jacobi-Bellman equation in the viscosity sense for both open and closed 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.