I am currently working on the optimal control of certain classes of stochastic processes and I have difficulties understanding the roles of verification theorems.

My problem is the following: I am not sure to understand whether this is purely a problem that arises from the use of partial differential equations for which we may need to consider viscosity solutions or whether this is something related to the connection between the optimal control problem and its solution expressed in terms of the value function which is a solution to the Hamilton-Jacobi-Bellman (HJB) equation, which is a PDE in many instances of those problems. Would not be the Bellman optimality principle ensures that the optimal control can be computed from the value function itself a solution of the HJB equation?

I am also wondering whether this a specificity of stochastic optimal control problems because I am not sure to have seen verification theorems in the deterministic setting.

Thanks and feel free to comment to ask for more details.