Non-equivalence of stochastic optimal control problems with open and closed loop controls

For an optimal control problem of Itô’s type stochastic differential equation, the process could be taken in open-loop or closed-loop forms. In standard literature, provided appropriate regularity, value functions under these two types controls are equal and unique (viscosity) solution to corresponding (path-dependent) HJB equation. this short note, we provide a counterexample path dependent setting showing that can different general.