“…Since the relationship between these two approaches is the one between the derivatives of the value function and the adjoint processes along the optimal state, or actually the one between HJB equations and stochastic Hamiltonian systems, and more generally, the one between PDEs and SDEs. For recent development of the relationship between dynamic programming and maximum principle for stochastic optimal control problems (without delay but including jump diffusions, Markov switching, singular control, or FBSDE systems), refer to Framstad et al [15], Shi and Wu [34], Donnelly [9], Zhang et al [42], Bahlali et al [4], Shi and Yu [35], Chighoub and Mezerdi [8]. Thereby, it is natural to ask the question: Are there any relations between these two extensively used and important approaches, for stochastic optimal control problems with time delay?…”