  In this manuscript we consider a class optimal control problem for stochastic
differential delay equations. First, we rewrite the problem in a suitable
infinite-dimensional Hilbert space. Then, using the dynamic programming
approach, we characterize the value function of the problem as the unique
viscosity solution of the associated infinite-dimensional
Hamilton-Jacobi-Bellman equation. Finally, we prove a $C^{1,\alpha}$-partial
regularity of the value function. We apply these results to path dependent
financial and economic problems (Merton-like portfolio problem and optimal
advertising).
