Optimal control of stochastic delay differential equations and applications to path-dependent financial and economic models

In this manuscript we consider a class of optimal control problems of 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-JacobiBellman 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).