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-Jacobi-Bellman equation. Finally, we prove a C1,\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).