Stochastic optimal control problems with delays in the state and in the control via viscosity solutions and applications to optimal advertising and optimal investment problems

In this manuscript we consider optimal control problems of stochastic differential equations with delays in the state and in the control. First, we prove an equivalent Markovian reformulation on Hilbert spaces of the state equation. Then, using the dynamic programming approach for infinite-dimensional systems, we prove that the value function is the unique viscosity solution of the infinite-dimensional Hamilton-Jacobi-Bellman equation. We apply these results to problems coming from economics: stochastic optimal advertising problems and stochastic optimal investment problems with time-to-build.