Stochastic Optimal Control of Delay Equations Arising in Advertising Models

In this chapter we consider a class of stochastic optimal control problems where the state equation is a stochastic delay differential equations (SDDEs). Such problems arise for instance in modeling optimal advertising under uncertainty for the introduction of a new product to the market, generalizing classical work of Nerlove and Arrow [30]. The main novelty in our model is that we deal also with delays in the control: this is interesting from the applied point of view and introduces new mathematical difficulties in the problem.