Optimal advertising under uncertainty with carryover eects

We consider a class of dynamic advertising problems under uncertainty in the presence of carryover and distributed forgetting eects, generalizing the classical model of Nerlove and Arrow. In particular, we allow the dynamics of the product goodwill to depend on its past values, as well as previous advertising levels. The optimal advertising model is formulated as an infinite dimensional stochastic control problem to which we associate, through the dynamic programming principle, a Hamilton-Jacobi-Bellman (HJB) equation. In the absence of carryover advertising eects, the value function of the problem and the optimal advertising policy can be characterized (in some simple cases even explicitly) in terms of the solution of the associated HJB equation. In the general case, due to the lack of a solvability theory for the corresponding HJB equation, only partial results can be obtained. In particular, we fully characterize the value function and the optimal state-control pair in two special cases where the HJB equation admits a closed-form solution.