  We consider a class of dynamic advertising problems under uncertainty in the
presence of carryover and distributed forgetting effects, generalizing a
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. Building on previous work of two of the authors, the
optimal advertising model is formulated as an infinite dimensional stochastic
control problem. We obtain (partial) regularity as well as approximation
results for the corresponding value function. Under specific structural
assumptions we study the effects of delays on the value function and optimal
strategy. In the absence of carryover effects, since the value function and the
optimal advertising policy can be characterized in terms of the solution of the
associated HJB equation, we obtain sharper characterizations of the optimal
policy.
