We consider a class of optimal advertising problems under uncertainty for the introduction of a new product into the market, on the line of the seminal papers of Vidale and Wolfe, 1957, and Nerlove and Arrow, 1962. The main features of our model are that, on one side, we assume a carryover effect (i.e. the advertisement spending affects the goodwill with some delay); on the other side we introduce, in the state equation and in the objective, some mean field terms which take into account the presence of other agents. We take the point of view of a planner who optimizes the average profit of all agents, hence we fall into the family of the so-called "Mean Field Control" problems. The simultaneous presence of the carryover effect makes the problem infinite dimensional hence belonging to a family of problems which are very difficult in general and whose study started only very recently, see Cosso et Al, 2023. Here we consider, as a first step, a simple version of the problem providing the solutions in a simple case through a suitable auxiliary problem.
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
Spread, then Target, and Advertise in Waves: Optimal Budget Allocation Across Advertising Channels
Yale University（耶鲁大学）；University of Pennsylvania（宾夕法尼亚大学）；Cornell University（康奈尔大学）；University of California, Davis（加州大学戴维斯分校）；Army Research Laboratory（陆军研究实验室）
