A model for optimal stopping in advertisement

In this work we propose a model for optimal advertisement in new product diffusion based on the Bass model and assuming that the effect of the environmental pressure in the diffusion of the product is subject to a stochastic dependence. The optimal stopping problem is reduced to a free boundary problem which is analyzed and solved numerically, in order to determine an optimal stopping rule for the advertisement campaign. The numerical solution is obtained through a policy iteration like contraction scheme, the convergence properties of which are studied in detail. Furthermore, the expected time until the optimal stopping of the campaign is estimated. Finally, a combined optimal stopping and control problem for the optimization of the advertisement effectiveness is also proposed and solved numerically. Our results are expected to provide useful guidelines for campaign managers, for the choice of effectiveness and duration of an advertisement campaign.