A Model for Optimally Advertising and Launching a Product

We formulate and solve a problem that combines the features of the so-called monotone follower of singular stochastic control theory with optimal stopping. In particular, we consider a stochastic system whose uncontrolled state dynamics are modelled by a general one-dimensional Itô diffusion. The aim of the problem that we solve is to maximise the utility derived from the system's state at the discretionary time when the system's control is terminated. This objective is reflected by the performance criterion that we maximise, which also penalises control expenditure as well as waiting. The model that we study is motivated by the so-called goodwill problem, a variant of which is concerned with how to optimally raise a new product's image, e.g., through advertising, and with determining the best time to launch the product into the market. In the presence of the rather general assumptions that we make, we fully characterise the optimal strategy, which can take one of three qualitatively different forms, depending on the problem data.