A singular control model with application to the goodwill problem

Andrew Jack, Timothy C. Johnson, Mihail Zervos

Research output: Contribution to journalArticlepeer-review

16 Citations (Scopus)


We consider a stochastic system whose uncontrolled state dynamics are modelled by a general one-dimensional Itô diffusion. The control effort that can be applied to this system takes the form that is associated with the so-called monotone follower problem of singular stochastic control. The control problem that we address aims at maximising a performance criterion that rewards high values of the utility derived from the system's controlled state but penalises any expenditure of control effort. This problem has been motivated by applications such as the so-called goodwill problem in which the system's state is used to represent the image that a product has in a market, while control expenditure is associated with raising the product's image, e.g., through advertising. We obtain the solution to the optimisation problem that we consider in a closed analytic form under rather general assumptions. Also, our analysis establishes a number of results that are concerned with analytic as well as probabilistic expressions for the first derivative of the solution to a second-order linear non-homogeneous ordinary differential equation. These results have independent interest and can potentially be of use to the solution of other one-dimensional stochastic control problems. © 2008 Elsevier B.V. All rights reserved.

Original languageEnglish
Pages (from-to)2098-2124
Number of pages27
JournalStochastic Processes and their Applications
Issue number11
Publication statusPublished - Nov 2008


  • Goodwill problem
  • Monotone follower problem
  • Second-order linear ODE's
  • Singular control


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