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We consider a stochastic system whose uncontrolled state dynamics aremodelled by a general one-dimensional It\^{o} diffusion. The control effortthat can be applied to this system takes the form that is associated with theso-called monotone follower problem of singular stochastic control. The controlproblem that we address aims at maximising a performance criterion that rewardshigh values of the utility derived from the system's controlled state butpenalises any expenditure of control effort. This problem has been motivated byapplications such as the so-called goodwill problem in which the system's stateis used to represent the image that a product has in a market, while controlexpenditure is associated with raising the product's image, e.g., throughadvertising. We obtain the solution to the optimisation problem that weconsider in a closed analytic form under rather general assumptions. Also, ouranalysis establishes a number of results that are concerned with analytic aswell as probabilistic expressions for the first derivative of the solution to asecond order linear non-homogeneous ordinary differential equation. Theseresults have independent interest and can potentially be of use to the solutionof other one-dimensional stochastic control problems.

A Singular Control Model with Application to the Goodwill Problem