A Singular Control Model with Application to the Goodwill Problem

TL;DR

This paper solves a singular stochastic control problem related to the goodwill of a product, providing explicit solutions and new analytical results that could benefit similar control challenges.

Contribution

It offers a closed-form solution to a singular control problem with applications to marketing, and introduces new analytical methods for differential equations in stochastic control.

Findings

Explicit solution for the control problem under general assumptions

Analytical expressions for derivatives of differential equation solutions

Potential applicability to other one-dimensional stochastic control problems

Abstract

We consider a stochastic system whose uncontrolled state dynamics are modelled by a general one-dimensional It\^{o} 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.