### Abstract

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 language | English |
---|---|

Pages (from-to) | 2098-2124 |

Number of pages | 27 |

Journal | Stochastic Processes and their Applications |

Volume | 118 |

Issue number | 11 |

DOIs | |

Publication status | Published - Nov 2008 |

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### Keywords

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

### Cite this

*Stochastic Processes and their Applications*,

*118*(11), 2098-2124. https://doi.org/10.1016/j.spa.2008.01.001

}

*Stochastic Processes and their Applications*, vol. 118, no. 11, pp. 2098-2124. https://doi.org/10.1016/j.spa.2008.01.001

**A singular control model with application to the goodwill problem.** / Jack, Andrew; Johnson, Timothy C.; Zervos, Mihail.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A singular control model with application to the goodwill problem

AU - Jack, Andrew

AU - Johnson, Timothy C.

AU - Zervos, Mihail

PY - 2008/11

Y1 - 2008/11

N2 - 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.

AB - 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.

KW - Goodwill problem

KW - Monotone follower problem

KW - Second-order linear ODE's

KW - Singular control

UR - http://www.scopus.com/inward/record.url?scp=38549110770&partnerID=8YFLogxK

U2 - 10.1016/j.spa.2008.01.001

DO - 10.1016/j.spa.2008.01.001

M3 - Article

VL - 118

SP - 2098

EP - 2124

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

SN - 0304-4149

IS - 11

ER -