### Abstract

We prove a verification theorem for a class of singular control problems which model optimal harvesting with density-dependent prices or optimal dividend policy with capital-dependent utilities. The result is applied to solve explicitly some examples of such optimal harvesting/optimal dividend problems. In particular, we show that if the unit price decreases with population density, then the optimal harvesting policy may not exist in the ordinary sense, but can be expressed as a “chattering policy”, i.e. the limit as Δ x and Δ t go to 0 of taking out a sequence of small quantities of size Δ x within small time periods of size Δ t.

Original language | English |
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Pages (from-to) | 427-442 |

Number of pages | 16 |

Journal | Afrika Matematika |

Volume | 27 |

Issue number | 3-4 |

DOIs | |

Publication status | Published - Jun 1 2016 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Afrika Matematika*,

*27*(3-4), 427-442. https://doi.org/10.1007/s13370-015-0357-0

}

*Afrika Matematika*, vol. 27, no. 3-4, pp. 427-442. https://doi.org/10.1007/s13370-015-0357-0

**Optimal multi-dimensional stochastic harvesting with density-dependent prices.** / Alvarez, Luis H.R.; Lungu, Edward; Øksendal, Bernt.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Optimal multi-dimensional stochastic harvesting with density-dependent prices

AU - Alvarez, Luis H.R.

AU - Lungu, Edward

AU - Øksendal, Bernt

PY - 2016/6/1

Y1 - 2016/6/1

N2 - We prove a verification theorem for a class of singular control problems which model optimal harvesting with density-dependent prices or optimal dividend policy with capital-dependent utilities. The result is applied to solve explicitly some examples of such optimal harvesting/optimal dividend problems. In particular, we show that if the unit price decreases with population density, then the optimal harvesting policy may not exist in the ordinary sense, but can be expressed as a “chattering policy”, i.e. the limit as Δ x and Δ t go to 0 of taking out a sequence of small quantities of size Δ x within small time periods of size Δ t.

AB - We prove a verification theorem for a class of singular control problems which model optimal harvesting with density-dependent prices or optimal dividend policy with capital-dependent utilities. The result is applied to solve explicitly some examples of such optimal harvesting/optimal dividend problems. In particular, we show that if the unit price decreases with population density, then the optimal harvesting policy may not exist in the ordinary sense, but can be expressed as a “chattering policy”, i.e. the limit as Δ x and Δ t go to 0 of taking out a sequence of small quantities of size Δ x within small time periods of size Δ t.

UR - http://www.scopus.com/inward/record.url?scp=84971278465&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84971278465&partnerID=8YFLogxK

U2 - 10.1007/s13370-015-0357-0

DO - 10.1007/s13370-015-0357-0

M3 - Article

VL - 27

SP - 427

EP - 442

JO - Afrika Matematika

JF - Afrika Matematika

SN - 1012-9405

IS - 3-4

ER -