### Abstract

We study a Merton type optimization problem under a reallocation constraint. Under this restriction, the stock holdings can not be liquidated faster than a certain rate. This is a common restriction in certain type of investment firms. Our main objective is to study the large time optimal growth rate of the expected value of the utility from wealth. We also consider a discounted infinite horizon problem as a step towards understanding the first problem. A numerical study is done by solving the dynamic programming equations. Under the assumption of a power utility function, an appropriate dimension reduction argument is used to reduce the original problem to a two dimensional one in a bounded domain with convenient boundary conditions. Computation of the optimal growth rate introduces additional numerical difficulties as the straightforward approach is unstable. In this direction, new analytical results characterizing the growth rate as the limit of a sequence of finite horizon problems with continuously derived utility are proved.

Original language | English (US) |
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Pages (from-to) | 551-585 |

Number of pages | 35 |

Journal | Mathematical Methods of Operations Research |

Volume | 71 |

Issue number | 3 |

DOIs | |

State | Published - Jun 1 2010 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Software
- Mathematics(all)
- Management Science and Operations Research

### Keywords

- Dynamic programming
- Merton problem
- Reallocation constraint

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## Cite this

*Mathematical Methods of Operations Research*,

*71*(3), 551-585. https://doi.org/10.1007/s00186-010-0306-5