### Abstract

Two major financial market complexities are transaction costs and uncertain volatil- ity, and we analyze their joint impact on the problem of portfolio optimization. When volatility is constant, the transaction costs optimal investment problem has a long history, especially in the use of asymptotic approximations when the cost is small. Under stochastic volatility, but with no transaction costs, the Merton problem under general utility functions can also be analyzed with asymptotic methods. Here, we look at the long-run growth rate problem when both complexities are present, using separation of time scales approximations. This leads to perturbation analysis of an eigenvalue problem. We find the first term in the asymptotic expansion in the time scale parameter, of the optimal long-term growth rate, and of the optimal strategy, for fixed small transaction costs. We give a proof of accuracy in the case of fast mean-reverting stochastic volatility, which is based on the finite time problem analyzed in the companion Part II of this paper.

Original language | English (US) |
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Pages (from-to) | 3799-3832 |

Number of pages | 34 |

Journal | SIAM Journal on Control and Optimization |

Volume | 55 |

Issue number | 6 |

DOIs | |

State | Published - Jan 1 2017 |

### All Science Journal Classification (ASJC) codes

- Control and Optimization
- Applied Mathematics

### Keywords

- Asymptotic analysis
- Optimal investment
- Stochastic volatility
- Transaction costs
- Utility maximization