Optimal Two-Mode Switching with Setup Costs for a Controlled Ornstein--Uhlenbeck Process
Abstract
We study an infinite-horizon discounted optimal switching problem for an Ornstein--Uhlenbeck process whose mean-reversion level can be switched between two values. Switching incurs positive, potentially asymmetric fixed costs, and the mode with the higher mean-reversion level entails an additional running cost. The revenue rate is proportional to the positive part of the state. We characterize the value function and identify parameter conditions under which an optimal policy either remains permanently in one mode after a possible costly initial switch or follows a threshold rule. The threshold formulation allows infinite boundaries and therefore includes one-way switching. A key technical contribution is to establish existence and uniqueness of the relevant solutions to the coupled nonlinear free-boundary equations arising from value-matching and smooth-fit conditions. In the corresponding parameter regimes, these solutions determine the finite switching thresholds and the coefficients of the candidate value functions. A verification argument then establishes the optimality of the resulting policies over all admissible switching strategies.
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BibTeX
@misc{xia2026optimal,
title = {{Optimal Two-Mode Switching with Setup Costs for a Controlled Ornstein--Uhlenbeck Process}},
author = {Hongkun Xia and Dacheng Yao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27552},
eprint = {2610.27552},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27552},
}
RIS
TY - UNPB
TI - Optimal Two-Mode Switching with Setup Costs for a Controlled Ornstein--Uhlenbeck Process
AU - Hongkun Xia
AU - Dacheng Yao
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.27552
UR - https://arxiv.org/abs/2610.27552
AB - We study an infinite-horizon discounted optimal switching problem for an Ornstein--Uhlenbeck process whose mean-reversion level can be switched between two values. Switching incurs positive, potentially asymmetric fixed costs, and the mode with the higher mean-reversion level entails an additional running cost. The revenue rate is proportional to the positive part of the state. We characterize the value function and identify parameter conditions under which an optimal policy either remains permanently in one mode after a possible costly initial switch or follows a threshold rule. The threshold formulation allows infinite boundaries and therefore includes one-way switching. A key technical contribution is to establish existence and uniqueness of the relevant solutions to the coupled nonlinear free-boundary equations arising from value-matching and smooth-fit conditions. In the corresponding parameter regimes, these solutions determine the finite switching thresholds and the coefficients of the candidate value functions. A verification argument then establishes the optimality of the resulting policies over all admissible switching strategies.
ER -