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Buying at Support: The Entry Problem under Multi-Skew Brownian Motion

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  • Jun Maeda

Abstract

We study when to buy a share that will later be sold optimally, when the price follows a geometric multi-skew Brownian motion whose skew levels model support and resistance. The reward for buying is the exit premium of the liquidation problem solved in a companion paper. This premium is strictly $r$-subharmonic inside the exit continuation region, so buying is optimal only at levels where the price is pushed upward, and the entry problem reduces to a finite one whose value is an upper concave hull of finitely many points. With one support and one resistance, the optimal rule is to buy exactly at the support, in every exit regime, with an explicit value.

Suggested Citation

  • Jun Maeda, 2026. "Buying at Support: The Entry Problem under Multi-Skew Brownian Motion," Papers 2610.04990, arXiv.org.
    • Handle: RePEc:arx:papers:2610.04990
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    References listed on IDEAS

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    1. Luis H. R. Alvarez E. & Paavo Salminen, 2017. "Timing in the presence of directional predictability: optimal stopping of skew Brownian motion," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 86(2), pages 377-400, October.
    2. Tim Leung & Xin Li, 2015. "Optimal Mean Reversion Trading With Transaction Costs And Stop-Loss Exit," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 18(03), pages 1-31.
    3. Dayanik, Savas & Karatzas, Ioannis, 2003. "On the optimal stopping problem for one-dimensional diffusions," Stochastic Processes and their Applications, Elsevier, vol. 107(2), pages 173-212, October.
    4. Tim Leung & Xin Li & Zheng Wang, 2014. "Optimal Starting-Stopping and Switching of a CIR Process with Fixed Costs," Papers 1411.6080, arXiv.org.
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