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Stochastic Control Problems with Infinite Horizon and Regime Switching Arising in Optimal Liquidation with Semimartingale Strategies

Author

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  • Xinman Cheng
  • Guanxing Fu
  • Xiaonyu Xia

Abstract

We study an optimal control problem on infinite time horizon with semimartingale strategies, random coefficients and regime switching. The value function and the optimal strategy can be characterized in terms of three systems of backward stochastic differential equations (BSDEs) with infinite horizon. One of them is a system of linear BSDEs with unbounded coefficients and infinite horizon, which seems to be new in literature. We establish the existence of the solutions to these BSDEs by BMO analysis and comparison theorem for multi-dimensional BSDEs. Next, we establish that the optimal control problem is well posed, in the sense that the value function is finite and the optimal strategy-when it exists-is unique. This is achieved by reformulating the cost functional as the sum of a quadratic functional and the candidate value function. The reformulation crucially relies on the well-established well-posedness results for systems of BSDEs. Finally, under additional assumptions, we obtain the unique optimal strategy.

Suggested Citation

  • Xinman Cheng & Guanxing Fu & Xiaonyu Xia, 2026. "Stochastic Control Problems with Infinite Horizon and Regime Switching Arising in Optimal Liquidation with Semimartingale Strategies," Papers 2602.20552, arXiv.org, revised Feb 2026.
    • Handle: RePEc:arx:papers:2602.20552
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    References listed on IDEAS

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    1. Xinman Cheng & Guanxing Fu & Xiaonyu Xia, 2025. "Long time behavior of optimal liquidation problems with semimartingale strategies and external flows," Mathematics and Financial Economics, Springer, volume 19, number 3, January.
    2. Ying Hu & Gechun Liang & Shanjian Tang, 2018. "Systems of ergodic BSDEs arising in regime switching forward performance processes," Papers 1807.01816, arXiv.org, revised Jun 2020.
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    5. Julia Ackermann & Thomas Kruse & Mikhail Urusov, 2020. "C\`adl\`ag semimartingale strategies for optimal trade execution in stochastic order book models," Papers 2006.05863, arXiv.org, revised Jul 2021.
    6. Julia Ackermann & Thomas Kruse & Mikhail Urusov, 2024. "Reducing Obizhaeva–Wang-type trade execution problems to LQ stochastic control problems," Finance and Stochastics, Springer, vol. 28(3), pages 813-863, July.
    7. Siu, Chi Chung & Guo, Ivan & Zhu, Song-Ping & Elliott, Robert J., 2019. "Optimal execution with regime-switching market resilience," Journal of Economic Dynamics and Control, Elsevier, vol. 101(C), pages 17-40.
    8. Guanxing Fu & Xiaomin Shi & Zuo Quan Xu, 2024. "A System of BSDEs with Singular Terminal Values Arising in Optimal Liquidation with Regime Switching," Papers 2412.19058, arXiv.org, revised Jan 2025.
    9. Julia Ackermann & Thomas Kruse & Mikhail Urusov, 2025. "Multi-asset optimal trade execution with stochastic cross-effects: An Obizhaeva-Wang-type framework," Papers 2503.05594, arXiv.org, revised Mar 2026.
    10. Moustapha Pemy & Qing Zhang & G. George Yin, 2008. "Liquidation Of A Large Block Of Stock With Regime Switching," Mathematical Finance, Wiley Blackwell, vol. 18(4), pages 629-648, October.
    11. Li, Xinying & Zhang, Yaqi & Fan, Shengjun, 2025. "Weighted solutions of random time horizon BSDEs with stochastic monotonicity and general growth generators and related PDEs," Stochastic Processes and their Applications, Elsevier, vol. 190(C).
    12. Antonis Papapantoleon & Dylan Possamai & Alexandros Saplaouras, 2016. "Existence and uniqueness results for BSDEs with jumps: the whole nine yards," Papers 1607.04214, arXiv.org, revised Nov 2018.
    13. Guanxing Fu & Ulrich Horst & Xiaonyu Xia, 2022. "A Mean-Field Control Problem of Optimal Portfolio Liquidation with Semimartingale Strategies," Papers 2207.00446, arXiv.org, revised Sep 2023.
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