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On optimal tracking portfolio in incomplete markets: The classical control and the reinforcement learning approaches

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  • Lijun Bo
  • Yijie Huang
  • Xiang Yu

Abstract

This paper studies an infinite horizon optimal tracking portfolio problem using capital injection in incomplete market models. We consider the benchmark process modelled by a geometric Brownian motion with zero drift driven by some unhedgeable risk. The relaxed tracking formulation is adopted where the portfolio value compensated by the injected capital needs to outperform the benchmark process at any time, and the goal is to minimize the cost of the discounted total capital injection. In the first part, we solve the stochastic control problem when the market model is known, for which the equivalent auxiliary control problem with reflections and the associated HJB equation with a Neumann boundary condition are studied. In the second part, the market model is assumed to be unknown, for which we consider the exploratory formulation of the control problem with entropy regularizer and develop the continuous-time q-learning algorithm for the stochastic control problem with state reflections. In an illustrative example, we show the satisfactory performance of the q-learning algorithm.

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  • Lijun Bo & Yijie Huang & Xiang Yu, 2023. "On optimal tracking portfolio in incomplete markets: The classical control and the reinforcement learning approaches," Papers 2311.14318, arXiv.org.
    • Handle: RePEc:arx:papers:2311.14318
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    References listed on IDEAS

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    1. Sun, Yeneng, 2006. "The exact law of large numbers via Fubini extension and characterization of insurable risks," Journal of Economic Theory, Elsevier, vol. 126(1), pages 31-69, January.
    2. Strub, O. & Baumann, P., 2018. "Optimal construction and rebalancing of index-tracking portfolios," European Journal of Operational Research, Elsevier, vol. 264(1), pages 370-387.
    3. Lijun Bo & Huafu Liao & Xiang Yu, 2020. "Optimal Tracking Portfolio with A Ratcheting Capital Benchmark," Papers 2006.13661, arXiv.org, revised Apr 2021.
    4. Chendi Ni & Yuying Li & Peter Forsyth & Ray Carroll, 2022. "Optimal asset allocation for outperforming a stochastic benchmark target," Quantitative Finance, Taylor & Francis Journals, vol. 22(9), pages 1595-1626, September.
    5. Yanwei Jia & Xun Yu Zhou, 2022. "q-Learning in Continuous Time," Papers 2207.00713, arXiv.org, revised Apr 2023.
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