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Mean Field Game of Optimal Tracking Portfolio

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  • Lijun Bo
  • Yijie Huang
  • Xiang Yu

Abstract

This paper studies the mean field game (MFG) problem arising from a large population competition in fund management, featuring a new type of relative performance via the benchmark tracking constraint. In the n-agent model, each agent can strategically inject capital to ensure that the total wealth outperforms the benchmark process, which is modeled as a linear combination of the population's average wealth process and an exogenous market index process. That is, each agent is concerned about the performance of her competitors captured by the floor constraint. With a continuum of agents, we formulate the constrained MFG problem and transform it into an equivalent unconstrained MFG problem with a reflected state process. We establish the existence of the mean field equilibrium (MFE) using the PDE approach. Firstly, by applying the dual transform, the best response control of the representative agent can be characterized in analytical form in terms of a dual reflected diffusion process. As a novel contribution, we verify the consistency condition of the MFE in separated domains with the help of the duality relationship and properties of the dual process.

Suggested Citation

  • Lijun Bo & Yijie Huang & Xiang Yu, 2025. "Mean Field Game of Optimal Tracking Portfolio," Papers 2505.01858, arXiv.org.
    • Handle: RePEc:arx:papers:2505.01858
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    References listed on IDEAS

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    1. Gilles-Edouard Espinosa & Nizar Touzi, 2015. "Optimal Investment Under Relative Performance Concerns," Mathematical Finance, Wiley Blackwell, vol. 25(2), pages 221-257, April.
    2. Guanxing Fu & Chao Zhou, 2023. "Mean field portfolio games," Finance and Stochastics, Springer, vol. 27(1), pages 189-231, January.
    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. Abraham, Romain, 2000. "Reflecting Brownian snake and a Neumann-Dirichlet problem," Stochastic Processes and their Applications, Elsevier, vol. 89(2), pages 239-260, October.
    5. 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.
    6. Daniel Lacker & Thaleia Zariphopoulou, 2019. "Mean field and n‐agent games for optimal investment under relative performance criteria," Mathematical Finance, Wiley Blackwell, vol. 29(4), pages 1003-1038, October.
    7. Guanxing Fu, 2023. "Mean field portfolio games with consumption," Mathematics and Financial Economics, Springer, volume 17, number 4, September.
    8. Panagiotis E. Souganidis & Thaleia Zariphopoulou, 2024. "Mean field games with unbounded controlled common noise in portfolio management with relative performance criteria," Mathematics and Financial Economics, Springer, volume 18, number 10, September.
    9. Gaivoronski, Alexei A. & Krylov, Sergiy & van der Wijst, Nico, 2005. "Optimal portfolio selection and dynamic benchmark tracking," European Journal of Operational Research, Elsevier, vol. 163(1), pages 115-131, May.
    10. 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.
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