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Mean field game of optimal relative investment with jump risk

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Abstract

In this paper, we study the n-player game and the mean field game under the constant relative risk aversion relative performance on terminal wealth, in which the interaction occurs by peer competition. In the model with n agents, the price dynamics of underlying risky assets depend on a common noise and contagious jump risk modeled by a multi-dimensional nonlinear Hawkes process. With a continuum of agents, we formulate the mean field game problem and characterize a deterministic mean field equilibrium in an analytical form under some conditions, allowing us to investigate some impacts of model parameters in the limiting model and discuss some financial implications. Moreover, based on the mean field equilibrium, we construct an approximate Nash equilibrium for the n-player game when n is sufficiently large. The explicit order of the approximation error is also derived.

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Acknowledgements

Lijun Bo was supported by Natural Science Basic Research Program of Shaanxi (Grant No. 2023-JC-JQ-05) and National Natural Science Foundation of China (Grant No. 11971368). Shihua Wang was supported by the Fundamental Research Funds for the Central Universities (Grant No. WK3470000024). Xiang Yu was supported by The Hong Kong Polytechnic University (Grant Nos. P0031417 and P0039251). The authors thank the referees for their helpful comments on the presentation of this paper.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Xidian University, Xi’an, 710126, China

    Lijun Bo

  2. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, China

    Shihua Wang

  3. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China

    Xiang Yu

Authors
  1. Lijun Bo
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  2. Shihua Wang
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  3. Xiang Yu
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Correspondence to Xiang Yu.

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Bo, L., Wang, S. & Yu, X. Mean field game of optimal relative investment with jump risk. Sci. China Math. 67, 1159–1188 (2024). https://doi.org/10.1007/s11425-021-2109-3

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  • DOI: https://doi.org/10.1007/s11425-021-2109-3

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