Competitive optimal portfolio selection under mean-variance criterion
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914142563074048 |
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| author | Shao, Guojiang Xu, Zuo Quan Zhang, Qi |
| author_facet | Shao, Guojiang Xu, Zuo Quan Zhang, Qi |
| contents | We investigate a portfolio selection problem involving multi competitive agents, each exhibiting mean-variance preferences. Unlike classical models, each agent's utility is determined by their relative wealth compared to the average wealth of all agents, introducing a competitive dynamic into the optimization framework. To address this game-theoretic problem, we first reformulate the mean-variance criterion as a constrained, non-homogeneous stochastic linear-quadratic control problem and derive the corresponding optimal feedback strategies. The existence of Nash equilibria is shown to depend on the well-posedness of a complex, coupled system of equations. Employing decoupling techniques, we reduce the well-posedness analysis to the solvability of a novel class of multi-dimensional linear backward stochastic differential equations (BSDEs). We solve a new type of nonlinear BSDEs (including the above linear one as a special case) using fixed-point theory. Depending on the interplay between market and competition parameters, three distinct scenarios arise: (i) the existence of a unique Nash equilibrium, (ii) the absence of any Nash equilibrium, and (iii) the existence of infinitely many Nash equilibria. These scenarios are rigorously characterized and discussed in detail. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_05270 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Competitive optimal portfolio selection under mean-variance criterion Shao, Guojiang Xu, Zuo Quan Zhang, Qi Optimization and Control Mathematical Finance Portfolio Management We investigate a portfolio selection problem involving multi competitive agents, each exhibiting mean-variance preferences. Unlike classical models, each agent's utility is determined by their relative wealth compared to the average wealth of all agents, introducing a competitive dynamic into the optimization framework. To address this game-theoretic problem, we first reformulate the mean-variance criterion as a constrained, non-homogeneous stochastic linear-quadratic control problem and derive the corresponding optimal feedback strategies. The existence of Nash equilibria is shown to depend on the well-posedness of a complex, coupled system of equations. Employing decoupling techniques, we reduce the well-posedness analysis to the solvability of a novel class of multi-dimensional linear backward stochastic differential equations (BSDEs). We solve a new type of nonlinear BSDEs (including the above linear one as a special case) using fixed-point theory. Depending on the interplay between market and competition parameters, three distinct scenarios arise: (i) the existence of a unique Nash equilibrium, (ii) the absence of any Nash equilibrium, and (iii) the existence of infinitely many Nash equilibria. These scenarios are rigorously characterized and discussed in detail. |
| title | Competitive optimal portfolio selection under mean-variance criterion |
| topic | Optimization and Control Mathematical Finance Portfolio Management |
| url | https://arxiv.org/abs/2511.05270 |