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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2512.00830 |
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| _version_ | 1866912860934766592 |
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| author | Cheng, Weilun Liang, Zongxia Wang, Sheng Xia, Jianming |
| author_facet | Cheng, Weilun Liang, Zongxia Wang, Sheng Xia, Jianming |
| contents | This paper studies a continuous-time portfolio selection problem under a general distribution of random risk aversion (RRA). We provide a complete characterization of all deterministic equilibrium strategies in closed form. Our results show that the structure of the solution depends crucially on the distribution of RRA: the equilibrium is unique (if exits) when the expectation of RRA is finite, whereas an infinite expectation leads either to infinitely many equilibria or to a unique trivial one (i.e. risk-free investment). To resolve this multiplicity of equilibria, we select, among all deterministic equilibria, the one that maximizes the objective functional at the initial time. We establish a necessary and sufficient condition for the existence of such an optimal equilibrium, which is then shown to be unique and uniformly optimal. Finally, we conduct a comparative statics. Using counterexamples based on two-point distributed RRA, we demonstrate that a larger risk aversion in the sense of first-order stochastic dominance does not necessarily lead to less risky investment. Within the two-point distribution framework, we further examine the single-crossing property of equilibrium strategies and the monotonicity of the crossing time. We show that a larger risk aversion under a stronger stochastic order -- the reverse hazard rate order -- always leads to less risky investment. In addition, we analyze how the convex combination of independent and identically distributed RRAs influences investment. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_00830 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Equilibrium Investment with Random Risk Aversion: (Non-)uniqueness, Optimality, and Comparative Statics Cheng, Weilun Liang, Zongxia Wang, Sheng Xia, Jianming Mathematical Finance This paper studies a continuous-time portfolio selection problem under a general distribution of random risk aversion (RRA). We provide a complete characterization of all deterministic equilibrium strategies in closed form. Our results show that the structure of the solution depends crucially on the distribution of RRA: the equilibrium is unique (if exits) when the expectation of RRA is finite, whereas an infinite expectation leads either to infinitely many equilibria or to a unique trivial one (i.e. risk-free investment). To resolve this multiplicity of equilibria, we select, among all deterministic equilibria, the one that maximizes the objective functional at the initial time. We establish a necessary and sufficient condition for the existence of such an optimal equilibrium, which is then shown to be unique and uniformly optimal. Finally, we conduct a comparative statics. Using counterexamples based on two-point distributed RRA, we demonstrate that a larger risk aversion in the sense of first-order stochastic dominance does not necessarily lead to less risky investment. Within the two-point distribution framework, we further examine the single-crossing property of equilibrium strategies and the monotonicity of the crossing time. We show that a larger risk aversion under a stronger stochastic order -- the reverse hazard rate order -- always leads to less risky investment. In addition, we analyze how the convex combination of independent and identically distributed RRAs influences investment. |
| title | Equilibrium Investment with Random Risk Aversion: (Non-)uniqueness, Optimality, and Comparative Statics |
| topic | Mathematical Finance |
| url | https://arxiv.org/abs/2512.00830 |