Dynamic Lagrange Multipliers in a Non-concave Utility Framework
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866913038247919616 |
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| author | Liu, Yang Schied, Alexander Shen, Zhenyu |
| author_facet | Liu, Yang Schied, Alexander Shen, Zhenyu |
| contents | In continuous-time portfolio selection for non-concave utility functions, the martingale duality approach is widely adopted in complete markets, while the dynamic programming approach may sometimes lead to singular solutions of the Hamilton-Jacobi-Bellman (HJB) equation. We propose "dynamic Lagrange multipliers" in a non-concave utility framework, bridging two approaches and demonstrating that the Lagrangian multiplier function (in the martingale duality approach) equals the conjugate dual point related to the value function (in dynamic programming), which is exactly its partial derivative with respect to wealth. Moreover, the dynamic multiplier process exhibits homogeneity via the optimal wealth and pricing kernel processes, offering intuitive economic interpretations as a dynamic shadow price of the envelope theorem. Finally, classical optimal results are recovered and numerically validated by non-concave utility examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_14924 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Dynamic Lagrange Multipliers in a Non-concave Utility Framework Liu, Yang Schied, Alexander Shen, Zhenyu Optimization and Control 91B16, 91G10 In continuous-time portfolio selection for non-concave utility functions, the martingale duality approach is widely adopted in complete markets, while the dynamic programming approach may sometimes lead to singular solutions of the Hamilton-Jacobi-Bellman (HJB) equation. We propose "dynamic Lagrange multipliers" in a non-concave utility framework, bridging two approaches and demonstrating that the Lagrangian multiplier function (in the martingale duality approach) equals the conjugate dual point related to the value function (in dynamic programming), which is exactly its partial derivative with respect to wealth. Moreover, the dynamic multiplier process exhibits homogeneity via the optimal wealth and pricing kernel processes, offering intuitive economic interpretations as a dynamic shadow price of the envelope theorem. Finally, classical optimal results are recovered and numerically validated by non-concave utility examples. |
| title | Dynamic Lagrange Multipliers in a Non-concave Utility Framework |
| topic | Optimization and Control 91B16, 91G10 |
| url | https://arxiv.org/abs/2604.14924 |