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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2404.14059 |
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| _version_ | 1866917024825868288 |
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| author | Fan, Shengjun Hu, Ying Tang, Shanjian |
| author_facet | Fan, Shengjun Hu, Ying Tang, Shanjian |
| contents | The dynamic concave utility (or the dynamic convex risk measure) of an unbounded endowment is studied and represented as the value process in the unique solution of a backward stochastic differential equation (BSDE) with an unbounded terminal value, with the help of our recent existence and uniqueness results on unbounded solutions of scalar BSDEs whose generators have a linear, super-linear, sub-quadratic or quadratic growth. Moreover, the infimum in the dynamic concave utility is proved to be attainable. The Fenchel-Legendre transform (dual representation) of convex functions, the de la Vallée-Poussin theorem, and Young's and Gronwall's inequalities constitute the main ingredients of the dual representation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_14059 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Unbounded Dynamic Concave Utilities via BSDEs Fan, Shengjun Hu, Ying Tang, Shanjian Probability The dynamic concave utility (or the dynamic convex risk measure) of an unbounded endowment is studied and represented as the value process in the unique solution of a backward stochastic differential equation (BSDE) with an unbounded terminal value, with the help of our recent existence and uniqueness results on unbounded solutions of scalar BSDEs whose generators have a linear, super-linear, sub-quadratic or quadratic growth. Moreover, the infimum in the dynamic concave utility is proved to be attainable. The Fenchel-Legendre transform (dual representation) of convex functions, the de la Vallée-Poussin theorem, and Young's and Gronwall's inequalities constitute the main ingredients of the dual representation. |
| title | Unbounded Dynamic Concave Utilities via BSDEs |
| topic | Probability |
| url | https://arxiv.org/abs/2404.14059 |