Variational proof of conditional expectations
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866912300356599808 |
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| author | Castañeda, Hugo Guadalupe Reyna Sandoval-Romero, María de los Ángeles |
| author_facet | Castañeda, Hugo Guadalupe Reyna Sandoval-Romero, María de los Ángeles |
| contents | In this paper, we show that the conditional expectation of a random variable with finite second moment given a $σ$-algebra is the unique critical point of an energy functional in Hilbert space $L^2$. Then, we extend by density the result to every integrable random variable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_22947 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Variational proof of conditional expectations Castañeda, Hugo Guadalupe Reyna Sandoval-Romero, María de los Ángeles Probability Optimization and Control 49J50, 60A10 In this paper, we show that the conditional expectation of a random variable with finite second moment given a $σ$-algebra is the unique critical point of an energy functional in Hilbert space $L^2$. Then, we extend by density the result to every integrable random variable. |
| title | Variational proof of conditional expectations |
| topic | Probability Optimization and Control 49J50, 60A10 |
| url | https://arxiv.org/abs/2503.22947 |