Risk measures on incomplete markets: a new non-solid paradigm
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917906320719872 |
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| author | Melnikov, Vasily |
| author_facet | Melnikov, Vasily |
| contents | We study risk measures $φ:E\longrightarrow\mathbb{R}\cup\{\infty\}$, where $E$ is a vector space of random variables which a priori has no lattice structure$\unicode{x2014}$a blind spot of the existing risk measures literature. In particular, we address when $φ$ admits a tractable dual representation (one which does not contain non-$σ$-additive signed measures), and whether one can extend $φ$ to a solid superspace of $E$. The existence of a tractable dual representation is shown to be equivalent, modulo certain technicalities, to a Fatou-like property, while extension theorems are established under the existence of a sufficiently regular lift, a potentially non-linear mechanism of assigning random variable extensions to certain linear functionals on $E$. Our motivation is broadening the theory of risk measures to spaces without a lattice structure, which are ubiquitous in financial economics, especially when markets are incomplete. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_05194 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Risk measures on incomplete markets: a new non-solid paradigm Melnikov, Vasily Risk Management Functional Analysis Probability Mathematical Finance 46E30, 46N30, 91G70 We study risk measures $φ:E\longrightarrow\mathbb{R}\cup\{\infty\}$, where $E$ is a vector space of random variables which a priori has no lattice structure$\unicode{x2014}$a blind spot of the existing risk measures literature. In particular, we address when $φ$ admits a tractable dual representation (one which does not contain non-$σ$-additive signed measures), and whether one can extend $φ$ to a solid superspace of $E$. The existence of a tractable dual representation is shown to be equivalent, modulo certain technicalities, to a Fatou-like property, while extension theorems are established under the existence of a sufficiently regular lift, a potentially non-linear mechanism of assigning random variable extensions to certain linear functionals on $E$. Our motivation is broadening the theory of risk measures to spaces without a lattice structure, which are ubiquitous in financial economics, especially when markets are incomplete. |
| title | Risk measures on incomplete markets: a new non-solid paradigm |
| topic | Risk Management Functional Analysis Probability Mathematical Finance 46E30, 46N30, 91G70 |
| url | https://arxiv.org/abs/2409.05194 |