Representation theorems for dynamic convex risk measures
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912974098137088 |
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| author | Zheng, Shiqiu |
| author_facet | Zheng, Shiqiu |
| contents | In this paper, we prove that under the domination condition: \begin{equation*} {\cal{E}}^{-μ,-ν}[-ξ|{\cal{F}}_t]\leqρ_t(ξ)\leq{\cal{E}}^{μ,ν}[-ξ|{\cal{F}}_t],\quad \forallξ\in \mathcal{L}^{\exp}_T\ (\text{resp.}\ L^2(\mathcal{F}_T)),\ \forall t\in[0,T], \end{equation*} where ${\cal{E}}^{μ,ν}$ is the $g$-expectation with generator $μ|z|+ν|z|^2, μ\geq0, ν\geq0$, the dynamic convex (resp. coherent) risk measure $ρ$ admits a representation as a $g$-expectation, whose generator $g$ is convex (resp. sublinear) in the variable $z$ and has a quadratic (resp. linear) growth. As an application, we show that such dynamic convex (resp. coherent) risk measure $ρ$ admits a dual representation, where the penalty term (resp. the set of probability measures) is characterized by the corresponding generator $g$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_20660 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Representation theorems for dynamic convex risk measures Zheng, Shiqiu Probability In this paper, we prove that under the domination condition: \begin{equation*} {\cal{E}}^{-μ,-ν}[-ξ|{\cal{F}}_t]\leqρ_t(ξ)\leq{\cal{E}}^{μ,ν}[-ξ|{\cal{F}}_t],\quad \forallξ\in \mathcal{L}^{\exp}_T\ (\text{resp.}\ L^2(\mathcal{F}_T)),\ \forall t\in[0,T], \end{equation*} where ${\cal{E}}^{μ,ν}$ is the $g$-expectation with generator $μ|z|+ν|z|^2, μ\geq0, ν\geq0$, the dynamic convex (resp. coherent) risk measure $ρ$ admits a representation as a $g$-expectation, whose generator $g$ is convex (resp. sublinear) in the variable $z$ and has a quadratic (resp. linear) growth. As an application, we show that such dynamic convex (resp. coherent) risk measure $ρ$ admits a dual representation, where the penalty term (resp. the set of probability measures) is characterized by the corresponding generator $g$. |
| title | Representation theorems for dynamic convex risk measures |
| topic | Probability |
| url | https://arxiv.org/abs/2510.20660 |