Nonlinear semigroups and limit theorems for convex expectations
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866909366903373824 |
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| author | Blessing, Jonas Kupper, Michael |
| author_facet | Blessing, Jonas Kupper, Michael |
| contents | Based on the Chernoff approximation, we provide a general approximation result for convex monotone semigroups which are continuous w.r.t. the mixed topology on suitable spaces of continuous functions. Starting with a family $(I(t))_{t\geq 0}$ of operators, the semigroup is constructed as the limit $S(t)f:=\lim_{n\to\infty}I(\frac{t}{n})^n f$ and is uniquely determined by the time derivative $I'(0)f$ for smooth functions. We identify explicit conditions for the generating family $(I(t))_{t\geq 0}$ that are transferred to the semigroup $(S(t))_{t\geq 0}$ and can easily be verified in applications. Furthermore, there is a structural link between Chernoff type approximations for nonlinear semigroups and law of large numbers and central limit theorem type results for convex expectations. The framework also includes large deviation results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_14096 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Nonlinear semigroups and limit theorems for convex expectations Blessing, Jonas Kupper, Michael Probability Primary 47H20, 60F05, Secondary 47J25, 60F10, 60G50 Based on the Chernoff approximation, we provide a general approximation result for convex monotone semigroups which are continuous w.r.t. the mixed topology on suitable spaces of continuous functions. Starting with a family $(I(t))_{t\geq 0}$ of operators, the semigroup is constructed as the limit $S(t)f:=\lim_{n\to\infty}I(\frac{t}{n})^n f$ and is uniquely determined by the time derivative $I'(0)f$ for smooth functions. We identify explicit conditions for the generating family $(I(t))_{t\geq 0}$ that are transferred to the semigroup $(S(t))_{t\geq 0}$ and can easily be verified in applications. Furthermore, there is a structural link between Chernoff type approximations for nonlinear semigroups and law of large numbers and central limit theorem type results for convex expectations. The framework also includes large deviation results. |
| title | Nonlinear semigroups and limit theorems for convex expectations |
| topic | Probability Primary 47H20, 60F05, Secondary 47J25, 60F10, 60G50 |
| url | https://arxiv.org/abs/2210.14096 |