Probabilistic approximation of fully nonlinear second-order PIDEs with convergence rates for the universal robust limit theorem
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918462358552576 |
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| author | Jiang, Lianzi Hu, Mingshang Liang, Gechun |
| author_facet | Jiang, Lianzi Hu, Mingshang Liang, Gechun |
| contents | This paper develops a probabilistic approximation scheme for a class of nonstandard, fully nonlinear second-order partial integro-differential equations (PIDEs) associated with nonlinear Levy processes under Peng's G-expectation framework. The PIDE features a supremum over a family of alpha-stable Levy measures, possibly degenerate diffusion coefficients, and a non-separable uncertainty set, which places it outside the scope of existing numerical theories for PIDEs.
We construct a recursive, piecewise-constant approximation of the viscosity solution and establish explicit error estimates for the scheme. As a key application, our results yield quantitative convergence rates for the universal robust limit theorem under sublinear expectations. This provides a unified treatment of Peng's robust central limit theorem and law of large numbers, as well as the alpha-stable limit theorem of Bayraktar and Munk, together with explicit Berry-Esseen-type bounds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_18374 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Probabilistic approximation of fully nonlinear second-order PIDEs with convergence rates for the universal robust limit theorem Jiang, Lianzi Hu, Mingshang Liang, Gechun Probability Numerical Analysis 60F05, 65M15, 60H30, 45K05 This paper develops a probabilistic approximation scheme for a class of nonstandard, fully nonlinear second-order partial integro-differential equations (PIDEs) associated with nonlinear Levy processes under Peng's G-expectation framework. The PIDE features a supremum over a family of alpha-stable Levy measures, possibly degenerate diffusion coefficients, and a non-separable uncertainty set, which places it outside the scope of existing numerical theories for PIDEs. We construct a recursive, piecewise-constant approximation of the viscosity solution and establish explicit error estimates for the scheme. As a key application, our results yield quantitative convergence rates for the universal robust limit theorem under sublinear expectations. This provides a unified treatment of Peng's robust central limit theorem and law of large numbers, as well as the alpha-stable limit theorem of Bayraktar and Munk, together with explicit Berry-Esseen-type bounds. |
| title | Probabilistic approximation of fully nonlinear second-order PIDEs with convergence rates for the universal robust limit theorem |
| topic | Probability Numerical Analysis 60F05, 65M15, 60H30, 45K05 |
| url | https://arxiv.org/abs/2506.18374 |