Multilevel Picard approximation algorithm for semilinear partial integro-differential equations and its complexity analysis
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866929754378076160 |
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| author | Neufeld, Ariel Wu, Sizhou |
| author_facet | Neufeld, Ariel Wu, Sizhou |
| contents | In this paper we introduce a multilevel Picard approximation algorithm for semilinear parabolic partial integro-differential equations (PIDEs). We prove that the numerical approximation scheme converges to the unique viscosity solution of the PIDE under consideration. To that end, we derive a Feynman-Kac representation for the unique viscosity solution of the semilinear PIDE, extending the classical Feynman-Kac representation for linear PIDEs. Furthermore, we show that the algorithm does not suffer from the curse of dimensionality, i.e. the computational complexity of the algorithm is bounded polynomially in the dimension $d$ and the reciprocal of the prescribed accuracy $\varepsilon$. We also provide a numerical example in up to 10'000 dimensions to demonstrate its applicability. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_09639 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Multilevel Picard approximation algorithm for semilinear partial integro-differential equations and its complexity analysis Neufeld, Ariel Wu, Sizhou Numerical Analysis Probability In this paper we introduce a multilevel Picard approximation algorithm for semilinear parabolic partial integro-differential equations (PIDEs). We prove that the numerical approximation scheme converges to the unique viscosity solution of the PIDE under consideration. To that end, we derive a Feynman-Kac representation for the unique viscosity solution of the semilinear PIDE, extending the classical Feynman-Kac representation for linear PIDEs. Furthermore, we show that the algorithm does not suffer from the curse of dimensionality, i.e. the computational complexity of the algorithm is bounded polynomially in the dimension $d$ and the reciprocal of the prescribed accuracy $\varepsilon$. We also provide a numerical example in up to 10'000 dimensions to demonstrate its applicability. |
| title | Multilevel Picard approximation algorithm for semilinear partial integro-differential equations and its complexity analysis |
| topic | Numerical Analysis Probability |
| url | https://arxiv.org/abs/2205.09639 |