Taming singular stochastic differential equations: A numerical method
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866913671862550528 |
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| author | Lê, Khoa Ling, Chengcheng |
| author_facet | Lê, Khoa Ling, Chengcheng |
| contents | We consider a generic and explicit tamed Euler--Maruyama scheme for multidimensional time-inhomogeneous stochastic differential equations with multiplicative Brownian noise. The diffusive coefficient is uniformly elliptic, Hölder continuous and weakly differentiable in the spatial variables while the drift satisfies the strict Ladyzhenskaya--Prodi--Serrin condition, as considered by Krylov and Röckner (2005). In the discrete scheme, the drift is tamed by replacing it by an approximation. A strong rate of convergence of the scheme is provided in terms of the approximation error of the drift in a suitable and possibly very weak topology. A few examples of approximating drifts are discussed in detail. The parameters of the approximating drifts can vary and -- under suitable conditions -- be fine-tuned to achieve a strong convergence rate which is arbitrarily close to the benchmark $0.5$ rate. The result is then applied to provide numerical solutions for stochastic transport equations with singular vector fields satisfying the aforementioned condition. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2110_01343 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Taming singular stochastic differential equations: A numerical method Lê, Khoa Ling, Chengcheng Probability Numerical Analysis We consider a generic and explicit tamed Euler--Maruyama scheme for multidimensional time-inhomogeneous stochastic differential equations with multiplicative Brownian noise. The diffusive coefficient is uniformly elliptic, Hölder continuous and weakly differentiable in the spatial variables while the drift satisfies the strict Ladyzhenskaya--Prodi--Serrin condition, as considered by Krylov and Röckner (2005). In the discrete scheme, the drift is tamed by replacing it by an approximation. A strong rate of convergence of the scheme is provided in terms of the approximation error of the drift in a suitable and possibly very weak topology. A few examples of approximating drifts are discussed in detail. The parameters of the approximating drifts can vary and -- under suitable conditions -- be fine-tuned to achieve a strong convergence rate which is arbitrarily close to the benchmark $0.5$ rate. The result is then applied to provide numerical solutions for stochastic transport equations with singular vector fields satisfying the aforementioned condition. |
| title | Taming singular stochastic differential equations: A numerical method |
| topic | Probability Numerical Analysis |
| url | https://arxiv.org/abs/2110.01343 |