Numerical stability revisited: A family of benchmark problems for the analysis of explicit stochastic differential equation integrators
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866917510404636672 |
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| author | Hudson, Thomas Helfert, Sarah Li, Xingjie Helen |
| author_facet | Hudson, Thomas Helfert, Sarah Li, Xingjie Helen |
| contents | We revisit the numerical stability of four well-established explicit stochastic integration schemes through a new generic benchmark stochastic differential equation designed to assess asymptotic statistical accuracy and stability properties. This one-parameter benchmark equation is derived from a general one-dimensional first-order SDE using spatio-temporal nondimensionalization and is employed to evaluate the performance of the (1) Euler-Maruyama, (2) Milstein, (3) Stochastic Heun, and (4) three-stage Runge-Kutta schemes. Our findings reveal that lower-order schemes can outperform higher-order ones over a range of time step sizes, depending on the benchmark parameters and application context. The theoretical results are validated through a series of numerical experiments, and we discuss their implications for more general applications, including a nonlinear example. Our results suggest that the insights obtained from the linear benchmark problem provide reliable guidance for time-stepping strategies when simulating nonlinear SDEs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_19203 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Numerical stability revisited: A family of benchmark problems for the analysis of explicit stochastic differential equation integrators Hudson, Thomas Helfert, Sarah Li, Xingjie Helen Numerical Analysis 60H35, 65L20 We revisit the numerical stability of four well-established explicit stochastic integration schemes through a new generic benchmark stochastic differential equation designed to assess asymptotic statistical accuracy and stability properties. This one-parameter benchmark equation is derived from a general one-dimensional first-order SDE using spatio-temporal nondimensionalization and is employed to evaluate the performance of the (1) Euler-Maruyama, (2) Milstein, (3) Stochastic Heun, and (4) three-stage Runge-Kutta schemes. Our findings reveal that lower-order schemes can outperform higher-order ones over a range of time step sizes, depending on the benchmark parameters and application context. The theoretical results are validated through a series of numerical experiments, and we discuss their implications for more general applications, including a nonlinear example. Our results suggest that the insights obtained from the linear benchmark problem provide reliable guidance for time-stepping strategies when simulating nonlinear SDEs. |
| title | Numerical stability revisited: A family of benchmark problems for the analysis of explicit stochastic differential equation integrators |
| topic | Numerical Analysis 60H35, 65L20 |
| url | https://arxiv.org/abs/2503.19203 |