Lattice Random Walk Discretisations of Stochastic Differential Equations
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arXiv
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| Hauptverfasser: | , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866911452107898880 |
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| author | Duffield, Samuel Aifer, Maxwell Melanson, Denis Belateche, Zach Coles, Patrick J. |
| author_facet | Duffield, Samuel Aifer, Maxwell Melanson, Denis Belateche, Zach Coles, Patrick J. |
| contents | We introduce a lattice random walk discretisation scheme for stochastic differential equations (SDEs) that samples binary or ternary increments at each step, suppressing complex drift and diffusion computations to simple 1 or 2 bit random values. This approach is a significant departure from traditional floating point discretisations and offers several advantages; including compatibility with stochastic computing architectures that avoid floating-point arithmetic in place of directly manipulating the underlying probability distribution of a bitstream, elimination of Gaussian sampling requirements, robustness to quantisation errors, and handling of non-Lipschitz drifts. We prove weak convergence and demonstrate the advantages through experiments on various SDEs, including state-of-the-art diffusion models. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_20883 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lattice Random Walk Discretisations of Stochastic Differential Equations Duffield, Samuel Aifer, Maxwell Melanson, Denis Belateche, Zach Coles, Patrick J. Numerical Analysis Emerging Technologies Computation We introduce a lattice random walk discretisation scheme for stochastic differential equations (SDEs) that samples binary or ternary increments at each step, suppressing complex drift and diffusion computations to simple 1 or 2 bit random values. This approach is a significant departure from traditional floating point discretisations and offers several advantages; including compatibility with stochastic computing architectures that avoid floating-point arithmetic in place of directly manipulating the underlying probability distribution of a bitstream, elimination of Gaussian sampling requirements, robustness to quantisation errors, and handling of non-Lipschitz drifts. We prove weak convergence and demonstrate the advantages through experiments on various SDEs, including state-of-the-art diffusion models. |
| title | Lattice Random Walk Discretisations of Stochastic Differential Equations |
| topic | Numerical Analysis Emerging Technologies Computation |
| url | https://arxiv.org/abs/2508.20883 |