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Lattice Random Walk Discretisations of Stochastic Differential Equations

Published: August 28, 2025 | arXiv ID: 2508.20883v1

By: Samuel Duffield , Maxwell Aifer , Denis Melanson and more

Potential Business Impact:

Makes computers understand complex math faster.

Business Areas:
A/B Testing Data and Analytics

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.

Page Count
16 pages

Category
Mathematics:
Numerical Analysis (Math)