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.

翻译：本文提出了一种用于随机微分方程（SDE）的格点随机游走离散化方案，该方案在每一步采样二元或三元增量，将复杂的漂移和扩散计算简化为简单的1或2比特随机值。该方法与传统浮点离散化方案有显著不同，并具有若干优势：包括与避免浮点运算、直接操作比特流底层概率分布的随机计算架构兼容，无需高斯采样，对量化误差具有鲁棒性，并能处理非Lipschitz漂移项。我们证明了该方法的弱收敛性，并通过在多种SDE（包括最先进的扩散模型）上的实验展示了其优势。