Lattice Random Walk Discretisations of Stochastic Differential Equations

Abstract

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.